Add rules and tests for MvNormalWeightedMeanPrecision node

[albertpod]

This PR encompasses the following contributions for the MvNormalWeightedMeanPrecision node:

1. Fix for average energy.
2. Addition of rules for the out interface (particularly useful for priors).
3. Inclusion of missing tests.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (f4835c2) 62.10% compared to head (b3444a2) 62.12%.
Report is 4 commits behind head on main.

Additional details and impacted files

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+ Coverage   62.10%   62.12%   +0.01%     
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  Files         180      181       +1     
  Lines        5927     5930       +3     
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+ Hits         3681     3684       +3     
  Misses       2246     2246              

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[bartvanerp]

Good catch! Would it make sense to add the method mean(chollogdet, ...) to BayesBase.jl? Then we can use the symmetry information to speed up this computation.

[bartvanerp]

I have some concerns as to whether this rule is actually correct (likely my bad). Just did the derivations again and seem to end up with
ndims(q_out) * log2π - mean(chollogdet, q_Λ) + tr(mean(q_Λ) * (v_out + v_mean + m_mean*m_mean'+ m_out * m_out')) - 2*m_out*m_mean ) / 2
I might have a mistake somewhere, but we better double check.

[albertpod]

I've changed it to a different form.

[bvdmitri]

Something is completely wrong with this PR, I disagree with the change as for now. I also think that the tests might be incorrect. Lets discuss. But tldr, we cannot simply call average energy of the mean-precision parametrization, because semantics of the inputs is different.

[albertpod]

Thanks, @bvdmitri and @bartvanerp, for spotting the mistakes. It's a little more difficult than I thought.

Anyway, just to re-iterate, I want to use this node as a prior mostly.