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Robust Sequential Approximate Bayesian Estimation

SUMMARY An approximation to the sequential updating of the distribution of location parameters of a linear time series model is developed for non-normal observations. The behaviour of the resulting non-linear recursive filtering algorithm is examined and shown to have certain desirable properties for a variety of non-normal error distributions. Illustrative examples are given and relationships with previous work on robustness and sequential estimation are mentioned. WE consider here the problem of sequential estimation of the location vector of a linear time series model, termed the Dynamic Linear Model by Harrison and Stevens (1976). The straightforward, exact analysis obtained by assuming normal error and prior structure is generally lost when alternative error distributions are adopted, and yet considerations of realism or robustness may strongly suggest such non-normal assumptions. The multi-state model of Harrison and Stevens provides an approximate analysis based on a discrete variance mixture of normal distributions, an approach which has been extensively investigated in the engineering literature under the name of Gaussian Sum approximations; see, for example, Alspach and Sorenson (1971). Our aim in this paper is to provide an approximate, tractible, recursive updating procedure for the location parameters, which is applicable to a wide range of symmetric, unimodal error distributions and is both more realistic and more elegant than the discrete Gaussian Sum approach. In particular, for heavy-tailed distributions our procedures provide approximate Bayesian methods for time series analysis which extend considerably the work of Masreliez and Martin (1977) and have close connections with classical robustness ideas such as M-estimation and influence functions.