I am trapped by such a problem. Assume the state variable $\theta$ is (prior) normally distributed $N(\eta, \sigma^{2}_{0})$. Now we have two independent signals about $\theta$. Signal 1 is $s_{1}=\theta+\epsilon_{1}$, in which $\epsilon_{1}$ is normally distributed $N(0,\sigma^{2}_{1})$ and $\epsilon_{1}$ is independent of $\theta$. Signal 2 is $s_{2}=\theta+\epsilon_{2}$, in which $\epsilon_{2}$ is normally distributed $N(0,\sigma^{2}_{2})$ and $\epsilon_{2}$ is independent of $\theta$ and $\epsilon_{1}$. Now
- please calculate the posterior distribution of $\theta$ after observing these two independent signals, i.e.$f(\theta|s_{1},s_{2})$.
- please calculate the mean and variance of $s_{2}$ given $s_{1}$ and prior distribution of $\theta$, i.e.$g(s_{2}|\theta, s_{1})$.
I have been calculating this for a week and could not find a solution to it. Guys, please help me to find a way to calculate them. Thanks very much ~
You have a prior distribution $$\pi_0(\theta) = \phi\left(\dfrac{\theta - \eta}{\sigma_0}\right)$$ and two likelihoods proportional to $\phi\left(\dfrac{s_1 - \theta}{\sigma_1}\right)$ and $\phi\left(\dfrac{s_2 - \theta}{\sigma_2}\right)$ so you want $$\pi(\theta|s_1,s_2)=\dfrac{\phi\left(\dfrac{\theta - \eta}{\sigma_0}\right)\phi\left(\dfrac{s_1 - \theta}{\sigma_1}\right)\phi\left(\dfrac{s_2 - \theta}{\sigma_2}\right)}{\displaystyle \int_\theta\phi\left(\dfrac{\theta - \eta}{\sigma_0}\right)\phi\left(\dfrac{s_1 - \theta}{\sigma_1}\right)\phi\left(\dfrac{s_2 - \theta}{\sigma_2}\right)d\theta}.$$ I will leave you to do the calculation.
For part 2 you can do something similar to find the posterior distribution of $\theta$ given $s_1$ and so a consequential distribution for $s_2$. Remember that if $X$ and $Y$ are independent then $E[X+Y]=E[X]+E[Y]$ and $Var(X+Y)=Var(X)+Var(Y)$.