Gibbs sampling is a Markov Chain Monte Carlo (MCMC)
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Jun 18, 2024
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About This Presentation
Gibbs sampling is a Markov Chain Monte Carlo (MCMC) algorithm used for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution when direct sampling is difficult. This method is particularly useful in Bayesian statistics and computational bio...
Slide Content
Gibbs Sampling
30.07.2022
30.07.2022
Gibbs Sampling
30.07.2022
2/22 DJSCE, Mumbai
30.07.2022
2/22 DJSCE, Mumbai
Outline
1Introduction
2Gibbs Sampling : Introduction
3Robust Modeling
4Binary Response Regression with a Probit Link
5Missing Data and Gibbs Sampling
6Proper Priors and Model Selection
7Estimating a Table of Means
8Predicting the Success of Future Students
3/22 DJSCE, Mumbai
1Introduction
2Gibbs Sampling : Introduction
3Robust Modeling
4Binary Response Regression with a Probit Link
5Missing Data and Gibbs Sampling
6Proper Priors and Model Selection
7Estimating a Table of Means
8Predicting the Success of Future Students
3/22 DJSCE, Mumbai
Introduction to Gibbs Sampling
What is Gibbs Sampling ?
Why is it important ?
4/22 DJSCE, Mumbai
What is Gibbs Sampling ?
Why is it important ?
4/22 DJSCE, Mumbai
Why is Gibbs Sampling Important ?
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Why is Gibbs Sampling Important ?
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Why is Gibbs Sampling Important ?
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Why is Gibbs Sampling Important ?
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Why is Gibbs Sampling Important ?
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Why is Gibbs Sampling Important ?
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used
for sampling from complex probability distributions.
It is particularly useful when it’s challenging to sample directly from the
joint distribution of multiple variables.
Gibbs Sampling allows us to explore high-dimensional spaces and
estimate complex probabilistic models.
It’s widely used in Bayesian statistics, machine learning, and data analysis.Gibbs Sampling is a fundamental tool for Bayesian inference, which helps
us make probabilistic inferences about unknown quantities.
5/22 DJSCE, Mumbai
Markov Chain Monte Carlo (MCMC) Technique
MCMC is a class of algorithms used for statistical sampling and
numerical integration.
It’s particularly useful for estimating complex probability distributions.Central to MCMC methods is the concept of a Markov chain, a sequence
of random variables where each depends only on the previous one.
6/22 DJSCE, Mumbai
MCMC is a class of algorithms used for statistical sampling and
numerical integration.
It’s particularly useful for estimating complex probability distributions.Central to MCMC methods is the concept of a Markov chain, a sequence
of random variables where each depends only on the previous one.
6/22 DJSCE, Mumbai
Markov Chain Monte Carlo (MCMC) Technique
MCMC is a class of algorithms used for statistical sampling and
numerical integration.
It’s particularly useful for estimating complex probability distributions.Central to MCMC methods is the concept of a Markov chain, a sequence
of random variables where each depends only on the previous one.
6/22 DJSCE, Mumbai
MCMC is a class of algorithms used for statistical sampling and
numerical integration.
It’s particularly useful for estimating complex probability distributions.Central to MCMC methods is the concept of a Markov chain, a sequence
of random variables where each depends only on the previous one.
6/22 DJSCE, Mumbai
Markov Chain Monte Carlo (MCMC) Technique
MCMC is a class of algorithms used for statistical sampling and
numerical integration.
It’s particularly useful for estimating complex probability distributions.Central to MCMC methods is the concept of a Markov chain, a sequence
of random variables where each depends only on the previous one.
6/22 DJSCE, Mumbai
MCMC is a class of algorithms used for statistical sampling and
numerical integration.
It’s particularly useful for estimating complex probability distributions.Central to MCMC methods is the concept of a Markov chain, a sequence
of random variables where each depends only on the previous one.
6/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Terms)Imagine you have a big bag of candies. You want to know the average
color of candies in the bag without checking every single one.
Statistical Sampling is like taking a handful of candies from the bag and
using that small group to estimate the average color of all the candies.
It’s a shortcut that gives you a good idea without checking every candy.
Now, let’s talk about Numerical Integration. Imagine you want to know
how much paint you need to cover a curved wall. You can’t measure it
directly, so you divide the wall into many small rectangles, find the area
of each rectangle, and add them up. This is a simple way to estimate the
total paint needed.
Numerical Integration is similar. It’s a method to estimate complicated
math problems that can’t be solved directly. You break them down into
simpler pieces, approximate each piece, and add them up to get a good
answer.
7/22 DJSCE, Mumbai
Estimating Candy Colors with MCMC
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
Estimating Candy Colors with MCMC
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
Estimating Candy Colors with MCMC
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
Estimating Candy Colors with MCMC
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
Estimating Candy Colors with MCMC
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
We still have our big bag of candies, and we want to know the average
color of candies inside it without checking every single one.
Suppose we have a special machine called the ”Candy Color Analyzer”
(MCMC). This machine can randomly sample a few candies from the bag
and determine their color.
