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I'm working with a hierarchical regression model of the following form similar to that presented in Peter D. Hoff's book, A First Course in Bayesian Statistical Methods:

$\boldsymbol{Y}_j \sim \text{multivariate normal}(\boldsymbol{X}_j\boldsymbol{\beta}_j,\sigma^2\boldsymbol{I}) \\ \boldsymbol{\beta}_1,...\boldsymbol{\beta}_m \sim \text{multivariate normal}(\boldsymbol{\theta},\boldsymbol{\Sigma}) \\ \boldsymbol{\theta} \sim \text{multivariate normal}(\boldsymbol{\mu}_0, \boldsymbol{\Omega}_0) \\ \boldsymbol{\Sigma} \sim \text{inverse-Wishart}(\eta_0,\boldsymbol{\text{S}}_0^{-1}) \\ \sigma^2 \sim \text{inverse-gamma}(\nu_0/2,\nu_0\sigma_0^2/2)$

After 100000 repetitions, when I look at the $\beta$ coefficients, the $\theta$ coefficients and $\sigma^2$, they appear to have converged. However, when I see the $\Sigma$ matrix, it appears that it has not converged. My question is, can I really say that the other components of the model have truly converged, especially since each conditional distribution depends on the other parameters? Should I trust the results until all of the parameters have converged?

Thanks!

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  • $\begingroup$ I am guessing you are using a Gibbs sampler with more than two blocks. In which case, it is important to remember that each marginal process, say for $\beta_1$ is not a Markov chain itself. So, theoretically, since the Markov chain is a chain in the vector of all parameters, convergence can only be discussed for the chain as a whole. $\endgroup$ – Greenparker Mar 7 '16 at 12:25

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