Show that the posterior probability of $H_0$ equals $\Phi\left(\sqrt{n}\frac{\theta_o -\bar{X}_n}{\sigma}\right)$

Exercise: Assume a Bayesian setup in which $X_1,\ldots,X_n\mid\Theta\stackrel{iid}{\sim}N(\theta,\sigma^2)$ and $\Theta$ gets assigned a prior distribution. Take a flat prior on $\Theta: f_\Theta(\theta)\propto 1$. We consider testing $H_0:\theta\leq \theta_0$ versus $H_1:\theta > \theta_0$. Show that the posterior probability of $H_0$ equals $$\Phi\bigg(\sqrt{n}\dfrac{\theta_o -\bar{X}_n}{\sigma}\bigg)$$

My approach: I need to show that $\mathbb{P}(\theta\leq \theta_0\mid X) = \Phi\bigg(\sqrt{n}\dfrac{\theta_o -\bar{X}_n}{\sigma}\bigg)$, and I know that $f_{\Theta\mid X}(\theta\mid x)\propto f_\Theta(\theta)\,f_{X\mid\Theta}(x\mid\theta) \propto\prod\limits_{i = 1}^n\exp\bigg(\dfrac{(x_i - \theta)^2}{\sigma^2}\bigg) = \exp\bigg(-\dfrac{\sum_{i = 1}^nx_i^2}{n\sigma^2} + \dfrac{2\theta\frac{1}{n}\sum_{i = 1}^n x_i}{2\sigma^2} - \dfrac{\theta^2}{\sigma^2}\bigg)$. If I'm not mistaken, I need to find $a$ and $b$ in $\Theta|X\sim N(a,b^2)$ so that I can normalise $\Theta$. Unfortunately I can't find a way to write $\exp\bigg(-\dfrac{\sum_{i = 1}^nx_i^2}{n\sigma^2} + \dfrac{2\theta\frac{1}{n}\sum_{i = 1}^n x_i}{2\sigma^2} - \dfrac{\theta^2}{\sigma^2}\bigg)$ as $\exp\bigg(-\dfrac{(\theta - a)^2}{b^2}\bigg)$.

Question: How do I solve this exercise/what am I missing?

The posterior distribution of $\theta$ (with $\sigma$ fixed) is proportional to the likelihood, which is given by $$\mathcal L(\theta \mid \boldsymbol x) \propto \exp \left( - \frac{1}{2\sigma^2} \sum_{i=1}^n (x_i - \theta)^2 \right).$$ We may ignore the improper prior $\pi(\theta) = 1$. We can ignore all other factors that do not depend on $\theta$. Now if we let $\bar x = n^{-1} \sum_{i=1}^n x_i$ be the sample mean, we have \begin{align*} \sum_{i=1}^n (x_i - \theta)^2 &= \sum_{i=1}^n (x_i - \bar x + \bar x - \theta)^2 \\ &= \sum_{i=1}^n (x_i - \bar x)^2 + 2(x_i - \bar x)(\bar x - \theta) + (\bar x - \theta)^2 \\ &= n(\bar x - \theta)^2 + 2(\bar x - \theta) \sum_{i=1}^n (x_i - \bar x) + \sum_{i=1}^n (x_i - \bar x)^2 \\ &= n(\bar x - \theta)^2 + 0 + \sum_{i=1}^n (x_i - \bar x)^2, \end{align*} where the second term is zero because the sample total equals $n$ times the sample mean. Hence $$f(\theta \mid \boldsymbol x) \propto \exp\left( - \frac{n(\bar x - \theta)^2}{2\sigma^2}\right)\exp\left(-\frac{1}{2\sigma^2} \sum_{i=1}^n (x_i - \bar x)^2\right),$$ and as this second exponential factor is independent of $\theta$, it too is just a constant of proportionality with respect to the posterior distribution of $\theta$. We conclude that the posterior density of $\theta$ is normal, with mean $\bar x$ and variance $\sigma^2/n$; thus the posterior probability of $H_0$ is $$\Pr[H_0 \mid \boldsymbol x] = \Pr[\theta \le \theta_0 \mid \boldsymbol x] = \Pr\left[\frac{\theta - \bar x}{\sigma/\sqrt{n}} \le \frac{\theta_0 - \bar x}{\sigma/\sqrt{n}} \mid \boldsymbol x \right] = \Pr\left[Z \le \frac{\theta_0 - \bar x}{\sigma/\sqrt{n}}\right] = \Phi\left(\sqrt{n}\frac{\theta_0 - \bar x}{\sigma}\right).$$
• Thanks for your reply! There is just one thing that I don't understand! Why is the mean of the posterior density of $\theta$ equal to $\bar{x}$? We have $f(\theta\mid x)\propto \exp\bigg(-\dfrac{n(\bar{x} - \theta)^2}{2\sigma^2}\bigg)$. If $\bar{x}$ was the mean of the posterior density wouldn't we have $f(\theta\mid x)\propto \exp\bigg(-\dfrac{n(\theta - \bar{x})^2}{2\sigma^2}\bigg)$? Could you just explain this final step? Thanks! – titusAdam Jan 24 '18 at 10:29
• $(a-b)^2 = (b-a)^2$. – heropup Jan 24 '18 at 14:22