Proposals

The classical Metropolis-Hastings algorithm uses a normal distribution for producing jump proposals. This can be quite inefficient for many likelihood surfaces. To improve convergence, multiple proposal classes are offered that can be used for different parameters in your model. These are:

• Normal

• Eigenvector

• BoundedEigenvector

• SSAdaptiveNormal

• AdaptiveNormal

• BoundedNormal

• SSAdaptiveBoundedNormal

• AdaptiveBoundedNormal

• Angular

• SSAdaptiveAngular

• AdaptiveAngular

• NormalDiscrete

• SSAdaptiveNormalDiscrete

• AdaptiveNormalDiscrete

• BoundedDiscrete

• SSAdaptiveBoundedDiscrete

• AdaptiveBoundedDiscrete

• NestedTransdimensional

• ATAdaptiveNormal

• ATAdaptiveBoundedNormal

• ATAdaptiveAngular

• AdaptiveEigenvector

• IsotropicSolidAngle

• AdaptiveIsotropicSolidAngle

Proposals with Adaptive in the name attempt to learn the shape of the likelihood surface during a burn-in period; see the documentation for each class for details.

By default, if no proposal class is provided for a parameter the Normal proposal is used with a standard deviation of 1. To use one of the other proposals (or to use a different standard deviation with the normal proposal), you must first initialize the proposal class giving the name(s) of the parameter(s) you wish to use it for. You then provide the list of proposals to use to the sampler’s proposals argument.

For example, say your model consists of two parameters, amp and phase, where phase is cyclic on \([0, 2\pi)\) and amp is in \([0, 1)\). You wish to use the BoundedNormal proposal for amp and the Angular proposal for phase. To do so, you first initialize the appropriate proposal for each parameter:

from epsie.proposals import (BoundedNormal, Angular)
amp_prop = BoundedNormal(['amp'], boundaries={'amp': (0, 1)})
phase_prop = Angular(['phase'])
proposals = [amp_prop, phase_prop]

You would then initialize the sampler passing the list of proposals to the sampler’s proposals keyword argument (both the MetropolisHastingsSampler and the ParallelTemperedSampler use this argument). For a detailed example, see the angular proposal tutorial.

Creating Custom Proposals

You may also define your own proposal classes. The only requirement is that the proposal class inherits from BaseProposal. This is an abstract base class that defines the standard API needed for all proposals. At bare minimum, you must provide:

• a parameters attribute: a list of the parameter names that the proposal produces jumps for.

• a _jump method: this should take a dictionary (fromx) that defines the current point in parameter space and returns another dictionary giving the new proposed point to jump to.

• a _logpdf method: this should take two dictionaries (xi and givenx) that define the proposed point and the current position, respectively, and returns the log of the pdf of jumping from the current position to the new point. This only used for non-symmetric proposals, but is good to define regardless.

• symmetric: a boolean indicating whether the proposal is symmetric or not (see symmetric() for details).

• state: a dictionary of any parameters that need to be set in order to produce a deterministic jump; see state for details.

• set_state: a method for setting the state when recovering from a checkpoint; see set_state() for details.

Note

The BaseProposal has a random_generator attribute, which is inherited from BaseRandom. This is a thin wrapper around an instance of numpy.random.Generator. You must use the random_generator attribute for producing random variates for jump proposals, or any other random variates you create in the proposal class. Use of any other psuedo-random number generator will cause the sampler to not work correctly in a parallel environment.

If you wish to create a proposal that makes use of the chain history — e.g., for creating an adaptive proposal — an update() method can be optionally implemented. This should expect to take an instance of Chain, which contains the history of the samples (up to the last time a clear() was called), as well as several other properties, such as the acceptance ratio. See the Chain API for more details.