PGMs - 1

Overview

@AishFenton @NetflixResearch

Why PGMs?

• Probabilistic models provide an elegant framework for modeling
• Assumptions are made explicit (and can be played with)
• Elegant way of incorporating “prior” knowledge
• Full machinery of prob lets you do more (explore/exploit, factor in uncertainty, etc)
• More whitebox than deep-learning; results are often interpretable
• Graphical models provide tools for producing complex hierarchical (deep?) probabilistic models

Example / Motivation

Example / Motivation

Example / Motivation

Layout of tutorial

• Intro
• Probability reminders
• Probabilistic modeling intro
• Mixture models
• Overstanding the model
• First look at Expectation Maximization (EM)
• Inference
• EM revisited
• Variational bayes
• Samplers
• Topic models

Probability Reminders

Basic Identities

Sum rule

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Product rule

Bayes Rule

Useful for swapping from: p(obs|event) -> p(event|obs)

Entropy

More uncertainty = more information

Kullback–Leibler Divergence

Kullback–Leibler Divergence

Distributions we’ll need

Bernoulli

Multinomial (1 draw)

A

B

C

p([0,1,0])=

0.7

0.3

0.2

Multinomial (n draws)

Beta / Dirichlet Distributions

Beta

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Dirichlet

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Bernoulli-Beta Conjugacy

= Beta(α+1, β)

Multinomial-Dirichlet Conjugacy

= Dir([α₁+x₁,...,α_i+x_i])

Bayes Nets

Directed Graphical Models

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B

C

D

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B

C

D

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B

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D

Directed graphical models

• In general, factorizes to:

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• Where each “CPD” has an associated distribution
• Cycles disallowed (i.e. DAG)

Plate Notation

C

P

2

A

α

D

B

C

β

Plate Notation

(Non-standard, but useful)

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2

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α

D

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β

Plate Notation

C

P

2

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α

D

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C

What’s missing?

β

Generative Story

for k = 1 to 2

D_k ~ Dir(β)

foreach ML conference, i:

A_i ~ Beta(α)

foreach paper, j:

B_ij ~ Bern(A_i)

C_ij ~ Mult(D_B)

// Ak & WK

// topic list

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// bias!

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// index D

// draw topic

What we observe...

C

P

2

A

α

D

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C

β

Observe:

• List of papers per conference
• Authors blinded out

What we observe...

C

P

2

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α

D

B

C

β

Can infer:

• Conference bias
• Where the paper likely came from
• AK vs. WK topic preferences

Conditional Independence

A ⫫ C | ∅

🌧

☂️

📺

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A ⫫ C | B

🌧

☂️

📺

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C

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A ⫫ C | B

🌬

⛄

❄️

A

C

B

A ⫫ C | ∅

💦

🌧

🚿

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C

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A ⫫ C | B

💦

🌧

🚿

A

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B

Markov Blanket

x_i

Bayesian Modeling

Maximum Likelihood (MLE)

Example: MLE of Multinomial

But…

• Point estimate of parameters
• Can overfit (why?)
• Still question on how to maximize p(X|θ)
• Closed form for many well known distributions
• Also can use standard gradient based optimizers
• But sometimes parameters are coupled. What to do then?

Going Bayesian…

Anatomy of a model

Conjugacy revisited

For example:

= Dir([α₁+x₁,...,α_i+x_i])

Why go Bayesian?

• Factors in uncertainty of parameters
• Marginalization gives a more robust estimate
• Can be used for explore/exploit
• Priors provide an elegant way to encode our prior assumptions (connection to regularization).

Posterior Predictive Distribution

Marginalize over θ

Monte Carlo Estimate

Modeling tips

• Latent variables are normally first, flow top to bottom, and bottom are observations.
• Discrete latent variables per observation, can act as “switches” between model components
• Common to add latent variable per observation, to model mixtures of well-known (i.e. easier to work with) distributions.
• Nice if parent & child are conjugate (more later)

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Markov Random Fields

(Undirected Graphical Models)