# Conditional distributions derivation

I am trying to solve the following problem:

Given $$n$$ independent observations $$Y_i$$ from a Normal$$(\theta, \tau^{-1})$$ distribution with unknown mean $$\theta$$ and unknown precision $$\tau$$, i.e

$$Y_i \approx Normal(\theta, \tau^{-1}) \ , \ i = {1, ..., n}$$

Assume for $$\theta$$ and $$\tau$$ the following non-informative priors:

$$\theta \approx Normal(0,10^6)$$ $$\tau \approx Gamma(0.001,0.001)$$

Given $$\textbf{y}$$ is the observations $$(y_1, ...,y_n)$$. Derive two conditional distributions $$p(\theta|\tau, y)$$ and $$p(\tau|\theta, y)$$. One should be written as a normal distribution and the other as a Gamma distribution.

I have started by calculating $$p(\theta|\tau, y)$$ as:

$$p(\theta|\tau, y) = p(\theta,\tau) \prod^{n}_{i=1} p(y_{i}|\theta, \tau) = \tau^{-1} \overbrace{\tau^{n/2}\exp[\frac{-\tau}{2} \sum^{n}_{i=1}(y_i - \theta)^2]}^{L(\theta,\tau;\textbf{y})} \\ = \tau^{\frac{n}{2} - 1} \exp [-\frac{\tau}{2} \sum^{n}_{i=1} (y_i - \bar{y} + \bar{y} -\theta)^2] \\ = \tau^{\frac{n}{2} - 1} \exp [-\frac{\tau(n-1)}{2} s^2 -\frac{\tau n}{2}(\bar{y} - \theta)^2 ]$$

In the last step $$s = \frac{1}{n-1} \sum^{n}_{i=1} (y_{i} -\bar{y})^2$$

I am new to Bayesian, not sure whether this is correct/best way to approach this question. Also I am unsure how could I find the other conditional distribution as Gamma Distribution.

My attempt at the derivation for the Gamma distribution is:

$$p(\tau|\theta,y) = p(\theta,\tau) p(y_{i}|\tau, \theta) \\ = \tau^{-1} \frac{y^{\theta}}{\Gamma(\theta)} \tau^{\theta -1} \exp^{-y \tau}$$

Thanks.

• @Xian, Sorry I am not very experience and may be confused, for the conditional posterior on $\theta$, I am not sure if it is correct but I assumed the joint prior distribution to be $\tau ^{-1}$ given as $p(\theta|\tau) = p(\theta|\tau)p(\tau)$. Could you please clarify of how to find this joint prior ? Mar 30, 2020 at 9:46
• @Xian for the conditional posterior om $\tau$, so the likelihood becomes $p(y_i|\tau, \theta) = Exp[\frac{\theta}{2} \sum^{n}_{i=1} (y_{i} - \tau)^2]$ ? I am a bit confused given we are asked to calculate one as a Normal Distribution and the other as a Gamma Distribution hence my choice for the likelihood here. I am very confused at this moment any clarifications are welcomed. Mar 30, 2020 at 9:49
• @Xi'an OK. I still don't understand what is wrong with my joint prior as you have mentioned above and by writing the likelihood exactly the same as for $\theta$ I am not answering the question that requires one to be given as normal and the other as a gamma distribution. Thanks for your pointers. Mar 30, 2020 at 11:36

For $$\theta$$, \begin{align*} p(\theta|\tau,\textbf{y}) &\propto \overbrace{\tau^{\frac{n}{2} - 1} \exp [-\frac{\tau(n-1)}{2} s^2}^\text{does not depend on \theta} -\frac{\tau n}{2}(\bar{y} - \theta)^2 ]\\ &\propto \exp[-\frac{\tau n}{2}(\bar{y} - \theta)^2 ] \end{align*} and spot the Normal density in $$\theta$$
For $$\tau$$, \begin{align*} p(\tau|\theta,\textbf{y}) &\propto p(\theta,\tau|\textbf{y})\\ &\propto \tau^{-1} \underbrace{\tau^{n/2}\exp[\frac{-\tau}{2} \sum^{n}_{i=1}(y_i - \theta)^2]}_{L(\theta,\tau;\textbf{y})} \end{align*} and spot the Gamma density in $$\tau$$.
• So my joint prior $p(\theta|\tau)$ and the powers in $\tau$ are correct? I am sorry but I am still confused given the requirement that one of the distros must be a Gamma distribution, they both look like normal distributions and effectively the same to me :/ Sorry for the confusion. Mar 30, 2020 at 13:36