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Suppose I've observed $x$ from a Student-t distribution with unknown $\mu$, and I'd now like to infer $\mu$. Since the t-distribution isn't exponential family, there's no conjugate prior available, and so I'll assume $\mu$ is Normally distributed.

$$\mu \sim N(\mu_0,\sigma_0^2)$$ $$x \sim t_\nu(\mu, \sigma^2)$$ (student-t parametrization as used by Kevin Murphy https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf)

I'm aware that there's probably no closed-form expression for the posterior itself, but is it possible to analytically solve for the posterior mean, $$\hat \mu = \mathbb{E}[\mu\mid x]$$

Thanks!

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  • $\begingroup$ The t-distribution can be written as a scale mixture of Gaussians, where the mixing distribution of the unknown scale is an inverse gamma distribution. This fact can be exploited to write an algorithm for sampling from the posterior distribution of your model and the posterior mean can be computed by Monte Carlo with high accuracy. $\endgroup$
    – R Hahn
    Commented Aug 21, 2019 at 1:17
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    $\begingroup$ Thanks for the answer. Unfortunately monte carlo simulation isn't sufficient for my purposes. I need an analytical solution, but I'm open-minded about what form the prior should take. I know there's no conjugate prior available, but I presume there must at least be some prior for which the posterior mean is analytically soluble? $\endgroup$
    – Luke
    Commented Aug 21, 2019 at 1:34
  • $\begingroup$ Is $\nu$ known, or to be estimated based on the data? $\endgroup$
    – Michael Hardy
    Commented Aug 21, 2019 at 20:10
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    $\begingroup$ $\nu$, $\mu_0$ and $\sigma_0$ are all known $\endgroup$
    – Luke
    Commented Aug 22, 2019 at 20:54

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Section 2.1 of this paper gives expressions for the posterior mean of location parameters. This may be helpful in your context.

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    $\begingroup$ Ooh, this is certainly relevant, I'll have a go at the integrals and see if this solves things for me. Thanks! $\endgroup$
    – Luke
    Commented Aug 22, 2019 at 20:57

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