On prediction and density estimation

Having observed the initial segment of a random sequence, subsequent values may be predicted by calculating the conditional distribution given what has been observed. In statistical applications, it is usually necesary to work with a parametric family of processes, so the predictive distribution depends on the parameter, which must be estimated from the data. This device is used in a finitedimensional parametric model, so that maximum-likelihood can be applied to density estimation. An exchangeable process is constructed by generating a random probability distribution having a smooth density, and subsequently generating independent and identically distributed components from that distribution. The random probability distribution is determined by the squared modulus of a complex Gaussian process,and the finite-dimensional joint densities of the resulting process are obtained in the form of matrix permanents. The conditional density of Yn+1 given (y1, . . . , yn) is obtained as a weighted baseline, in which the permanent is the weight function. For prediction in the sense of density estimation, the permanent plays a role similar to that of the likelihood function in parametric inference.