Conditional probability table

In statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to demonstrate conditional 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 conditional probability values for $P(x_{1}\mid x_{2},x_{3})$. This table has $K^{3}$ cells. In general, for $M$ variables $x_{1},x_{2},\ldots ,x_{M}$ with $K$ states, the CPT has size $K^{M}.$

With only two variables, a CPT 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 the rows sum to 1; i.e. $\sum _{j}T_{ij}=1$ for all i.