Conditional probability table

In statistics, the Conditional Probability Table (CPT) is defined for a set of discrete (not independent) random variable 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 math values for $P(x_{1}|x_{2},x_{3})$. Clearly, this table has $O(K^{3})$ number of rows. In general, for $M$ number of variables $x_{1},x_{2},...,x_{M}$ with $K$ states, the CPT has the size of $O(K^{M})$.

CPT table can be put into a matrix form. For example, the values of $P(x_{j}|x_{i})=T_{ij}$ create a matrix. This matrix is Stochastic matrix since its row sum is equals to 1; i.e. $\sum _{j}T_{ij}$.