1 Graphical Models

• From course Probabilistic Models and Inference Algorithms for Machine Learning, Prof. Dahua Lin
• All contents here are from the course and self understandings.

Basic Concepts

• The key idea behind graphical models is factorization
• A graphical model generally refers to a family of joint distributions over multiple variables that factorize according to the structure of the underlying graph.
• 可以從兩方面來理解 graphical models:
  
  • 是個 數據結構(data structure), 這個數據結構呢能通過分解的方式描述聯合分佈（a joint distribution in a factorized manner.）
  • 一種緊湊的一系列條件獨立(conditional independencies)的分佈（a family of distributions.）的表示方法
  • 以上這兩點實際上是等價的。

• Graphical Models 的類別：

  • Bayesian Networks (Directed Acyclic Graphs)
  • Markov Random Fields (Undirected Graphs)
  • Chain Graphs (Directed acyclic graphs over undirected components)
  • Factor Graphs

• Directed Acyclic Graph

  • A graph G is called a directed acyclic graph (DAG)
    if it has no directed cycles. (即每月 自環的現象)
  • 因爲有向圖(directed graph)是有方向的，所以對圖中的有向邊來說，是要分parent 和 child 的
  • A vertex s is called an ancestor of t and t an descendant of s, denoted as s ≺ t, if there exists
    a directed path from s to t.
  • Topological Ordering： A topological ordering of a directed graph G = (V, E) is a linear ordering of vertices such that for each edge (s, t) ∈ E, s always comes before t.
  • A finite directed graph is acyclic if and only if it has a topological ordering.

Bayesian Networks

• Given a DAG G = (V, E), we say a joint distribution over XV factorizes according to G, if its density p can be expressed as
  
  p(xV)=∏s∈Vps(xs|xπ(s))
  
  
  • Such a model is called a Bayesian Network over G.
  • π(s) is the set of s `s parents, which can be empty
  • example:
    

Markov Random Fields

• 考慮一個無向圖 G = (V, E)

  • clique: is a fully connected subset of vertices
  • A clique is called maximal if it is not properly contained in another clique. (指當另一個點加進來的時候，這個clique 就變得 不 clique 了)
  • C(G) denotes the set of all maximal cliques.
  • example
    

• Markov Random Fields

  • Consider an undirected graph G=(V,E) , we say a joint distribution of XV factorizes according to G if its density p can be expressed as
    
    p(xV)=1X∏C∈CψC(xC)
  • This is called a Markov Random Field over G.
  • ψC:XC→R+ are called factors
  • The normalizing constant Z is usually needed to ensure the distribution is properly normalized:
    
    Z=∫∏C∈C(G)ψC(xC)μ(dx)
  • ψC 稱爲 compatibility functions，它不需要服從marginal or conditional distributions.

Analysis of Conditional Independence

• The graphical structure also encodes a set of conditional independencies among the variables.
• Consider a joint distribution over (X, Y, Z), X and Y are called conditionally independent given Z, denoted by X⊥Y|Z if
  Pr(X∈A&Y∈B|Z)=Pr(X∈A|Z)Pr(Y∈B|Z)
  
  or more generally
  
  EX,Y|Z[f(X)g(Y)]=EX|Z[f(X)]EY|Z[g(Y)]
  
  
  • Suppose the conditional distributions X|Z and Y|Z have densities pX|z and pY|z , then X⊥Y|Z , if the following equality holds almost
    surely:
    p(X,Y)|z(x,y)=pX|z(x)pY|z(y)

Factor Graphs

• An MRF does not always fully reveal the factorized structure of a distribution.
• A factor graph can sometimes give a more accurate characterization of a family of distributions.
• A factor graph is a bipartite graph with links between two types of nodes: variables and factors.
• A variable x and a factor f is linked in a factor graph, if the factor involves x as an argument.