Contents

Graphical models¶

A graphical model is a way to represent a joint distribution by making CI assumptions, in particular, the nodes in the graph represent random variables, and the (lack of) edges represent CI assumptions.

Directed graphical moels¶

The edges in the graph are directed.

Markov random field¶

The edges in the graph have no orientation.

Inference¶

There are two common types of questions we ask.

Marginal inference¶

What is the probability of a given variable in our model after we sum everything else out: $\( p(y=1) =\sum_{x1} \sum_{x_2} \cdots \sum_{x_n} p(y=1|x_1, x_2,\cdots,x_n) \)$

Maximum a posteriori (MAP) inference¶

What is the most likely assignment to the variables in the model:

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

In general an exact answer is NP-hard. Tractable computation depends on the structure of the graph that describes that probability. In cases when exact inference is not tractable we result to approximate inference.

Approximate inference¶

MAD Manifold learning