Re: How to derive Covariance of Dirichlet distirubtion from its variance

On Tue, Dec 19, 2023 at 1:18 PM marc nicole <[email protected]> wrote:

> How you manage to write  cov_dirichlet ?
> Based on which definition of the dirichlet function ?
>

A straightforward implementation of the math of the formula that I
referenced, but following the conventions of `scipy.stats.dirichlet`.


> (np.diag and np.multiply.outer got me confused) could you explain more?
>

Given a vector `x`, `np.diag(x)` will make a diagonal matrix with that
vector on the diagonal and 0s elsewhere.

`np.multiply.outer(alpha, alpha)` computes the outer product of the vector
`alpha`; that is, the matrix `P` such that `P[i, j] = alpha[i] * alpha[j]`.

The covariance formula has a form where all the entries can be formed by an
outer product of the `alpha` vector, except that the diagonal has an extra
term added onto it.


> also the function is supposed to take two random variables Xi and Xj to
> measure their covariance not just alpha (what is alpha anyways?)
>

That isn't what you asked for originally. If you think you need those
because of the formula I referenced, I think you are misreading it. You do
not need an explicit `X` anywhere in the code to compute the covariance of
a Dirichlet distribution specified by the `alpha` vector. The `cov(X_i,
X_j)` on the left hand side is just traditional math notation to specify
the `[i, j]` element of the covariance matrix.

Also could alpha be multi dimensional ?
>

By which you mean that you would like to write a version that computes the
covariance matrices for a number of different `alpha` vectors in batch? One
could write a version, but I won't.

-- 
Robert Kern

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