Causal Bayesian Networks

Causal Bayesian Networks

Today:

  • Causal Bayesian Networks
  • How do we reason about independence in Bayesian Networks?

Review: Distributions of Discrete R.V.s

Joint

\(P(X=x, Y=y)\)

Shorthand: \(P(x, y)\)

\(P(X, Y)\)

Single number

"Probability that \(X=x\) and \(Y=y\)"

"Joint distribution of \(X\) and \(Y\)"

A table

Conditional

A collection of tables for each \(y\)

\(P(X=x\mid Y=y)\)

Shorthand: \(P(x \mid y)\)

\(P(X \mid Y)\)

Single number

"Probability that \(X=x\) if \(Y=y\)"

"Conditional distribution of \(X\) given \(Y\)"

Marginal

\(P(X=x)\)

Shorthand: \(P(x)\)

\(P(X)\)

Single number

A table

"Probability that \(X=x\)"

"Marginal distribution of \(X\)"

Causal Bayesian Networks

A Bayesian Network compactly represents a joint probability distributions using two components:

  1. Structure: a directed acyclic graph (DAG), where each node is a R.V.
  2. Parameters: Numerical values that determine a conditional distribution at each node

At each node, \(P(X \mid pa(X))\)

In a Causal Bayesian Network, arrows denote causation.

\(B\) is a result of \(A\) (and some aleatory uncertainty)

Chain rule for Bayesian Networks

\[P(X_{1:n}) = \prod_{i=1}^n P(X_i \mid \text{pa}(X_i))\]

Simple Causal Bayes Net Example

Naive Inference on Bayes Nets

Bayes Net with 3 Random Variables: \(A \rightarrow C \rightarrow B\)

Want to find \(P(\underbrace{A=a}_\text{query} \mid \underbrace{B=b}_\text{evidence})\)

  1. \(P(A, B=b, C) = P(B=b \mid C) \, P(C \mid A) \, P(A)\).
  2. Marginalize over hidden and query variables to get \[P(A=a, B=b) = \sum_c P(A=a, B=b, C=c)\] and
    \[P(B=b) = \sum_{a, c} P(A=a, B=b, C=c)\]
  3. \(P(A=a \mid B=b) = \frac{P(A=a, B=b)}{P(B=b)}\)

\(C\) is a "hidden variable"

(Book introduces unnormalized "factors", but process is the same.)

Next Year: Reverse, like solutions to 2026 Exam 1

Conditional Independence in Bayes Nets

Next year: make sure to have example

Conditional Independence: Fork

\(B \perp C \mid A\)  ?

Yes

https://kunalmenda.com/2019/02/21/causation-and-correlation/

Conditional Independence: Inverted Fork

\(B \perp C \mid A\)  ?

https://kunalmenda.com/2019/02/21/causation-and-correlation/

Next year, swap examples

More Complex Example

\((B\perp D \mid A)\) ?

\((B\perp D \mid E)\) ?

Yes!

Inconclusive

Why is this relevant to decision making?

d-Separation

Let \(\mathcal{C}\) be a set of random variables.

An undirected path between \(A\) and \(B\) is d-separated by \(\mathcal{C}\) if any of the following exist along the path:

Separators (a.k.a "inactive triples"):

  1. Chain: \(X \rightarrow Y \rightarrow Z\) such that \(Y \in \mathcal{C}\)
  2. Fork: \(X \leftarrow Y \rightarrow Z\) such that \(Y \in \mathcal{C}\)
  3. Inverted Fork (v-structure): \(X \rightarrow Y \leftarrow Z\) such that \(Y \notin \mathcal{C}\) and no descendant of \(Y\) is in \(\mathcal{C}\).

Also:

  • d-separated = "inactive"
  • not d-separated = "active"

\(B \leftarrow A \rightarrow C \leftarrow D\) 

\(B \leftarrow C \rightarrow A \leftarrow D\) 

Break

Are these paths d-separated by \(\mathcal{C} = \{C\}\)?

d-Separation for Bayes Nets

We say that \(A\) and \(B\) are d-separated by \(\mathcal{C}\) if all acyclic paths between \(A\) and \(B\) are d-separated by \(\mathcal{C}\).

If \(A\) and \(B\) are d-separated by \(\mathcal{C}\) then \(A \perp B \mid \mathcal{C}\)

*short for "directionally separated"

In other words, if there is any active path w.r.t. \(\mathcal{C}\) between \(A\) and \(B\), we cannot conclude that \(A \perp B \mid \mathcal{C}\) based on the structure alone.

