MetropolisHastings¶

class MetropolisHastings(*args)¶

Base class for Metropolis-Hastings algorithms.

Refer to Bayesian calibration, The Metropolis-Hastings Algorithm.

See also

RandomWalkMetropolisHastings

Notes

Metropolis-Hastings algorithms are Markov Chain Monte-Carlo algorithms: they can sample from a target distribution defined as having a PDF proportional to some known function.

If setLikelihood() is used, then a likelihood factor is applied to the PDF of the target distribution: in Bayesian terms, the initially targeted distribution is the prior, and multiplying its PDF by the likelihood yields the PDF of the posterior up to a multiplicative constant.

Methods

`computeLogLikelihood`(state)	Compute the logarithm of the likelihood w.r.t.
`computeLogPosterior`(currentState)	Compute the logarithm of the unnormalized posterior density.
`getAcceptanceRate`()	Get acceptance rate.
`getBurnIn`()	Get the length of the burn-in period.
`getClassName`()	Accessor to the object's name.
`getConditional`()	Get the conditional distribution.
`getCovariates`()	Get the parameters.
`getDimension`()	Accessor to the dimension of the RandomVector.
`getHistory`()	Get the history storage.
`getId`()	Accessor to the object's id.
`getImplementation`()	Accessor to the underlying implementation.
`getInitialState`()	Get the initial state.
`getLinkFunction`()	Get the model.
`getName`()	Accessor to the object's name.
`getObservations`()	Get the observations.
`getRealization`()	Compute one realization of the RandomVector.
`getTargetDistribution`()	Get the target distribution.
`getTargetLogPDF`()	Get the target log-pdf.
`getTargetLogPDFSupport`()	Get the target log-pdf support.
`getThinning`()	Get the thinning parameter.
`setBurnIn`(burnIn)	Set the length of the burn-in period.
`setHistory`(strategy)	Set the history storage.
`setLikelihood`(*args)	Set the likelihood.
`setName`(name)	Accessor to the object's name.
`setThinning`(thinning)	Set the thinning parameter.

getMarginalIndices

__init__(*args)¶

computeLogLikelihood(state)¶

Compute the logarithm of the likelihood w.r.t. observations.

Parameters:

currentStatesequence of float: Current state.

Returns:

logLikelihoodfloat: Logarithm of the likelihood w.r.t. observations $(\vect{y}^1, \dots, \vect{y}^n)$ .

computeLogPosterior(currentState)¶

Compute the logarithm of the unnormalized posterior density.

Parameters:

currentStatesequence of float: Current state.

Returns:

logPosteriorfloat: Target log-PDF plus log-likelihood if the log-likelihood is defined

getAcceptanceRate()¶

Get acceptance rate.

Returns:

acceptanceRatefloat: Global acceptance rates over all the MCMC iterations performed.

getBurnIn()¶

Get the length of the burn-in period.

Returns:

burninint: Length of the burn-in period, that is the number of first iterates of the MCMC chain which will be thrown away when generating the sample.

getClassName()¶

Accessor to the object’s name.

Returns:

class_namestr: The object class name (object.__class__.__name__).

getConditional()¶

Get the conditional distribution.

Returns:

conditionalDistribution: The conditional argument provided to setLikelihood()

getCovariates()¶

Get the parameters.

Returns:

parametersPoint: Fixed parameters of the model $g$ required to define the likelihood.

getDimension()¶

Accessor to the dimension of the RandomVector.

Returns:

dimensionpositive int: Dimension of the RandomVector.

getHistory()¶

Get the history storage.

Returns:

historyHistoryStrategy: Used to record the chain.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getImplementation()¶

Accessor to the underlying implementation.

Returns:

implImplementation: A copy of the underlying implementation object.

getInitialState()¶

Get the initial state.

Returns:

initialStatesequence of float: Initial state of the chain

getLinkFunction()¶

Get the model.

