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).

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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

Thanks in advance.

  • $\begingroup$ 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 $k\times k$. $\endgroup$
    – plop
    Aug 4, 2020 at 10:36
  • $\begingroup$ Thank you very much, that is exactly what I was looking for. $\endgroup$
    – salto
    Aug 4, 2020 at 13:52
  • $\begingroup$ @plop, can you write that as an answer? We discourage writing answers in the comments. Thank you! $\endgroup$
    – D.W.
    Aug 4, 2020 at 17:00

1 Answer 1



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.


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