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