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How can I derive the posterior distribution of a likelihood function $N({\bf{t}}|\phi(x)^{T}{\bf{w}}, \beta^{-1})$ and a prior $N({\bf{w}}|{\bf{m_{0}}}, \beta^{-1}{\bf{S_{0}}})Gam(\beta, a_{0},b_{0})$?

I'd like to use something similar to the general relations for conditional distributions in the normal case:

$$p(x|y) \propto p(y|x)p(x) = N(y|Ax+b, L^{-1})N(x|\mu, \Lambda^{-1})$$

which gives the result of:

$$p(x|y) \propto N(x|\Sigma\{A^{T}L(y-b)+\Lambda \mu\}, \Sigma)$$

However, this doesn't work with a normal-gamma distribution as a prior, and I'm not sure if there is such equivalent relation.

Any thoughts about how to approach this?

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I assume you are trying to conduct Bayesian linear regression using a conjugate prior. The calculations are explained in the wikipedia entry Bayesian linear regression.

The only difference is that you are working with the precision $\beta$ instead of the variance $\sigma^2$, which are equivalent up to a change of variable.

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  • $\begingroup$ Thanks. There is another difference in the use of the inverse Gamma distribution instead of the Gamma distribution, although I don't think it will create issue. Let me go through that proof to make sure. By the way, is there a set of general formulas for conditional distributions in the way I described above? $\endgroup$ – Robert Smith Apr 3 '13 at 20:53
  • $\begingroup$ Actually the model you are describing is exactly the same as the one described in the wikipedia entry since $\sigma^2 = \beta^{-1}$ (recall that if $X$ is gamma distributed, then $1/X$ is inverse-gamma distributed). You are then using the prior $p({\bf w},\beta)=p({\bf w}\vert\beta)p(\beta)$, which is a hierarchical prior. The posterior is $p({\bf w},\beta \vert {\bf t}) = N(...)p({\bf w},\beta)$. Then, yes, it is analogous to the normal-gamma case. $\endgroup$ – Sastre Apr 3 '13 at 20:56

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