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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

They all must implement the following methods

Base.rand(G::ObservationSchemes.StartingPtPrior, [z, ρ=0.0])
ObservationSchemes.start_pt(z, G::ObservationSchemes.StartingPtPrior, P)
ObservationSchemes.start_pt(z, G::ObservationSchemes.StartingPtPrior)
ObservationSchemes.logpdf(G::ObservationSchemes.StartingPtPrior, y)

and should also implement

ObservationSchemes.inv_start_pt(y, G::ObservationSchemes.StartingPtPrior, 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

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

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.

ObservationSchemes.rand

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

ObservationSchemes.inv_start_pt

and start_pt is the reverse operation

ObservationSchemes.start_pt

!!! tip To see how to define your own priors over starting points see a How-to-guide on [Defining custom priors over starting points](@ref how_to_custom_prior)