Returns:

linkFunctionFunction: The linkFunction argument provided to setLikelihood()

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getObservations()¶

Get the observations.

Returns:

observationsSample: The observations argument provided to setLikelihood()

getRealization()¶

Compute one realization of the RandomVector.

Returns:

aRealizationPoint: Sequence of values randomly determined from the RandomVector definition. In the case of an event: one realization of the event (considered as a Bernoulli variable) which is a boolean value (1 for the realization of the event and 0 else).

See also

getSample

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.0], [1.0, 1.0], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getRealization())
[0.608202,-1.26617]
>>> print(randomVector.getRealization())
[-0.438266,1.20548]

getTargetDistribution()¶

Get the target distribution.

Returns:

targetDistributionDistribution: The targetDistribution argument provided to the constructor

getTargetLogPDF()¶

Get the target log-pdf.

Returns:

targetLogPDFFunction: The targetLogPDF argument provided to the constructor

getTargetLogPDFSupport()¶

Get the target log-pdf support.

Returns:

supportInterval: The support argument provided to the constructor

getThinning()¶

Get the thinning parameter.

Returns:

thinningint: Thinning parameter: storing only every $k^{th}$ point after the burn-in period.

Notes

When generating a sample of size $q$ , the number of MCMC iterations performed is $l+1+(q-1)k$ where $l$ is the burn-in period length and $k$ the thinning parameter.

setBurnIn(burnIn)¶

Set the length of the burn-in period.

Parameters:

burninint: Length of the burn-in period, that is the number of first iterates of the MCMC chain which will be thrown away when generating the sample.

setHistory(strategy)¶

Set the history storage.

Parameters:

historyHistoryStrategy: Used to record the chain.

setLikelihood(*args)¶

Set the likelihood.

Parameters:

conditionalDistribution: Required distribution to define the likelihood of the underlying Bayesian statistical model.
observations2-d sequence of float: Observations $\vect y^i$ required to define the likelihood.
linkFunctionFunction, optional: Function $g$ that maps the chain into the conditional distribution parameters. If provided, its input dimension must match the chain dimension and its output dimension must match the conditional distribution parameter dimension. Else it is set to the identity.
covariates2-d sequence of float, optional: Parameters $\vect c^i$ of the linkFunction for each observation $\vect y^i$ . If provided, their dimension must match the parameter dimension of linkFunction.

Notes

Once this method is called, the class no longer samples from the distribution targetDistribution or from the distribution defined by targetLogPDF and support, but considers that distribution as being the prior. Let $\pi(\vect \theta)$ be the PDF of the prior at the point $\vect \theta$ . The class now samples from the posterior, whose PDF is proportional to $L(\vect\theta) \, \pi(\vect\theta)$ , the likelihood $L(\vect \theta)$ being defined from the arguments of this method.

The optional parameters linkFunction and covariates allow several options to define the likelihood $L(\vect \theta)$ . Letting $f$ be the PDF of the distribution conditional:

Without linkFunction and covariates the likelihood term reads:

$L(\vect \theta) = \prod_{i=1}^n f(\vect{y}^i|\vect{\theta})$
If only the linkFunction is provided:

$L(\vect \theta) = \prod_{i=1}^n f(\vect{y}^i|g(\vect{\theta}))$
If both the linkFunction and covariates are provided:

$L(\vect \theta) = \prod_{i=1}^n f(\vect{y}^i|g_{\vect c^i}(\vect{\theta}))$

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setThinning(thinning)¶

Set the thinning parameter.

Parameters:

thinningint, $k \geq 0$: Thinning parameter: storing only every $k^{th}$ point after the burn-in period.

Notes

When generating a sample of size $q$ , the number of MCMC iterations performed is $l+1+(q-1)k$ where $l$ is the burn-in period length and $k$ the thinning parameter.

Examples using the class¶

Linear Regression with interval-censored observations

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MetropolisHastings¶

Examples using the class¶