Priors over starting points

On top of decorating each observation with appropriate information, for stochastic processes we need to provide additional information that describes the starting position of the process (as, for instance, no observation may be made at the initial time). This is done through prior distributions.

All priors over starting points inherit from

ObservationSchemes.StartingPtPrior — Type

StartingPtPrior{T}

Types inheriting from the abstract type StartingPtPrior indicate the prior that is put on the starting point of the observed path of some stochastic process. T denotes the DataType of the starting point.

They all must implement the following methods

Base.rand — Method

Base.rand(G::StartingPtPrior, [z, ρ=0.0])

Sample a new starting point according to its prior distribution. An implementation with arguments z, ρ implements a preconditioned Crank-Nicolson scheme with memory parameter ρ and a current non-centered variable z. z is also referred to as the driving noise.

ObservationSchemes.start_pt — Method

start_pt(z, G::StartingPtPrior, P)

Compute a new starting point from the white noise for a given posterior distribution obtained from combining prior G and the likelihood encoded by the object P.

ObservationSchemes.start_pt — Method

start_pt(z, G::StartingPtPrior)

Compute a new starting point from the white noise for a given prior distribution G

Distributions.logpdf — Method

logpdf(G::StartingPtPrior, y)

log-probability density function evaluated at y of a prior distribution G

and should also implement

ObservationSchemes.inv_start_pt — Method

inv_start_pt(y, G::StartingPtPrior, P)

Compute the driving noise that is needed to obtain starting point y under prior G and the likelihood in P

for MCMC setting. In this package we provide implementations for the following types of starting points

Known, fixed starting points
Gaussian priors over starting points

Known starting point

This is the simplest setting in which the starting point is assumed to be known.

ObservationSchemes.KnownStartingPt — Type

struct KnownStartingPt{T} <: StartingPtPrior{T}
    y::T
end

Indicates that the starting point is known and stores its value in y

KnownStartingPt(y::T) where T

Base constructor.

It can be defined with

x0 = [1.0, 2.0]
x0_prior = KnownStartingPt(x0)

a call to rand or start_pt will simply return the fixed starting point and logpdf(x0_prior, y) evaluates to 0 so long as x0 == y.

Gaussian priors

A Gaussian prior over the starting point.

ObservationSchemes.GsnStartingPt — Type

struct GsnStartingPt{T,S,TM} <: StartingPtPrior{T} where {S}
    μ::T
    Σ::S
    Λ::S
    μ₀::T
    Σ₀::UniformScaling
end

Indicates that the starting point is equipped with a Gaussian prior with mean μ and covariance matrix Σ (and pre-computed precision Λ:=Σ⁻¹). Sampling is always done via non-centred parametrization, by sampling white noise z according to Gaussian with zero mean and identity covariance: μ₀ and Σ₀, and then transforming z to a variable with mean and covariance μ and Σ.

GsnStartingPt(μ::T, Σ::S)

Base constructor. It initialises the mean μ and covariance Σ parameters and Λ is set according to Λ:=Σ⁻¹.

Can be defined with

μ, Σ = [1.0, 2.0], [1.0 0.0; 0.0 1.0]
x0_prior = GsnStartingPt(μ, Σ)

to set the mean and covariance to μ and Σ respectively. The underlying idea behind Gaussian starting point priors is that of non-centred parametrisation, so that a possibility of local updates is granted. More precisely any sampling is done with z∼N(0,Id) variables, which are then transformed to N(μ,Σ) via linear transformations. In particular, sampling with rand can be done with local perturbations via Crank-Nicolson scheme.

Base.rand — Function

Base.rand(G::StartingPtPrior, [z, ρ=0.0])

Sample a new starting point according to its prior distribution. An implementation with arguments z, ρ implements a preconditioned Crank-Nicolson scheme with memory parameter ρ and a current non-centered variable z. z is also referred to as the driving noise.

rand([rng::Random.AbstractRNG], G::GsnStartingPt, z, ρ=0.0)

Sample new white noise using Crank-Nicolson scheme with memory parameter ρ and a previous value of the white noise stored inside object G

rand([rng::Random.AbstractRNG], G::GsnStartingPt)

Sample new starting point according to its prior distribution.

rand([rng::Random.AbstractRNG], G::KnownStartingPt, args...)

Starting point is known. Nothing can be sampled. Returning known starting point.

Base.rand([rng::Random.AbstractRNG], o::LinearGsnObs, X)

Sample an observation according to

\[V ∼ L X + ξ,\quad \mbox{where } ξ∼N(μ,Σ),\]

with $L$, $μ$ and $Σ$ defined in o.

inv_start_pt returns the non-centrally parametrised noise z that produces a given starting point x0:

ObservationSchemes.inv_start_pt — Function

inv_start_pt(y, G::StartingPtPrior, P)

Compute the driving noise that is needed to obtain starting point y under prior G and the likelihood in P

inv_start_pt(y, G::GsnStartingPt, P)

Compute the driving noise that is needed to obtain starting point y under prior G and the likelihood in P

inv_start_pt(y, G::KnownStartingPt, P)

Starting point known, no need for dealing with white noise, use convention of returning y

and start_pt is the reverse operation

ObservationSchemes.start_pt — Function

start_pt(z, G::StartingPtPrior, P)

Compute a new starting point from the white noise for a given posterior distribution obtained from combining prior G and the likelihood encoded by the object P.

start_pt(z, G::StartingPtPrior)

Compute a new starting point from the white noise for a given prior distribution G

start_pt(z, G::GsnStartingPt, P)

Compute a new starting point from the white noise for a given posterior distribution obtained from combining prior G and the likelihood encoded by the object P.

start_pt(z, G::GsnStartingPt)

Compute a new starting point from the white noise for a given prior distribution G

start_pt(z, G::KnownStartingPt, P)

Return starting point

start_pt(G::KnownStartingPt, P)

Return starting point

Tip

To see how to define your own priors over starting points see a How-to-guide on Defining custom priors over starting points