跳转至

Lecture 9: Reasoning (I)

Question

We have several random variables.

We can observe some of them(Temperature \(P\), Date(\(D\))).

We want to know the probability of others (Sunshine \(S\), Rain \(R\)).

Modeling: assume that a joint distribution exists between variables:

\[ p(S,R,T,D) \]

Inference:

  • condition on some evidence (\(T = 36.5 ^\circ \text C, D = 2021/07/31\))
  • interested in some other things (\(p(R \mid T = 36.5 ^\circ \text C, D = 2021/07/31)\))

Formally: Probabilistic Reasoning

We want to use random variable model to learn, to achieve the predict job.

  • \(X = \{x_1, \cdots, x_D\}\) is a set of \(D\) random variables.
  • query set \(R\) and condition set \(C\) are subsets of \(X\)

Modeling: How to specify a joint distribution \(p(x_1, \cdots, x_D)\) compactly(紧凑地)1?

Inference: How to compute \(p(R \mid C)\) effciently?

Probabilistic Graphical Model

概率图模型。

We try to use graphs to describe the correlation and dependence relationships between random variables.

  • Vertex: Random variable
  • Edge: Their relationship

Bayesian Network: DAG version

Bayesian Network:

  • a DAG graph
  • \(p(x_1, \cdots, x_K) = \prod_{s=1}^{K} p(x_s \mid \bm x_{\Gamma(s)})\), \(\Gamma(s)\) is the set of parents of \(x_s\)
    • as this graph is a DAG, there must be a order(topology order) that we can calculate from basic to the hole joint probability.

Example: The Alarm Network

We can use our prior knowledge to reduce edges(finding independent variables).

Example: in Medical

Use observable variables to infer hidden ones.

Pro: Every conditional distribution of a random variable is reasonable.

A white box!

Example: Polynomial Regression

Markov Random Field

undirected graph \(G\)

\[ p(\bm x) = \frac{1}{Z} \prod_C \psi_C(\bm x_C) \]
  • \(C\) is all (maximal) cliques in \(G\)
  • \(\bm x = \{x_1, \cdots, x_n\}\)
  • \(\psi\) is a non-negative potential function defined on those cliques
    • usually \(\psi_C(\bm x_C) = \exp\{-E(\bm x_c)\}\), \(E\) is energy function.
    • low energy => high potential => high probability
  • \(Z\) is a normalized factor

Illustration: Image Denoising

Transformation method

Bayesian network can be transformed into Markov Random Field by

  • Linking the parents of each node;
  • remove directions.

Independence in Bayesian Network

Marginal Independence

Conditional Independence

D-separation

connected if:

  1. no head-to-head edge
  2. head-to-head edge belongs to C or C's ancestor

D-separation \(\iff\) conditional independence

Markov Blanket

Question

How to infer a node with variable \(x_i\) from the remaining variables?

Solution:

Find \(S_1 \subset \bm x_\{j \neq i\}\) that

\[ x_i \perp (\bm x_{\{j \neq i\}} \setminus S_1) \mid S_1 \]

which means \(S_1\) contains all information that is needed to infer \(x_i\).

\(S_1\) is the Markov blanket of \(x_i\).

Note

Markov Boundary: Minimal Markov blanket.

In Bayesian network, it includes parents, children and other parents of all its children (co-parents).

Independence in Markov Random Field

\(x,y\) are dependent if they are connected by a path of unobserved variables

Global Markov Property

If any path from \(\bm A\) to \(\bm B\) passes through \(\bm C\), which denotes as \(\text{sep}_G(\bm A, \bm B \mid \bm C)\), then \(( \bm X_A \perp \bm X_B ) \mid \bm X_C\)

Local Markov Property

For any node \(x\), \(N(x)\) denotes neighbours of \(x\), \(N[x] = x \cup N(x)\), then \(X_x \perp \bm X_{V \setminus N[x]} \mid \bm X_{N(x)}\)

Pairwise Markov Property

If \(u\) and \(v\) are not linked, then \(X_u \perp X_v \mid \bm X_{V\setminus\{u,v\}}\) .

Marginal Inference

\[ p(y=1) = \sum_{x_1}\sum_{x_2} \cdots \sum_{x_n} p(y=1, x_1, \cdots, x_n) \]

  1. 我的理解是，计算的过程能尽可能的快和简单。 ↩