Forward rule for Bernoulli node under beta message should be removed

[ThijsvdLaar]

The rule towards out for the Bernoulli node under a Beta message,

@rule Bernoulli(:out, Marginalisation) (m_p::Beta,) = begin
    @logscale 0
    return Bernoulli(mean(m_p))
end

should be removed.

This forward message does not explicitly mention that the update is derived under a "tentative decision" approximation, where the integration over the parameter is replaced by the mean of the parameter.

More specifically, given a model

$$ p(\theta) = \mathcal{B}\!eta(\theta|a,b) $$ $$ p(y|\theta) = \mathcal{B}\!er(y|\theta) $$

the computation of the predictive distribution,

$$ p(y) = \int p(y|\theta)p(\theta) \mathrm{d}\!\theta $$

does not give the result of the specified rule, unless the beta disttribution is approximated by a pointmass on its mean.

[bvdmitri]

I think your explanation makes sense. @bartvanerp you remember why do we need this rule (ref) ? Was it for the mixture/log-scale addons experiments? If removed perhaps it breaks this example (just guessing)?

[bartvanerp]

Oei oei oei, git blame me. I took the rule from here: https://biaslab.github.io/pdf/sips2022/scalefactor.pdf. I double checked where this one came from and came across it in our holy document. @ThijsvdLaar I think you should discuss this with @ThijsvdLaar :-)

[ThijsvdLaar]

@bartvanerp Thanks for quoting the scripture, it seems the derivation is correct and I should not be opening issues after 5pm.