Marginal distribution

In probability theory, given a joint probability density function of two parameters or variables x and y, the marginal distribution of x is the probability density function of x after information about y has been averaged out. For example from a Bayesian probability perspective, when doing parameter estimation we can consider the joint probability density function as a joint inference which characterizes our uncertainty 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 out. We can say that, in this case, we are considering y as a nuisance parameter.

For a continuous probability density function (pdf), an associated marginal pdf can be written as my(x|I). Such that

$m_{y}(x|I) = \int_y p(x,y|I) \, dy = \int_y p(x|yI) \, p(y|I) \, dy$

where p(x,y|I) gives the joint probability density function of x and y, and p(x|yI) gives the conditional probability density function for x given y. The second integral was formulated by use of the Bayesian product rule. Note that the marginal distribution has the form of an expectation value with respect to the prior distribution of y. This marginalization procedure readily generalizes for an arbitrary number of variables or parameters. Also note that all densities are considered conditioned on common background or prior information I.

For a discrete probability mass function (pmf), the marginal probability for xk can be written as pk Such that

 pk = ∑ pkj = ∑ pjpk | j j j

where the j index spans all indices of the discrete y. The notation pkj here means the joint probability value when x has the value xk and y has the value yj while pk|j here references the conditional probability value for xk for y fixed at the value yj. With k fixed in the above summation and pk,j considered as a matrix, this can be thought of as summing over all columns in the kth row. Similarly, the marginal mass function for yj (say qj) can be computed by summing over all rows in column j. When all of the pk are determined this way for all k, this set of pk constitute the pmf for the all relevant discrete values of x, in this particular case calculated as a marginal mass function from an original joint probability mass function.