Mv Student T scale matrix

Questions v5
1

Hi
Im looking for some advice please.

I need to define a bivariate Student T with the correlation between the 2 variables constrained to be <0. So IIUC I cant use the LKJ prior (not enough control).

But Im not clear how to use the scale parameter in pm.MvStudentT() - is this defined using the correlation matrix, or covariance matrix, or a scaled version of one of these, e.g. corr * (ν / ν - 2), where ν is the degrees of freedom?

Thanks

2

The scale is related to the covariance matrix. But you also have the option of parameterizing with the analogous to the MVN precision (tau) or cholesky factorization.

A MVT is MVN scaled by a common gamma based on the degrees of freedom, which is why the parametrizations are not exactly the covariance/precision/cholesky.

3

So would the following be correct, or is there a better way?
Would it be better to use MvN & Gamma?
(ndims = 2)

p0 = np.zeros(ndim)
one = pt.as_tensor_variable(1.0, dtype='float64')
with pm.Model() as mdl:

    ρ = pm.HalfNormal('corr', sigma=0.25)
    corrm = pt.stack([one, -ρ, -ρ, one]).reshape((ndim, ndim))

    sigs = pm.Gamma('sigs', mu=1.0, sigma=0.025, dims=('ndims'))  
    varm = pt.diag(sigs)
    cov = varm.dot(corrm).dot(varm)
    
    nu = pm.Exponential('nu0', lam=0.5) 
    scale = pm.Deterministic('scale', cov * (nu / (nu - 2)))

    T = pm.MvStudentT('T', mu=p0, scale=scale, nu=nu, dims=('trials', 'ndims'))

Thanks!