07 Bayesian

Probability And Naïve Bayes¶

Kindly go through Bayes' Theorem in Probability and Statistics

Bayesian Network¶

Represents the dependence between variables

• Nodes - represents variables
• Links - X points to Y, implies X has direct influence over Y or X is a parent of Y
• CPT - each node has a conditional probability distribution which determines the effect of the parent on that node

Joint Probability Distribution¶

\(P(X_i|X_{i-1},........., X_1) = P(X_i |Parents(X_i ))\)

Independence in BN¶

\(X {\perp \!\!\! \perp} Y | Z\) is read as \(X\) is conditionally independent of \(Y\) given \(Z\)

1. Casual chains

  graph LR;
    A((A))-->B((B));
    B((B))-->C((C));

1. Common Cause

  graph TD;
    B((B))-->A((A));
    B((B))-->C((C));

In such a configuration, A is guaranteed to be independent of C given B

1. Common effect

  graph TD;
    A((A))-->C((C));
    B((B))-->C((C));

In such a configuration, A is guaranteed to be independent of B.
But A and B are not independent given C.

Active Triples¶

  graph LR;
    A((" "))-->B((" "));
    B((" "))-->C((" "));

  graph TD;
    B((" "))-->A((" "));
    B((" "))-->C((" "));

  graph TD;
    A((" "))-->C((" "));
    B((" "))-->C((" "));
    style C fill : #808080

  graph TD;
    A((" "))-->C((" "));
    B((" "))-->C((" "));
    C((" "))-.->D((" "))
    style D fill : #808080

Inactive Triples¶

  graph LR;
    A((" "))-->B((" "));
    B((" "))-->C((" "));
    style B fill : #808080

  graph TD;
    B((" "))-->A((" "));
    B((" "))-->C((" "));
    style B fill : #808080

  graph TD;
    A((" "))-->C((" "));
    B((" "))-->C((" "));

NOTE : All possible configurations for active and inactive triples are listed above

Notes on d-seperation and variable elimination need to be added

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