Definition of Factors

  • A factor really is a function, or a table.
    • It takes a bunch of arguments. In this case, a set of random variables \(X_1\) up to \(X_k\), and just like any function it gives us a value for every assignment to those random variables.
    • So it takes all possible assignments in the cross products space of \(X_1\) up to \(X_k\). That is all possible combinations of assignments and in this case it gives us a real value for each such combination.
    • The set of variables \(X_1\) up to \(X_k\) is called the scope of the factor. That is it’s the set of arguments, that a factor takes.

In short,

  • A factor \(\phi(X_1, ..., X_k)\)

\(\phi\): Val\((X_1, ..., X_k)\) \(\longrightarrow\) \(R\)

  • Scope = \({X_1, ..., X_k}\)

Examples

\(\Longrightarrow\) every combination of values to the variables I and D, we have a probability distribution over G.

For example (in the last row of table of Factors - CPD), if I have an intelligent student I a difficult class, which is this last line over here, this tells us that the probability of getting an A is 0.5 (i.e. the process of renormalization of .06 / (0.06 + .036 + .024)), B is 0.3 and a C is 0.2 and as we can see, these numbers sum to 1 as they should because this is a probability distribution over G for this particular condition and context.

Factor Product

What is the scope of the factor product \(\phi(A,B,C) \times \phi(C,D)\)?
\(\Rightarrow\) The scope of a factor is the variables in its domain. Let \(f\) be the factor product, which means that \(f(A,B,C,D)\) = \(\phi(A,B,C) \times \phi(C,D)\). Since \(f\) is a function over \(A, B, C, D\) its scope is \(\{A,B,C,D\}\).

Factor Marginalization

Factor Reduction

Why factors?

  • Fundamental building block for defining distributions in high-dimensional spaces.

    \(\Longrightarrow\) That is the way in which we’re going to define an exponentially large probability distribution over N random variables is by taking a bunch of little pieces and putting them together by multiplying factors in order to define these high dimensional probability distributions.

  • Set of basic operations for manipulating these probability distributions in order to give us a set of basic inference algorithms.