The Candy Color Analyzer is like a detective. It starts with a guess for
the average color, and it keeps sampling candies and adjusting its guess
to get closer and closer to the true average color.
The more candies the machine samples and adjusts its guess, the more
accurate its estimate becomes. It’s like playing a guessing game, but it’s
getting better with each turn.
In the end, the Candy Color Analyzer (MCMC) provides a pretty good
estimate of the average color without checking every single candy
individually.
8/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Statistical Sampling and Numerical Integration (Layman’s
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Examples)Imagine you have a garden full of different types of flowers. You want to
know the average height of all the flowers in your garden. Counting each
flower is a lot of work !
Statistical Sampling is like picking a few flowers from different parts of
your garden and measuring their height. You use these measurements to
estimate the average height of all the flowers. It’s like getting a taste of
your whole garden without having to check every single flower.
Now, let’s talk about Numerical Integration. Suppose you’re throwing a
big party, and you want to know how much food to buy. The number of
guests keeps changing throughout the day.
Numerical Integration is like keeping track of the number of guests at
different times during the party. You add up the number of guests at
each hour to estimate the total number of guests for the whole day,
helping you decide how much food to prepare.
9/22 DJSCE, Mumbai
Estimating Average Flower Height with MCMC
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Estimating Average Flower Height with MCMC
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Estimating Average Flower Height with MCMC
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Estimating Average Flower Height with MCMC
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Estimating Average Flower Height with MCMC
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Estimating Average Flower Height with MCMC
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Imagine you have a garden full of different types of flowers, and you want
to know the average height of all the flowers in your garden. Counting
each flower is a lot of work !
Instead of measuring every single flower, you have a magic garden
”Height Estimator” (MCMC) tool.
The Height Estimator starts with a guess for the average flower height.
It’s like taking an educated guess based on a few flowers.
Now, the magic happens. The Height Estimator randomly selects a few
flowers, measures their height, and updates its guess for the average
height.
It repeats this process, picking different flowers each time. The more
flowers it measures and updates its guess, the closer it gets to the true
average height.
Eventually, the Height Estimator gives you a very good estimate of the
average flower height in your garden without having to measure every
single flower.
10/22 DJSCE, Mumbai
Estimating Food for a Fluctuating Party with MCMC
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Estimating Food for a Fluctuating Party with MCMC
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Estimating Food for a Fluctuating Party with MCMC
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Estimating Food for a Fluctuating Party with MCMC
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Estimating Food for a Fluctuating Party with MCMC
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Estimating Food for a Fluctuating Party with MCMC
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Suppose you’re throwing a big party, and you want to know how much
food to buy. The number of guests keeps changing throughout the day.
It’s like trying to figure out how many slices of pizza to order when you
don’t know how many friends will drop by.
Imagine you have a special ”Party Guest Counter” (MCMC) tool. This
tool helps you estimate the number of guests at any given time.
The Party Guest Counter starts with an initial estimate of guests, and it’s
continuously updating its guess as people arrive and leave.
It’s like a party detective – every so often, it takes a quick headcount,
adjusts its guess, and becomes better at estimating the number of guests.
As the party progresses, the Party Guest Counter (MCMC) provides you
with a good estimate of the number of guests, helping you order just the
right amount of pizza and snacks, even if the guest list keeps changing.
11/22 DJSCE, Mumbai
Examples of Complex Probability Distributions
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Examples of Complex Probability Distributions
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Examples of Complex Probability Distributions
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Examples of Complex Probability Distributions
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Examples of Complex Probability Distributions
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Examples of Complex Probability Distributions
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Examples of Complex Probability Distributions
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex probability distributions are those that cannot be easily
expressed using standard probability density functions.
In many real-world scenarios, such as Bayesian modeling, it’s common to
encounter these complex distributions.
Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are
valuable tools for sampling from such distributions.
Multivariate Normal Distributions : When dealing with multiple correlated
variables, the joint distribution can become complex. Gibbs Sampling can
be used for sampling from multivariate normal distributions.
Bayesian Hierarchical Models : Hierarchical models with multiple layers of
parameters can lead to complex posterior distributions. Gibbs Sampling
simplifies sampling in such cases.
Non-conjugate Priors : When the prior and likelihood are not conjugate,
the posterior distribution may be difficult to compute analytically. Gibbs
Sampling provides a solution.
12/22 DJSCE, Mumbai
Complex Probability Distributions (In Everyday Terms)
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
Complex Probability Distributions (In Everyday Terms)
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
Complex Probability Distributions (In Everyday Terms)
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
Complex Probability Distributions (In Everyday Terms)
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
Complex Probability Distributions (In Everyday Terms)
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
We’ve seen how to estimate things like candy colors, flower heights, and
party guests using tools like MCMC. But what if the problem becomes
even more challenging ?
Complex Probability Distributions are like really tricky problems where
you can’t simply use basic tools. They’re like puzzles that are hard to
solve.
For example, think about estimating the probability of finding a specific
rare flower in your garden based on its color, size, and location. That’s a
complex probability distribution problem.
Complex probability distributions are like the most challenging party
guest scenarios. Not only do guests come and go, but their behavior and
preferences change too. It’s hard to predict and manage.
