RandomVectorMetropolisHastings¶

class RandomVectorMetropolisHastings(*args)¶

Simple Metropolis-Hastings sampler defined from a random variable.

Refer to Bayesian calibration, The Metropolis-Hastings Algorithm.

Parameters:

randomVectorRandomVector: The random variable from which to update the chain
initialStatesequence of float: Initial state of the chain
marginalIndicessequence of int, optional: Indices of the components to be updated. If not specified, all components are updated. The number of updated components must be equal to the dimension of proposal.
linkFunctionFunction, optional: Link between the state of the chain and the parameters of randomVector. If not provided, then the parameters of randomVector are not updated, which means that samples from randomVector are produced independently from the state of the Markov chain.

See also

Gibbs, RandomWalkMetropolisHastings

Notes

This class creates a Markov chain from a RandomVector. It updates the parameters of the random vector based on the current state of the Markov chain, gets a realization from the random vector with the updated parameters, and then uses it to update the Markov chain.

If no likelihood is set with the setLikelihood() method, then it can be viewed as trivial Metropolis-Hastings algorithm where the proposal distribution is equal to the target distribution, so the Metropolis-Hastings ratio is always equal to 1 and the candidate is always accepted.

If a likelihood is set, then the Metropolis-Hastings ratio is the ratio of the likelihoods of the new and of the current state.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)

Build a random vector and choose the initial state of the Markov chain:

>>> randomVector = ot.RandomVector(ot.Normal())
>>> initialState = [3.0]

We can also link the parameters of the random vector to the state of the chain, Let us link the parameters of the random vector to the state of the chain. Here the parameters of the random vector are the parameters of its distribution:

>>> linkFunction = ot.SymbolicFunction(['x'], ['x', '0.1'])

The link function makes the first parameter of the normal distribution (the mean) equal to the current value of the Markov chain. Its standard deviation remains constant: $0.1$ . That way we construct a random walk with normal steps of standard deviation $0.1$ .

>>> sampler = ot.RandomVectorMetropolisHastings(randomVector, initialState, [0], linkFunction)
>>> x = sampler.getSample(10)

Methods

`computeLogLikelihood`(state)	Compute the logarithm of the likelihood w.r.t.
`computeLogPosterior`(state)	Compute the logarithm of the unnormalized posterior density.
`getAcceptanceRate`()	Get acceptance rate.
`getAntecedent`()	Accessor to the antecedent RandomVector in case of a composite RandomVector.
`getBurnIn`()	Get the length of the burn-in period.
`getClassName`()	Accessor to the object's name.
`getConditional`()	Get the conditional distribution.
`getCovariance`()	Accessor to the covariance of the RandomVector.
`getCovariates`()	Get the parameters.
`getDescription`()	Accessor to the description of the RandomVector.
`getDimension`()	Accessor to the dimension of the RandomVector.
`getDistribution`()	Accessor to the distribution of the RandomVector.
`getDomain`()	Accessor to the domain of the Event.
`getFunction`()	Accessor to the Function in case of a composite RandomVector.
`getHistory`()	Get the history storage.
`getId`()	Accessor to the object's id.
`getInitialState`()	Get the initial state.
`getLinkFunction`()	Get the model.
`getMarginal`(*args)	Get the random vector corresponding to the $i^{th}$ marginal component(s).
`getMarginalIndices`()	Get the indices of the sampled components.
`getMean`()	Accessor to the mean of the RandomVector.
`getName`()	Accessor to the object's name.
`getObservations`()	Get the observations.
`getOperator`()	Accessor to the comparaison operator of the Event.
`getParameter`()	Accessor to the parameter of the distribution.
`getParameterDescription`()	Accessor to the parameter description of the distribution.
`getProcess`()	Get the stochastic process.
`getRandomVector`()	Random vector accessor.
`getRealization`()	Compute one realization of the RandomVector.
`getSample`(size)	Compute realizations of the RandomVector.
`getShadowedId`()	Accessor to the object's shadowed id.
`getTargetDistribution`()	Get the target distribution.
`getTargetLogPDF`()	Get the target log-pdf.
`getTargetLogPDFSupport`()	Get the target log-pdf support.
`getThinning`()	Get the thinning parameter.
`getThreshold`()	Accessor to the threshold of the Event.
`getVerbose`()	Tell whether the verbose mode is activated or not.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`isComposite`()	Accessor to know if the RandomVector is a composite one.
`isEvent`()	Whether the random vector is an event.
`setBurnIn`(burnIn)	Set the length of the burn-in period.
`setDescription`(description)	Accessor to the description of the RandomVector.
`setHistory`(strategy)	Set the history storage.
`setLikelihood`(*args)	Set the likelihood.
`setName`(name)	Accessor to the object's name.
`setParameter`(parameters)	Accessor to the parameter of the distribution.
`setRandomVector`(randomVector)	Random vector accessor.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setThinning`(thinning)	Set the thinning parameter.
`setVerbose`(verbose)	Set the verbose mode.
`setVisibility`(visible)	Accessor to the object's visibility state.

__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(state)¶

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.

getAntecedent()¶

Accessor to the antecedent RandomVector in case of a composite RandomVector.

Returns:

antecedentRandomVector: Antecedent RandomVector $\vect{X}$ in case of a CompositeRandomVector such as: $\vect{Y}=f(\vect{X})$ .

