bayes_machine_learning_book for data scientist

bayes theory.
which is very important in machine learning.
we can use this for ml

1. BML lecture #1: Bayesics
http://github.com/rbardenet/bml-course
R´emi Bardenet
[email protected]
CNRS & CRIStAL, Univ. Lille, France
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2. What comes toyourmind when you hear ”Bayesian ML”?
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3. Course outline
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4. Outline
1A warmup: Estimation in regression models
2ML as data-driven decision-making
3Subjective expected utility
4Specifying joint models
550 shades of Bayes
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5. Quotes from Gelman et al., 2013 on Bayesian methods
▶[...] practical methods for making inferences from data, using
probability models for quantities we observe
which we wish to learn.
▶The essential characteristic of Bayesian methods is their
of probability for quantifying uncertainty
statistical data analysis.
▶Three steps:
1Setting up a full probability model,
2Conditioning on observed data, calculating and interpreting the
appropriate “posterior distribution”,
3Evaluating the fit of the model and the implications of the resulting
posterior distribution. In response, one can alter or expand the
model and repeat the three steps.
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6. Notation that I will try to stick to
▶y1:n= (y1, . . . ,yn)∈ Y
n
denote observable data/labels.
▶x1:n∈ X
n
denote covariates/features/hidden states.
▶z1:n∈ Z
n
denote hidden variables.
▶θ∈Θ denote parameters.
▶Xdenotes anX-valued random variable. Lowercasexdenotes either
a point inXor anX-valued random variable.
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7. More notation
▶Whenever it can easily be made formal, we write densities for our
random variables and let the context indicate what is meant. So if
X∼ N(0, σ
2
), we write
Eh(X) =
Z
h(x)
e
−x
2
/2σ
2
σ
√
2π
dx=
Z
h(x)p(x)dx.
Similarly, forX∼ P(λ), we write
Eh(X) =
∞
X
k=0
h(k)e
−λ
λ
k
k!
=
Z
h(x)p(x)dx
▶All pdfs are denoted byp, so that, e. g.
Eh(Y, θ) =
Z
h(y, θ)p(y, θ)dydθ
=
Z
h(y, θ)p(y,x, θ)dxdydθ
=
Z
h(y, θ)p(y, θ|x)p(x)dxdydθ
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8. Outline
1A warmup: Estimation in regression models
2ML as data-driven decision-making
3Subjective expected utility
4Specifying joint models
550 shades of Bayes
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9. Outline
1A warmup: Estimation in regression models
2ML as data-driven decision-making
3Subjective expected utility
4Specifying joint models
550 shades of Bayes
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10. Inference in regression models
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11. Inference in regression models
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12. Inference in regression models
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13. Inference in regression models
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14. Inference in regression models
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15. Outline
1A warmup: Estimation in regression models
2ML as data-driven decision-making
3Subjective expected utility
4Specifying joint models
550 shades of Bayes
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16. Describing a decision problem under uncertainty
▶A state spaceS,
Every quantity you need to consider to make your decision.
▶ActionsA ⊂ F(S,Z),
Making a decision means picking one of the available actions.
▶A reward spaceZ,
Encodes how you feel about having picked a particular action.
▶A loss functionL:A × S →R+.
How much you would suffer from picking actionain states.
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17. Classification as a decision problem
▶S=X
n
× Y
n
× X × Y, i.e.s= (x1:n,y1:n,x,y).
▶Z={0,1}.
▶A={ag:s7→1
y̸=g(x;x1:n,y1:n),g∈ G}.
▶L(ag,s) = 1
y̸=g(x;x1:n,y1:n).
PAC bounds; see e.g. (Shalev-Shwartz and Ben-David, 2014)
Let (x1:n,y1:n)∼P
⊗n
, and independently (x,y)∼P, we want an
algorithmg(·;x1:n,y1:n)∈ Gsuch that ifn⩾n(δ, ε),
P
⊗n
Θ
E
(x,y)∼PL(ag,s)⩽ε
Λ
⩾1−δ.
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18. Regression as a decision problem
▶S=
▶Z=
▶A=
▶
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19. Estimation as a decision problem
▶S=
▶Z=
▶A=
▶
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20. Clustering as a decision problem
▶S=
▶Z=
▶A=
▶
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21. Outline
1A warmup: Estimation in regression models
2ML as data-driven decision-making
3Subjective expected utility
4Specifying joint models
550 shades of Bayes
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22. SEU is what defines the Bayesian approach
The subjective expected utility principle
1ChooseS,Z,Aand a loss functionL(a,s),
2Choose poverS,
3Take the the corresponding
a
⋆
∈arg min
a∈A
Es∼pL(a,s). (1)
Corollary: minimize the posterior expected loss
Now partitions= (sobs,su), then
a
⋆
∈arg min
a∈A
Esobs
E
su|sobs
L(a,s).
In ML,A={ag}, withg=g(sobs), so that (1) is equivalent to
a
⋆
=ag
⋆, with
g
⋆
(sobs)≜arg min
g
E
su|sobs
L(a,s).
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23. Outline
1A warmup: Estimation in regression models
2ML as data-driven decision-making
3Subjective expected utility
4Specifying joint models
550 shades of Bayes
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24. A recap on probabilistic graphical models 1/2
▶PGMs (aka “Bayesian” networks) represent the dependencies in a
joint distributionp(s) by G= (E,V).
▶Two important properties:
p(s) =
Y
v∈V
p(sv|s
pa(v)) and yv⊥y
nd(v)|y
pa(v).
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25. A recap on probabilistic graphical models 2/2
Also good to know how to determine whetherA⊥B|C; see (Murphy,
2012, Section 10.5).
d-blocking
An undirected pathPinGisd-blocked byE⊂Vif at least one of the
following conditions hold.
▶Pcontains a “chain”a→b→candb∈E.
▶Pcontains a Ψent”a←b→candb∈E.
▶Pcontains a “v-structure”a→b←cand neitherbnor any of its
descendants are inE.
Theorem
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26. Exercise
▶Doesx2⊥x6|x5,x1?
▶Doesx2⊥x6|x1?
▶Write the joint distribution as factorized over the graph.
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27. Estimation as a decision problem: point estimates
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28. Estimation as a decision problem: credible intervals
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29. Choosing priors (see Exercises)
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30. Classification as a decision problem
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31. Regression as a decision problem 1/2
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32. Regression as a decision problem 2/2
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33. Dimensionality reduction as a decision problem
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34. Clustering as a decision problem
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35. Topic modelling as a decision problem
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36. Outline
1A warmup: Estimation in regression models
2ML as data-driven decision-making
3Subjective expected utility
4Specifying joint models
550 shades of Bayes
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37. 50 shades of Bayes
An issue (or is it?)
Depending on how they interpret and how they implement SEU, you will
meet many types of Bayesians (46656, according to Good).
A few divisive questions
▶Using data or the likelihood to choose your prior; see Lecture #5.
▶Using MAP estimators for their computational tractability, like in
inverse problems
ˆxλ∈arg min∥y−Ax∥+λΩ(x).
▶When and how should you revise your model (likelihood or prior)?
▶MCMC vs variational Bayes (more in Lectures #2 and #3)
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38. References
[1] Bayesian data analysis.
[2] Machine learning: a probabilistic perspective.
Press, 2012.
[3] Understanding machine
learning: From theory to algorithms.
2014.
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