All,
I see from The Matrix Cookbook Section 8.1.8 that there is a nice formula for the product of two multivariate Gaussian PDFs as a scalar times a Gaussian PDF. That's crisp. My issue is massaging an equation into the required form. It's basically part of a weird filtering problem I've encountered. I am going to end up integrating over all $x_k$ in the following equation $$ L = \int f(x_k)\cdot \mathcal{N}_{x_k}(\mu,\Sigma_1)\cdot \mathcal{N}_{x_{k+1}}(\mathbf{A}x_k + \mathbf{b},\Sigma_2) \,\text{d}x_k \tag{1}$$ and I am hoping (praying) that $L$ will also be proportional to a Gaussian PDF.
I know that the PDF of $x_{k+1}$ is just $\mathcal{N}(\mathbf{A}\mu + \mathbf{b}, \Sigma_2 + \mathbf{A}^T\Sigma_1\mathbf{A})$, which is what I'd get if I did not have $f(x_k)$ (I think). But with $f(x_k)$, I am not sure.
In short: Can anyone help derive the scaling, mean, and covariance of the Gaussian PDF represented by (1)? I tried for a few hours but got nowhere. Thanks!
Okay, so if I've understood correctly, you have $x_k \sim N(\mu, \Sigma_1)$, $x_{k+1} \mid x_k \sim N(Ax_k + b, \Sigma_2)$, and you want to know the distribution of $x_{k} \mid x_{k+1}$, so you can compute expressions like $\mathbb E[f(x_k) \mid x_{k+1}]$.
In general, this is just a conditional normal distribution. We have
$$\begin{pmatrix}x_k \\ x_{k+1}\end{pmatrix} \sim N\left(\begin{pmatrix}\mu_1\\\mu_2\end{pmatrix}, \begin{pmatrix}\Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22}\end{pmatrix}\right),$$ where $\mu_1 = \mu$, $\mu_2 = A\mu + b$, $\Sigma_{11}=\Sigma_1$, $\Sigma_{12} = \Sigma_{21}^T = \Sigma_1A^T$, and $\Sigma_{22} = \Sigma_2 + A\Sigma_1A^T.$ The conditional distribution is then $$x_k \mid x_{k+1} \sim N(\bar\mu, \bar\Sigma),$$ where $\bar\mu = \mu_1 +\Sigma_{12}\Sigma_{22}^{-1}(x_{k+1}-\mu_2)$, and $\bar\Sigma =\Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$.
You can then compute $$L = \mathbb E[f(x_k)\mid x_{k+1}]\ p(x_{k+1}),$$ where $p$ is the density of $x_{k+1}$, by integrating over $x_{k} \mid x_{k+1}$. Whether or not $L$ is proportional to a normal density will depend on the function $f$; it is not true in general.