Support parametrization of Beta (Mean and variance Beta(μ, σ²))

[DominiqueMakowski]

Folllow up to this thread: https://discourse.julialang.org/t/reparametrized-beta-in-turing-how-to-set-a-prior-that-depends-on-another-parameter/116753/1

Mean-variance Beta parametrization is very common in regression modelling. Althoug the definition is pretty trivial, it would be convenient to natively support it, for instance by adding a method through keywords arguments:

Beta(; μ=0.5, σ²=0.1)

Or simply through a "new" distribution, for instance: MeanVarBeta()

Here's the definition:

function MeanVarBeta(μ, σ²)
    if σ² <= 0 || σ² >= μ * (1 - μ)
        error("Variance σ² must be in the interval (0, μ*(1-μ)=$(μ*(1-μ))).")
    end

    ν = μ * (1 - μ) / σ² - 1
    α = μ * ν
    β = (1 - μ) * ν

    return Beta(α, β)
end

[DominiqueMakowski]

Actually, A parametrization based on precision phi is much more trivial and allows for parameter independence:

BetaMuPhi(μ, ϕ) = Beta(μ * ϕ, (1 - μ) * ϕ)

The former is probably pointless.