Define a multi-variate Gaussian as $f(\pmb y|\pmb x)=\mathcal N(\pmb x,.5)$, where $\pmb x$ is a sample from a pre-defined multi-variate Gaussian function $g(\pmb x)=\mathcal N(\pmb \mu, \pmb \Sigma)$. How should I compute the log probability of $\pmb y\sim\mathcal f(\pmb y|\pmb x)$?
I can think of the following answers but I'm not so sure if I take them right
- For a given $\pmb x$, it's $\log f(\pmb y|\pmb x)$
- Otherwise, it's $\int \log \mathcal f(\pmb y|\pmb x)+\log g(\pmb x)d\pmb x$
I’m answering the question as clarified in the comments.
$\mathbf y$ is the sum of the two multivariate Gaussian variables with distributions $\mathcal N(\mathbf\mu,\mathbf\Sigma)$, and $\mathcal N(0,0.5)$ (where I presume that by $0.5$ you mean $0.5\cdot\mathbf 1$). The sum of the two multivariate Gaussian variables is again a multivariate Gaussian variable, with mean the sum of the means and variance the sum of the variances (see e.g. this question). So $\mathbf y\sim\mathcal N(\mathbf \mu,\mathbf\Sigma+0.5\cdot\mathbf1)$.