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()

getCovariance()¶

Accessor to the covariance of the RandomVector.

Returns:

covarianceCovarianceMatrix: Covariance of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.5], [1.0, 1.5], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getCovariance())
[[ 1    0    ]
 [ 0    2.25 ]]

getCovariates()¶

Get the parameters.

Returns:

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

getDescription()¶

Accessor to the description of the RandomVector.

Returns:

descriptionDescription: Describes the components of the RandomVector.

getDimension()¶

Accessor to the dimension of the RandomVector.

Returns:

dimensionpositive int: Dimension of the RandomVector.

getDistribution()¶

Accessor to the distribution of the RandomVector.

Returns:

distributionDistribution: Distribution of the considered UsualRandomVector.

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.getDistribution())
Normal(mu = [0,0], sigma = [1,1], R = [[ 1 0 ]
 [ 0 1 ]])

getDomain()¶

Accessor to the domain of the Event.

Returns:

domainDomain: Describes the domain of an event.

getFunction()¶

Accessor to the Function in case of a composite RandomVector.

Returns:

functionFunction: Function used to define a CompositeRandomVector as the image through this function of the antecedent $\vect{X}$ : $\vect{Y}=f(\vect{X})$ .

getHistory()¶

Get the history storage.

Returns:

historyHistoryStrategy: Used to record the chain.

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

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()

getMarginal(*args)¶

Get the random vector corresponding to the $i^{th}$ marginal component(s).

Parameters:

iint or list of ints, $0\leq i < dim$: Indicates the component(s) concerned. $dim$ is the dimension of the RandomVector.

Returns:

vectorRandomVector: RandomVector restricted to the concerned components.

Notes

Let’s note $\vect{Y}=\Tr{(Y_1,\dots,Y_n)}$ a random vector and $I \in [1,n]$ a set of indices. If $\vect{Y}$ is a UsualRandomVector, the subvector is defined by $\tilde{\vect{Y}}=\Tr{(Y_i)}_{i \in I}$ . If $\vect{Y}$ is a CompositeRandomVector, defined by $\vect{Y}=f(\vect{X})$ with $f=(f_1,\dots,f_n)$ , $f_i$ some scalar functions, the subvector is $\tilde{\vect{Y}}=(f_i(\vect{X}))_{i \in I}$ .

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.getMarginal(1).getRealization())
[0.608202]
>>> print(randomVector.getMarginal(1).getDistribution())
Normal(mu = 0, sigma = 1)

getMarginalIndices()¶

Get the indices of the sampled components.

Returns:

marginalIndicesIndices: The marginalIndices argument provided to the constructor

getMean()¶

Accessor to the mean of the RandomVector.

Returns:

meanPoint: Mean of the considered UsualRandomVector.

Examples

>>> import openturns as ot
>>> distribution = ot.Normal([0.0, 0.5], [1.0, 1.5], ot.CorrelationMatrix(2))
>>> randomVector = ot.RandomVector(distribution)
>>> ot.RandomGenerator.SetSeed(0)
>>> print(randomVector.getMean())
[0,0.5]

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()

getOperator()¶

Accessor to the comparaison operator of the Event.

Returns:

operatorComparisonOperator: Comparaison operator used to define the RandomVector.

getParameter()¶

Accessor to the parameter of the distribution.

Returns:

parameterPoint: Parameter values.

getParameterDescription()¶

Accessor to the parameter description of the distribution.

Returns:

descriptionDescription: Parameter names.

getProcess()¶

Get the stochastic process.

Returns:

processProcess: Stochastic process used to define the RandomVector.

getRandomVector()¶

Random vector accessor.

Returns:

randomVectorRandomVector: The random variable from which to update the chain

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

getRealization

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.getSample(3))
    [ X0        X1        ]
0 : [  0.608202 -1.26617  ]
1 : [ -0.438266  1.20548  ]
2 : [ -2.18139   0.350042 ]

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

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.

getThreshold()¶

Accessor to the threshold of the Event.

Returns:

thresholdfloat: Threshold of the RandomVector.

getVerbose()¶

Tell whether the verbose mode is activated or not.

Returns:

isVerbosebool: The verbose mode is activated if it is True, desactivated otherwise.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

hasVisibleNamebool: True if the name is not empty and not the default one.

isComposite()¶

Accessor to know if the RandomVector is a composite one.

Returns:

isCompositebool: Indicates if the RandomVector is of type Composite or not.

isEvent()¶

Whether the random vector is an event.

Returns:

isEventbool: Whether it takes it values in {0, 1}.

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.

setDescription(description)¶

Accessor to the description of the RandomVector.

Parameters:

descriptionstr or sequence of str: Describes the components of the RandomVector.

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.

setParameter(parameters)¶

Accessor to the parameter of the distribution.

Parameters:

parametersequence of float: Parameter values.

setRandomVector(randomVector)¶

Random vector accessor.

Parameters:

randomVectorRandomVector: The random variable from which to update the chain

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

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.

setVerbose(verbose)¶

Set the verbose mode.

Parameters:

isVerbosebool: The verbose mode is activated if it is True, desactivated otherwise.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

Examples using the class¶

Gibbs sampling of the posterior distribution

Linear Regression with interval-censored observations

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RandomVectorMetropolisHastings¶

Examples using the class¶