Conditional probability table

In statistics, the conditional probability table (CPT) is defined for a set of discrete (not independent) random variables to demonstrate marginal probability of a single variable with respect to the others. For example, assume there are three random variables $x_1,x_2, x_3$ where each have $K$ states. Then, the conditional probability table of $x_1$ provides the marginal probability values for $P(x_1\mid x_2,x_3)$. Clearly, this table has K³ cells. In general, for $M$ number of variables $x_1,x_2,\ldots,x_M$ with $K$ states, the CPT has size K^M.

CPT table can be put into a matrix form. For example, the values of $P(x_j\mid x_i)=T_{ij}$ create a matrix. This matrix is a stochastic matrix since sum of all its elements is equals to 1; i.e. $\sum_j T_{ij}$.