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This post pertains to Bayesian pdf manipulation.

  • Firstly, assuming a prior probability specified as Gamma distribution such that $\alpha = \mu_{0}^{2}/\sigma_{0}^{2}$ and $\beta = \mu_{0}/\sigma_{0}^{2}$; is it correct to assume, substituting into the Gamma distribution, that the prior has the following form:

$$\pi(\mu) \propto \mu^{(\mu_{0}^{2}/\sigma_{0}^{2})-1}\text{exp}\{-(\mu_0/\sigma_{0}^{2})\mu\}1_{\{\mu>0\}}$$

  • Secondly, as an aside, assuming the prior adhered not to a Gamma distribution, but, in fact, to a Beta distribution, would it be correct to assume the following:

Given that the pdf for a Beta distribution has the form:

$$\frac{\mu^{\alpha-1}(1-\mu)^{\beta-1}}{B(\alpha, \beta)}$$

The prior distribution would have the following form:

$$\pi(\mu) \propto \mu^{\alpha-1}(1-\mu)^{\beta-1}$$

And, given that the likelihood remains normally distributed, the posterior would have the form:

$$\pi(\mu|\textbf{x}) \propto \mu^{\alpha-1}(1-\mu)^{\beta-1}\text{exp}\{-\frac{1}{2\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\mu)^{2}\}$$

With $\frac{\alpha}{\alpha+\beta}=\mu_{0}$ and $\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)} \space{ } = \sigma_{0}^{2}$

Owing to:

$E[\mu] = \frac{\alpha}{\alpha+\beta}$, $Var[\mu] = \frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}$

  • Finally, is it correct to suggest that $\pi(\mu|\textbf{x})$ in the above example (Beta/Normal posterior) is a member of the exponential family given that it can be rewritten in the general form:

$$h(\mu)^{n}\text{exp}\{\sum t(x_{i})\psi(\mu)\}$$

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  • $\begingroup$ A first question about the statistical model: you have a random sample from $\mathrm{N}(\mu,\sigma^2)$. Is $\sigma^2$ known? $\endgroup$ – Zen Oct 11 '12 at 20:14
  • $\begingroup$ Apologies, I should have been more clear: yes, for the likelihood, $\text{N}(\mu, \sigma^{2})$, $\sigma^{2}$ is known. Thanks for pointing that out. $\endgroup$ – user9171 Oct 11 '12 at 20:24
  • $\begingroup$ You first consider a (gamma) prior for $\mu$ that has support $[0,\infty)$. After that, you use a (beta) prior for $\mu$ that has support $[0,1]$. Can you tell us why it is sensible in your problem to consider those supports, given that in your statistical your model $\mu\in(-\infty,\infty)$? $\endgroup$ – Zen Oct 11 '12 at 20:36
  • $\begingroup$ Partly following on from Zen's statement, as a technical note you should have an indicator function in both your beta prior and posterior $1[\mu \in (0,1)]$. $\endgroup$ – Sam Livingstone Oct 11 '12 at 20:52
  • $\begingroup$ @Zen, to provide some background: I'm just beginning Bayesian analysis and I'm attempting to grasp the mechanics of the prior/likelihood manipulations. The above problem is entirely hypothetical, and, as such, there is no "sense" to the above supports. The prior/likelihood was selected purely on the basis that I hadn't yet encountered this combination; however, in hindsight, perhaps I should have chosen a more appropriate combination. $\endgroup$ – user9171 Oct 11 '12 at 20:58

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