# What does this expression mean?

## Normal distribution with condition

I am reading a research paper and found the following expression (Eq.28 in the paper below).

It means a Gaussian distribution, but the mean component seems conditional probability-like expression $$\it{\bf{s}}_t | \it{\bf{m}}_{b, t, m}^{(j)}$$. I have never seen this expression before and cannot find any info about it.

The variables $$\it{\bf{s}}_t$$ and $$\it{\bf{m}}_{b, t, m}^{(j)}$$ are both vectors and $$\bf{\Sigma}_{b}$$ is a covariance matrix.

Does anybody have an idea of what this expression means?

## Original paper where the expression is.

The original paper can be found here: https://eprints.soton.ac.uk/437941/1/08340823.pdf

• The $m$ is the vector of means and $\Sigma$ the covariance of the multivariate normal distribution. The $s$ is in the place of the variable of the density function. So, $\mathcal{N}(x,m,\Sigma)=(2\pi)^{-k/2}|\Sigma|^{-1/2}\exp(-2^{-1}(x-m)^T\Sigma^{-1}(x-m))$, where $k$ is the dimension of the vectors $x,m$ and $\Sigma$ is $k\times k$.
– plop
Aug 4, 2020 at 10:36
• Thank you very much, that is exactly what I was looking for. Aug 4, 2020 at 13:52
• @plop, can you write that as an answer? We discourage writing answers in the comments. Thank you!
– D.W.
Aug 4, 2020 at 17:00

$$\mathcal{N}(x|m,\Sigma)=(2\pi)^{-k/2}|\Sigma|^{-1/2}\exp\left(-2^{-1}(x-m)^T\Sigma^{-1}(x-m)\right)$$
where $$x,m$$ are vectors of dimension $$k$$, $$\Sigma$$ is a $$k\times k$$ matrix, $$|\Sigma|$$ is its determinant. The vector $$m$$ represents the mean of the multivariate normal, and $$\Sigma$$ the covariance matrix, which must be positive definite.