Quadrature Compound: An approximating family of distributions

Compound distributions allow construction of a rich set of distributions. Typically they involve an intractable integral. Here we use a quadrature approximation to that integral to define the quadrature compound family. Special care is taken that this approximation is suitable for computation of gradients with respect to distribution pa- rameters. This technique is applied to discrete (Poisson LogNormal) and continuous distributions. In the continuous case, quadrature compound family naturally makes use of parameterized transformations of unparameterized distributions (a.k.a “reparame- terization”), allowing for gradients of expectations to be estimated as the gradient of a sample mean. This is demonstrated in a novel distribution, the diffeomixture, which is is a reparameterizable approximation to a mixture distribution.

• Machine Intelligence

• Algorithms and Theory