VLBILikelihoods

abstract type AbstractVLBIDistributions <: Distributions.Distribution{Distributions.Multivariate, Distributions.Continuous}

Abstract type that detail that the distribution is the likelihood for some VLBI dataproduct. One key difference between an AbstractVLBIDistribution and a regular Distributions.jl one is that by default we expect on the fields of the struct defining the distribution to be lognorm which gives the log-normalization constant for the distribution.

This prevents the model from constantly having to compute the log-normalization constant everytime the density is evaluated, which is often the most expensive part of the computation.

To implement a AbstractVLBIDistributions a user then just needs to define

• unormed_logdensity(d::AbstractVLBIDistribution, x)

which takes in the new distribution type d and the point you wish to evaluate the density at. Internally VLBILikelihood adds in the normalization constant.

If the user wishes to change the default lognorm behavior they can also overload the lognorm function to opt-out of storing the normalization constant in the struct. For example

struct MyNormal <: AbstractVLBIDistributions end

unnormed_logdensity(d::MyNormal, x) = -x^2/2 lognorm(d::MyNormal) = -log(2π)/2

AmplitudeLikelihood(μ, Σ::Union{AbstractVector, AbstractMatrix})

Forms the likelihood for amplitudes from the mean vector μ and the covariance matrix Σ. If Σ is vector or a diagonal matrix then we assume that the argument is the diagonal covariance matrix. If Σ is a full matrix then we assume that a dense covariance was passed

ClosurePhaseLikelihood(μ, Σ)

Construct the Gaussian approximate closure phase likelihood distribution. The distribution used here is not the exact distribution for closure phases since this becomes quite complicated (especially for correlated covariances). For our approximation we take the Gaussian around the complex exponential of the phase, that is the unormalized likelihood is given by

\[ \log\mathcal{L} = -\frac{1}{2}(e^{i\theta} - e^{i\mu})^T\Sigma^{-1}(e^{i\theta} - e^{i\mu})\]

If Σ is diagonal then this reduces to a bunch of independent Von Mises distributions, and if Σ is dense, this becomes a complicated multi-variate extension of the Von Mises distribution^[1].

Parameters

• μ: The mean closure phase, which is usually computed from some VLBI model
• Σ: The measurement covariance matrix, which is usually computed directly from the data. Note that Σ can either be a matrix or a vector. If Σ is a vector then we interpret Σ to represent the diagonal of the covariance matrix.

ComplexVisLikelihood(μ::AbstractVector{<:StaticMatrix{2,2}}, Σ::AbstractVector{<:StaticMatrix{2,2, <:Real}})

Creates the coherency matrix likelihood distribution which is a Gaussian over the space of 2×2 complex visibilities.

Paramters

• μ: The mean coherency matrix stored as a vector of 2x2 static matrices, which is usually computed from some VLBI model
• Σ: The measurement covariance matrix, which is usually computed directly from the data. Note that Σ must vector of 2x2 real static matrices.

ComplexVisLikelihood(μ::AbstractVector{<:Complex}, Σ::AbstractVector{<:Real})

Creates the complex visibility likelihood distribution which is a diagonal complex Gaussian with strictly real covariance matrix.

Paramters

• μ: The mean complex visibility, which is usually computed from some VLBI model
• Σ: The measurement covariance matrix, which is usually computed directly from the data. Note that Σ must be a real element vector and is interpreted at the diagonal of the covariance matrix.

RiceAmplitudeLikelihood(μ, Σ::Union{AbstractVector, Diagonal})

Forms the likelihood for amplitudes from the mean vector μ and the diagonal covariance matrix Σ. Σ can either be a Diagonal matrix or a vector whose entries are the variance for each data point.

lognorm(d::AbstractVLBIDistributions)

Compute the log-normalizatoin constant of the distribution d.