Proving Conditional Independence

Separators

  1. Chain: \(X \rightarrow Y \rightarrow Z\) such that \(Y \in \mathcal{C}\)
  2. Fork: \(X \leftarrow Y \rightarrow Z\) such that \(Y \in \mathcal{C}\)
  3. Inverted Fork (v-structure): \(X \rightarrow Y \leftarrow Z\) such that \(Y \notin \mathcal{C}\) and no descendant of \(Y\) is in \(\mathcal{C}\).
  1. Enumerate all (non-cyclic) paths between nodes in question
  2. Check all paths for d-separation
  3. If all paths d-separated, then C.I.

Example: \((B \perp D \mid C, E)\) ?

Markov Blanket

The Markov Blanket of \(\mathcal{X}\) is the minimal set of nodes that, if their values were known, would make \(\mathcal{X}\) conditionally independent of all other nodes.

A Markov blanket of a particular node consists of its parents, its children, and the other parents of its children.

If \(\mathcal{B}\) is the Markov blanket of \(\mathcal{X}\), you can treat analyze \(\mathcal{B} \cup \mathcal{X}\) alone, and ignore any other nodes.

Exercise

\(D \perp C \mid B\) ?

\(D \perp C \mid E\) ?

  1. The path contains a chain \(X \rightarrow Y \rightarrow Z\) such that \(Y \in \mathcal{C}\)
  2. The path contains a fork \(X \leftarrow Y \rightarrow Z\) such that \(Y \in \mathcal{C}\)
  3. The path contains an inverted fork (v-structure) \(X \rightarrow Y \leftarrow Z\) such that \(Y \notin \mathcal{C}\) and no descendant of \(Y\) is in \(\mathcal{C}\).

Approximate Inference

Approximate Inference: Direct Sampling

Analogous to

unweighted particle filtering

Approximate Inference: Weighted Sampling

Analogous to

weighted particle filtering

Approximate Inference: Gibbs Sampling

Markov Chain Monte Carlo (MCMC)

Sampling from a Bayesian Network

Given a Bayesian network, how do we sample from the joint distribution it defines?

  1. Topoligical Sort (If there is an edge \(A \rightarrow B\), then \(A\) comes before \(B\))
  2. Sample from conditional distributions in order of the topological sort

Analogous to Simulating a (PO)MDP

Recap

Review

Independence

\(P(X, Y) = P(X)\, P(Y)\)

Conditional Independence

\(P(X, Y \mid Z) = P(X \mid Z) \, P(Y \mid Z)\)

 2022 Quiz 1

Joint Distribution Complexity

Binary Random Variables \(X_1\), \(X_2\), \(X_3\)

How many independent parameters (\(\theta\)) to specify joint distribution?

7

For \(n\) binary R.V.s, \(2^n-1\) independent parameters specify the joint distribution.

In general \[\dim(\theta) = \prod_{i=1}^n |\text{support}(X_i)| - 1\]

Bayesian Network

Bayesian Network: Directed Acyclic Graph (DAG) that represents a joint probability distribution

  • Node:
  • Edges encode:

Random Variable

\[P(X_{1:n}) = \prod_{i=1}^n P(X_i \mid \text{pa}(X_i))\]

Full Story: "Causality: Models, Reasoning and Inference" by Judea Pearl

Counting Parameters

For discrete R.V.s:

\[\text{dim}(\theta_X) = \left(|\text{support}(X)|-1\right)\prod_{Y \in Pa(X)} |\text{support}(Y)|\]

Inference

Inputs

  • Bayesian network structure
  • Bayesian network parameters
  • Values of evidence variables

Outputs

  • Posterior distribution of query variables

Given that you have detected a trajectory deviation, and the battery has not failed what is the probability of a solar panel failure?

\(P(S=1 \mid D=1, B=0)\)

Exact

Approximate

Exact Inference

Exact Inference

\[P(S{=}1 \mid D{=}1, B{=}0)\]

\[= \frac{P(S{=}1, D{=}1, B{=}0)}{P(D{=}1, B{=}0)}\]

\[P(S{=}1, D{=}1, B{=}0) = \sum_{e, c}P(B{=}0, S{=}1, E{=}e, D{=}1, C{=}c)\]

\[P(B{=}0, S{=}1, E, D{=}1, C)\]

\[= P(B{=}0)\,P(S{=}1)\,P(E\mid B{=}0, S{=}1)\,P(D{=}1\mid E)\,P(C{=}1\mid E)\]

Exact Inference

Product

Condition

Marginalize

Exact Inference

Exact Inference: Variable Elimination

Start with

Eliminate \(D\) and \(C\) (evidence) to get \(\phi_6(E)\) and \(\phi_7(E)\)

Eliminate \(E\)

Eliminate \(S\)

vs

Choosing optimal order is NP-hard

Next Year: Skip

Break

025 Bayesian Networks

By Zachary Sunberg

025 Bayesian Networks

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