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I would like to construct a ProbabilityDistribution with a variable number of dimensions. I tried to use $Assumptions = x ∈ Vectors[d, Reals] so that the variable x would be an unspecified vector with $d$ dimensions. However, I found that I could not use {x[[1]], -∞, ∞} to define the ranges of each element of the vector.

I'm seeking to define a distribution similar to the multivariate T distribution, but want flexibility to modify the definition. So I like to use ProbabilityDistribution for the definition, rather than using a built-in distribution.

My question is what's the best way to define a vector variable, which can be used in ProbabilityDistribution as a variable for the range of each dimensions defined?

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    $\begingroup$ Something like: n = 3; ProbabilityDistribution[pdf, Sequence @@ Table[{x[i], -\[Infinity], \[Infinity]}, {i, n}]] ? $\endgroup$
    – JimB
    Mar 8, 2019 at 17:28
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    $\begingroup$ But I think that if you don't want to specify the dimension explicitly, then you'll probably need to do it with Product, Sum, etc. commands rather than ProbabilityDistribution. But I could certainly be wrong about that. Others can chime in. If leaving the dimensions unspecified is necessary, then maybe explaining why that's necessary would help someone make a constructive suggestion. $\endgroup$
    – JimB
    Mar 8, 2019 at 18:55

2 Answers 2

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Vector variables are not supported in the Probability & Statistics framework as of now. So the dimensionality needs to be specified.

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Thank you JimB, your suggestion worked well. Use of the Sequence function which strips a layer of lists is part of what I was having difficulty with. Here is a simple example of a multivariate normal with independent dimensions.

n = 3; 
IndMultiNormal = ProbabilityDistribution[
    Exp[-Table[x[i], {i, n}].Table[x[i], {i, n}]], 
    Sequence @@ Table[{x[i], -\[Infinity], \[Infinity]}, {i, n}]
]
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