Marginal distribution

In probability theory, given a joint probability density function of two parameters x and y, the marginal distribution of x is the probability distribution of x after information about y has been averaged over. From a Bayesian probability perspective, we can consider the joint probability density as a joint inference about the true values of the two parameters and the marginal distribution of (say) x, as our inference about x after the uncertainty about y had been averaged over. We can say that in this case, we are considering y as a nuisance parameter.

For continuous probability densities, this marginal probability density function can be written as m_y(x). Such that

where p(x,y) gives the joint distribution of x and y, and c(x|y) gives the conditional distribution for x given y. Note that the marginal distribution has the form of an expectation.

For a discrete probability mass function, the marginal probability for the k^th ordinate can be written as p_k Such that

where the j index spans all values of the discrete y. With k fixed here and p_k,j considered as a matrix, then this can be thought of as summing over all columns in the k^th row. Similarly, the marginal mass function for y can be computed by summing over all rows in a particular column. When all of the p_k are determined this way for all k, this set of p_k constitute the discrete probability mass function for the relevant discrete values of x, in this particular case calculated as a marginal mass function from an original joint probability mass function.