Bayesian Networks

Graphical Models

• Bayesian networks

• Conditional random fields

• etc.

Topic=Course

“Grading” “Instructor” “Syllabus” “Vapnik”“Exams”

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ijnj cCxXPcCPxXxXcCP Naive Bayes:

Bayesian networks

• Idea is to represent dependencies (or causal relations) for all the variables so that space and computation-time requirements are minimized.

Topic=RussianMathematicians

Topic=Course

Vapnik Syllabus

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Vapnik Syllabus

Semantics of Bayesian networks

• If network is correct, can calculate full joint probability distribution from network.

where parents(Xi) denotes specific values of parents of Xi.

Compare with Naïve Bayes.

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Example

• Calculate

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ExamplesWhat is unconditional (marginal) probability that Vapnik is true?

What is the unconditional (marginal) probability that Russian Mathematicians is true?

Causal inferenceEvidence is cause, inference is probability of effect

Example: Instantiate evidence Course= true. What is P(Syllabus | Course)?

Different types of inference in Bayesian Networks

Diagnostic inference

Evidence is effect, inference is probability of cause

Example: Instantiate evidence Syllabus = true. What is P(Course | Syllabus)?

Example: What is P(Course|Vapnik)?

Inter-causal inference

Explain away different possible causes of effect

Example: What is P(Course|Vapnik,RussianMath)?

Why is P(Course|Vapnik,RussianMath) < P(Course|Vapnik)?

Complexity of Bayesian Networks

For n random Boolean variables:

• Full joint probability distribution: 2n entries

• Bayesian network with at most k parents per node:

– Each conditional probability table: at most 2k entries

– Entire network: n 2k entries

What are the advantages of Bayesian networks?

• Intuitive, concise representation of joint probability distribution (i.e., conditional dependencies) of a set of random variables.

• Represents “beliefs and knowledge” about a particular class of situations.

• Efficient (?) (approximate) inference algorithms

• Efficient, effective learning algorithms

Issues in Bayesian Networks

• Building / learning network topology

• Assigning / learning conditional probability tables

• Approximate inference via sampling

• In general, however, exact inference in Bayesian networks is too expensive.

Approximate inference in Bayesian networks

Instead of enumerating all possibilities, sample to estimate probabilities.

X1 X2 X3 Xn

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General question: What is P(X|e)?

Notation convention: upper-case letters refer to random variables; lower-case letters refer to specific values of those variables

Direct Sampling

• Suppose we have no evidence, but we want to determine P(C,S,R,W) for all C,S,R,W.

• Direct sampling:

– Sample each variable in topological order, conditioned on values of parents.

– I.e., always sample from P(Xi | parents(Xi))

1. Sample from P(Cloudy). Suppose returns true.

2. Sample from P(Sprinkler | Cloudy = true). Suppose returns false.

3. Sample from P(Rain | Cloudy = true). Suppose returns true.

4. Sample from P(WetGrass | Sprinkler = false, Rain = true). Suppose returns true.

Here is the sampled event: [true, false, true, true]

Example

• Suppose there are N total samples, and let NS (x1, ..., xn) be the observed frequency of the specific event x1, ..., xn.

• Suppose N samples, n nodes. Complexity O(Nn).

• Problem 1: Need lots of samples to get good probability estimates.

• Problem 2: Many samples are not realistic; low likelihood.

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Markov Chain Monte Carlo Sampling

• One of most common methods used in real applications.

• Uses idea of “Markov blanket” of a variable Xi:

– parents, children, children’s parents

Illustration of “Markov Blanket”

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• Recall that: By construction of Bayesian network, a node is conditionaly independent of its non-descendants, given its parents.

• Proposition: A node Xi is conditionally independent of all other nodes in the network, given its Markov blanket.

Markov Chain Monte Carlo Sampling Algorithm

• Start with random sample from variables: (x1, ..., xn). This is the current “state” of the algorithm.

• Next state: Randomly sample value for one non-evidence variable Xi , conditioned on current values in “Markov Blanket” of Xi.

Example

• Query: What is P(Rain | Sprinkler = true, WetGrass = true)?

• MCMC: – Random sample, with evidence variables fixed:

[true, true, false, true]

– Repeat:

1. Sample Cloudy, given current values of its Markov blanket: Sprinkler = true, Rain = false. Suppose result is false. New state:

[false, true, false, true]

2. Sample Rain, given current values of its Markov blanket:

Cloudy = false, Sprinkler = true, WetGrass = true. Suppose

result is true. New state: [false, true, true, true].

• Each sample contributes to estimate for query

P(Rain | Sprinkler = true, WetGrass = true)

• Suppose we perform 100 such samples, 20 with Rain = true and 80 with Rain = false.

• Then answer to the query is

Normalize (20,80) = .20,.80

• Claim: “The sampling process settles into a dynamic equilibrium in which the long-run fraction of time spent in each state is exactly proportional to its posterior probability, given the evidence.”

• That is: for all variables Xi, the probability of the value xi of Xi appearing in a sample is equal to P(xi | e).

.

Claim (again)

• Claim: MCMC settles into behavior in which each state is sampled exactly according to its posterior probability, given the evidence.