When we encounter complex probability distributions, MCMC comes to
the rescue. It’s like having a magical problem solver for the most
complicated puzzles and unpredictable parties.
13/22 DJSCE, Mumbai
Complex Probability Distributions with a Coin Toss
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Complex Probability Distributions with a Coin Toss
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Complex Probability Distributions with a Coin Toss
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Complex Probability Distributions with a Coin Toss
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Complex Probability Distributions with a Coin Toss
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Complex Probability Distributions with a Coin Toss
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Let’s dive into a numerical example. Imagine we have a biased coin.
The probability of getting a head (H) is 0.3, and the probability of
getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin.
Now, let’s say we want to know the probability of getting two heads
(HH) in a row when we flip this coin twice.
To calculate this, we need to deal with complex probability distributions.
It’s not just straightforward like flipping a fair coin.
Here’s where MCMC comes in. It can simulate thousands of coin flips,
accounting for the biased probabilities, and estimate the probability of
getting HH.
Using MCMC, we might find that the probability of HH is approximately
0.09 for this biased coin.
14/22 DJSCE, Mumbai
Bayesian Inference : MCMC is commonly used for Bayesian parameter
estimation. For example, in a Bayesian regression model, we can use
MCMC to estimate the posterior distribution of regression coefficients.
Image Segmentation : In computer vision, MCMC can be employed for
image segmentation, where each pixel’s class label is sampled based on
neighboring pixels.
Phylogenetics : MCMC is applied to infer evolutionary trees from genetic
data, helping us understand the relationships between species.
15/22 DJSCE, Mumbai
estimation. For example, in a Bayesian regression model, we can use
MCMC to estimate the posterior distribution of regression coefficients.
Image Segmentation : In computer vision, MCMC can be employed for
image segmentation, where each pixel’s class label is sampled based on
neighboring pixels.
Phylogenetics : MCMC is applied to infer evolutionary trees from genetic
data, helping us understand the relationships between species.
15/22 DJSCE, Mumbai
Complex Probability Distributions with Playing Cards
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Complex Probability Distributions with Playing Cards
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Complex Probability Distributions with Playing Cards
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Complex Probability Distributions with Playing Cards
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Complex Probability Distributions with Playing Cards
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Complex Probability Distributions with Playing Cards
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Consider a standard deck of playing cards with 52 cards. Now, let’s say
you want to know the probability of drawing two red cards (hearts or
diamonds) in a row without replacement.
To calculate this probability, we need to deal with complex probability
distributions. Unlike flipping a fair coin, it’s not straightforward.
In this case, the probability of drawing a red card on the first draw is
26/52, and on the second draw (without replacement), it becomes 25/51.
Complex probability distributions arise when we want to find the
probability of drawing two specific cards in a row from a deck with
different suits and ranks.
This is where MCMC comes to the rescue. It can simulate thousands of
card draws, account for the changing probabilities with each draw, and
estimate the probability of drawing two red cards in a row.
Using MCMC, we might find that the probability of drawing two red
cards in a row is approximately 0.244.
15/22 DJSCE, Mumbai
Robust Modeling
Discuss the concept of robust modeling.
How is Gibbs Sampling used in robust modeling ?
16/22 DJSCE, Mumbai
Discuss the concept of robust modeling.
How is Gibbs Sampling used in robust modeling ?
16/22 DJSCE, Mumbai
Binary Response Regression with a Probit Link
Explain binary response regression with a probit link.
Show practical applications and examples.
17/22 DJSCE, Mumbai
Explain binary response regression with a probit link.
Show practical applications and examples.
17/22 DJSCE, Mumbai
Missing Data and Gibbs Sampling
How does Gibbs Sampling handle missing data ?
Provide examples of its application.
18/22 DJSCE, Mumbai
How does Gibbs Sampling handle missing data ?
Provide examples of its application.
18/22 DJSCE, Mumbai
Proper Priors and Model Selection
Explain the importance of proper priors in Gibbs Sampling.
Discuss how Gibbs Sampling aids in model selection.
19/22 DJSCE, Mumbai
Explain the importance of proper priors in Gibbs Sampling.
Discuss how Gibbs Sampling aids in model selection.
19/22 DJSCE, Mumbai
Estimating a Table of Means
Show how Gibbs Sampling can be used to estimate a table of means.
Compare a flat prior over the restricted space with a hierarchical
regression prior.
20/22 DJSCE, Mumbai
Show how Gibbs Sampling can be used to estimate a table of means.
Compare a flat prior over the restricted space with a hierarchical
regression prior.
20/22 DJSCE, Mumbai
Predicting the Success of Future Students
Discuss the application of Gibbs Sampling in predicting the success of
future students.
Share the significance of this application.
21/22 DJSCE, Mumbai
Discuss the application of Gibbs Sampling in predicting the success of
future students.
Share the significance of this application.
21/22 DJSCE, Mumbai
Conclusion
Summarize the key points.
Highlight the importance of Gibbs Sampling in various domains.
22/22 DJSCE, Mumbai
Summarize the key points.
Highlight the importance of Gibbs Sampling in various domains.
22/22 DJSCE, Mumbai
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