Inference Using Full Joint Distributions

Inference using full joint distributions involves computing probabilities for a set of variables from a joint distribution of those variables. Given a joint distribution, we can compute any marginal distribution, conditional distribution, or joint distribution of a subset of the variables.

The joint distribution of a set of variables can be represented by a joint probability table, which lists the probabilities of all possible combinations of values of the variables. Computing marginal probabilities from a joint probability table involves summing over all the values of the other variables. Computing conditional probabilities involves dividing the joint probabilities by the marginal probabilities of the conditioning variables.

Inference using full joint distributions can become computationally infeasible for large or complex domains. This is because the number of entries in the joint probability table grows exponentially with the number of variables in the distribution. As a result, we often resort to other inference techniques such as approximate inference or sampling methods.

In addition to joint probability tables, there are also graphical models such as Bayesian networks and Markov networks that can be used to represent joint distributions compactly and efficiently. These models represent the joint distribution in terms of a graph, where the nodes represent variables and the edges represent dependencies between the variables. Inference in graphical models can be performed more efficiently than in joint probability tables because the graph encodes the conditional independence relationships between the variables.