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I'm working with a Python implementation of the Multivariate T distribution, and I've noticed when I evaluate the PDF at certain points, the likelihood returned is > 1. This is causing issues in other parts of my project.

The implementation looks right to me, I've read through it multiple times and compared to the PDF listed on the wiki page.

I'm wondering if the multivariate t distribution can have density > 1 at any point? There are so many free parameters (dimension of the data, degrees of freedom, covariance matrix, etc) that its hard to get an intuitive idea if it is possible. I know the single variate t distribution does not ever have density > 1, so I'm assuming the same holds here. Looking for some confirmation here though..

Thanks in advanced!

Addison
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    Even the multivariate T in one dimension can have arbitrarily high densities. Indeed, any scale family of distributions will exhibit arbitrarily large densities: just make the scale small enough. The densities are inversely proportional to the scale. – whuber Jul 06 '22 at 17:09
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    More specifically for th t-distribution: The density can get arbitrarily large for $x=\mu$ (and consequently also in more or less large neighbourhoods) if eigenvalues of $\Sigma$ are arbitrarily small. In the one-dimensional case, $\Sigma$ is just a single number, and you can easily convince yourself of my claim letting $\Sigma\searrow 0$. – Christian Hennig Jul 06 '22 at 17:10
  • Thanks for the response @whuber, intuitively I can convince myself of this based on the PDF function. However when I plug in to Geogebra, the behaviour is strange.. as I decrease the degrees of freedom parameter, density goes to 0. I narrowed down the issue and it seems that gamma((d+1)/2)/gamma(d/2) does not limit to 1 as d approaches 0 from the positive side. Do you think this is just a numerical quirk of their implementation, or is this something special about the gamma function? Thanks so much for the help! – Addison Jul 06 '22 at 18:53
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    That limit is $0.$ As the df approaches zero, the Student t distribution flattens down into nonexistence: the limiting density at all values of $x$ is zero. – whuber Jul 06 '22 at 19:31
  • I see, but didn't you originally say that as the scale gets small enough (i.e. as the df gets small enough) the density gets arbitrarily large? – Addison Jul 06 '22 at 19:45
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    That's correct. But the df parameter is not a scale parameter! $\Sigma$ is the scale parameter. See https://stats.stackexchange.com/a/49794/919 for an explanation of scale parameters. – whuber Jul 06 '22 at 20:04

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