Predictive Distribution with Normal Prior

Question

Given $\Theta = \theta$, let $X_1, X_2, \dots, X_n, X_{n+1} \sim \mathcal{N}(\theta, \sigma^2)$ be independent.

$\Theta \sim \mathcal{N}(\theta_0, \tau^2)$.

What is the easiest way to find the distribution of $X_{n+1}\mid X_1, \dots, X_n$?

I am looking only for hints. How I would start this is $$f_{X_{n+1} \mid X_1, \dots, X_n}(t \mid x_1, \dots, x_n) = \dfrac{f_{X_1, \dots, X_{n+1}}(x_1, \dots, x_{n}, t)}{f_{X_1, \dots, X_{n}}(x_1, \dots, x_n)}$$ and conditioned on $\Theta$, we know that the $X$s are independent, so $$f_{X_1, \dots, X_{n+1}}(x_1, \dots, x_{n}, t) = \int_{-\infty}^{\infty}f_{X_1, \dots, X_{n+1}\mid \Theta}(x_1, \dots, x_n, t\mid \theta)\pi(\theta)\text{ d}\theta$$ and we can write $$f_{X_1, \dots, X_{n+1}\mid \Theta}(x_1, \dots, x_n, t\mid \theta) = f_{X_1 \mid \Theta}(x_1\mid\theta)\cdots f_{X_n\mid \Theta}(x_n\mid \theta)f_{X_{n+1}\mid \Theta}(t\mid \theta)$$ by independence, and similarly for the denominator, but this looks disgusting.

Answer 1

Hints:

1. To get the joint distribution of $X_1, X_2,\ldots, X_{n+1}$, compute the moment generating function. This will get you the mean vector and covariance matrix. Additional hint, if you need it:

Condition on $\Theta$.

2. To get the conditional distribution of $X_{n+1}$ given $X_1,\ldots,X_n$, Google "multivariate normal conditional distribution". (The general formula is tricky to prove, so maybe you can cite the formula without proof.)

If you need to write out the conditional density of $X_{n+1}$ given the rest, it might help to use

the Sherman-Morrison formula.