Approximate Predictive Integrals for Dynamic Generalized Linear Models

Abstract

This paper contributes to approximate analysis of dynamic generalized linear models (DGLM). The issue is approximation of predictive integrals for DGLM. Such integrals arise e.g. when computing the likelihood function of data conditional on a fixed hyperparameter. Recall that in dynamic linear modelling it is assumed that each observation y _t of a (univariate) time series y ₁,...,y _N is a realization from a distribution of the exponential family with parameters changing over time ([14]). This change is explained through an unobservable state vector x _t ∈ ℝ^r which follows a Gaussian process. Conditional on a known value of x _t the distribution of y _t |x _t is assumed to belong to the exponential family. By predictive integral we now mean integrals of the type

$$p(y_t|y^s) = \int p(y_t |\textbf{\textit{x}}_t)p(\textbf{\textit{x}}_t|y^s)d\textbf{\textit{x}}_t$$

(1)

with t > s and $y^s = \{y_i \cdots, y_s\}$. p(y _t|y ^s) and p(y _t|x _t) denote the densities of the distribution of y _t|y ^s and y _t |x _t with respect to the Lebesgues (metric time series) or to the counting measure (discrete time series). Integrals of this type can be thought of as an infinite mixture of conditional (‘mixed’) densities with a mixing measure P _s which has ‘mixing’ Lebesgue density $p^s(\textbf{\textit{x}}_t)= p(\textbf{\textit{x}}_t|y^s)$:

$$p(y_t|y^s) = \int p(y_t|\textbf{\textit{x}}_t)dP_s(\textbf{\textit{x}}_t)$$

(2)

The problem of approximation of predictive integrals is most often discussed for i.i.d. observations arising from models with static parameters x _t = x (e.g. [8], [11], [6]). The methods suggested in these papers approximate the infinite mixture (2) by a finite number of mixed densities where the choice of the conditional parameter is based on the mixing density:

$$p(y_t|y^s) \approx \sum_{i=1}^{M} p(y_t|\textbf{\textit{x}}_s^{(i)})w_s^{(i)}$$

(3)