PolyaGamma(b::Real, c::Real) <: ContinuousUnivariateDistribution

Arguments

• b::Real
• c::Real exponential tilting

Create a PolyaGamma sampler with parameters b and c. Note that sampling will differ if b is a Int or a Real.

NegativeMultinomial(x₀::Real, p::AbstractVector)

Negative Multinomial distribution defined as

\[ p(\boldsymbol{x}|x_0, \boldsymbol{p}) = \Gamma\left(\sum_{i=0}^M x_i \right)\frac{p_0^{x_0}}{\Gamma(x_0)}\prod_{i=1}^M \frac{p_i^{x_i}}{x_i!}\]

where $p_0= 1-\sum_{i=1}^M p_i$.

For a detailed understanding of this distribution, see "Negative multinomial distribution" - Sibuya et al. - 1964

PolyaGammaPoisson(y::Real, c::Real, λ::Real)

A bivariate distribution, used as hierachical prior as:

\[ p(\omega, n) = \operatorname{PG}(\omega|y + n, c)\operatorname{Po}(n|\lambda).\]

Random samples as well as statistics from the distribution will returned as NamedTuple : (;ω, n).

This structured distributions is needed for example for the PoissonLikelihood.

PolyaGammaNegativeMultinomial(y::BitVector, c::AbstractVector{<:Real}, p::AbstractVector{<:Real})

A multivariate distribution, used as hierachical prior as:

\[ p(\boldsymbol{\omega}, \boldsymbol{n}) = \operatorname{NM}(\boldsymbol{n}|1, \boldsymbol{p})\prod_{i=1}^K\operatorname{PG}(\omega|y_i + n_i, c).\]

Random samples as well as statistics from the distribution will returned as a NamedTuple : (;ω, n).

This structured distributions is needed for the CategoricalLikelihood with a LogisticSoftMaxLink.

LaplaceLikelihood(β::Real)

Likelihood with a Laplace distribution:

\[ p(y|f,\beta) = \frac{1}{2\beta}\exp\left(-\frac{|y-f|}{\beta}\right)\]

Arguments

• β::Real, scale parameter

StudentTLikelihood(ν::Real, σ::Real)

Likelihood with a Student-T likelihood:

\[ p(y|f,\sigma, \nu) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\sqrt{\pi\nu}\sigma}\left(1 + \frac{1}{\nu}\left(\frac{x-\nu}{\sigma}\right)^2\right)^{-\frac{\nu+1}{2}}.\]

Arguments

• ν::Real, number of degrees of freedom, should be positive and larger than 0.5 to be able to compute moments
• σ::Real, scaling of the inputs.

AbstractNTDist is an abstract type for a wrapper type around measure(s) and distributions(s). The main idea is that instead of rand, mean and other statistical tools wrapped objects return NamedTuples or TupleVector when having a collection of them.

The following API has to be implemented: Given π::AbstractNTDist and Π::AbstractVector{<:AbstractNTDist}

Necessary

• ntrand(rng, π) -> NamedTuple
• ntmean(π) -> NamedTuple
• MeasureBase.logdensity_def(π, x::NamedTuple) -> Real
• Distributions.kldivergence(π₀, π₁) -> Real

Optional

• tvrand(rng, Π) -> TupleVector
• tvmean(Π) -> TupleVector

NTDist(d) -> NTDist{typeof(d),:ω}
NTDist{S}(d) -> NTDist{typeof(d),S}

Wrapper around a single distribution to be compatible with the ntrand, ntmean interface. One can pass the wanted symbol via S while the default will be :ω.

ntrand([rng::AbstractRNG,] d) -> NamedTuple

Return a sample as a NamedTuple.

ntmean(d::Distribution) -> NamedTuple

Return the mean as a NamedTuple.

tvrand([rng::AbstractRNG,] d::For) -> TupleVector
tvrand([rng::AbstractRNG,] d::AbstractVector{<:AbstractNTDist}) -> TupleVector

Return a collection of samples as a TupleVector

tvmean(d::AbstractVector{<:AbstractNTDist}) -> TupleVector
tvmean(d::For)

Return a collection of mean as a TupleVector.