Answers are grounded in this paper and cite their sources.
James H. Albert, Siddhartha Chib, Bayes Inference via Gibbs Sampling of Autoregressive Time Series Subject to Markov Mean and Variance Shifts, Journal of Business & Economic Statistics, Vol. 11, No. 1 (Jan., 1993), pp. 1-15
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Albert et al. (1993) studied this question.
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