RandomWalkMetropolisHastings¶

class RandomWalkMetropolisHastings(*args)¶

Random Walk Metropolis-Hastings method.

Refer to Bayesian calibration, The Metropolis-Hastings Algorithm.

Available constructor:

RandomWalkMetropolisHastings(targetDistribution, initialState, proposal, marginalIndices)

RandomWalkMetropolisHastings(targetLogPDF, support, initialState, proposal, marginalIndices)

Parameters

targetDistributionDistribution: Target distribution sampled
targetLogPDFFunction: Target log-density up to an additive constant
supportDomain: Support of the target when defined with targetLogPDF
initialStatesequence of float: Initial state of the chain
proposalDistribution: Distribution of the steps of the random walk
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.

See also

Gibbs, RandomVectorMetropolisHastings

Notes

The random walk Metropolis-Hastings algorithm is a Markov Chain Monte-Carlo algorithm. It draws candidates for the next state of the chain as follows: denoting the current state by $\vect{\theta}^k$ , the candidate $\vect{c}^k$ for $\vect{\theta}^{k+1}$ can be expressed as $\vect{c}^k = \vect{\theta}^k +\vect{\delta}^k$ where the distribution of $\vect{\delta}^k$ is the provided proposal distribution.

Examples

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

Sample from a target distribution defined through its log-PDF (defined up to some additive constant) and its support:

>>> log_density = ot.SymbolicFunction('x', 'log(2 + sin(x)^2) - (2 + cos(3*x)^3 + sin(2*x)^3) * x')
>>> support = ot.Interval([0.0], [2.0 * m.pi])
>>> proposal = ot.Normal(0.0, 2.0)
>>> initialState = [3.0]
>>> sampler = ot.RandomWalkMetropolisHastings(log_density, support, initialState, proposal)
>>> sampler.setBurnIn(20)
>>> sampler.setThinning(2)
>>> x = sampler.getSample(10)

The target distribution can also be a Distribution. Since all Distribution objects have a getSample() method, this is only useful in Bayesian settings where we also define a likelihood. The likelihood function penalizes the initially provided target distribution (now viewed as the prior) in order to get the posterior distribution. We sample from the posterior.

>>> prior = ot.ComposedDistribution([ot.Uniform(-100.0, 100.0)] * 2)
>>> proposal = ot.Normal([0.0] * 2, [0.5, 0.05], ot.IdentityMatrix(2))
>>> initialState = [0.0] * 2
>>> sampler = ot.RandomWalkMetropolisHastings(prior, initialState, proposal)
>>> conditional = ot.Bernoulli()
>>> data = ot.Sample([[53, 1], [57, 1], [58, 1], [63, 1], [66, 0], [67, 0],
... [67, 0], [67, 0], [68, 0], [69, 0], [70, 0], [70, 0], [70, 1], [70, 1],
... [72, 0], [73, 0], [75, 0], [75, 1], [76, 0], [76, 0], [78, 0], [79, 0], [81, 0]])
>>> observations = data[:, 1]
>>> covariates = data[:, 0]
>>> fun = ot.SymbolicFunction(['alpha', 'beta', 'x'], ['exp(alpha + beta * x) / (1 + exp(alpha + beta * x))'])
>>> linkFunction = ot.ParametricFunction(fun, [2], [0.0])
>>> sampler.setLikelihood(conditional, observations, linkFunction, covariates)
>>> alpha_beta = 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.
`getAdaptationExpansionFactor`()	Get the expansion factor.
`getAdaptationFactor`()	Get the adaptation factor.
`getAdaptationPeriod`()	Get the calibration step.
`getAdaptationRange`()	Get the range.
`getAdaptationShrinkFactor`()	Get the shrink factor.
`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.
`getProposal`()	Get the proposal distribution.
`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.
`setAdaptationExpansionFactor`(expansionFactor)	Set the expansion factor.
`setAdaptationPeriod`(period)	Set the calibration step.
`setAdaptationRange`(range)	Set the range.
`setAdaptationShrinkFactor`(shrinkFactor)	Set the shrink factor.
`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.
`setProposal`(proposal)	Set the proposal distribution.
`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.

getAdaptationExpansionFactor()¶

Get the expansion factor.

Returns

expansionFactorfloat: Expansion factor $e$ for the adaptation

getAdaptationFactor()¶

Get the adaptation factor.

Returns

factorfloat: Current adaptation factor, for inspection.

getAdaptationPeriod()¶

Get the calibration step.

Returns

periodpositive int: Number of samples before the adaptation occurs

getAdaptationRange()¶

Get the range.

Returns

rangeInterval of dimension 1: Range $[m,M]$ in the description of the method computeUpdateFactor().

getAdaptationShrinkFactor()¶

Get the shrink factor.

Returns

shrinkFactorfloat: Shrink factor $s$ for the adaptation

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.

getProposal()¶

Get the proposal distribution.

Returns

proposalDistribution: The distribution from which the transition kernels of the random walk Metropolis-Hastings algorithm is defined

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}.

setAdaptationExpansionFactor(expansionFactor)¶

Set the expansion factor.

Parameters

expansionFactorfloat, $e > 1$: Expansion factor $e$ for the adaptation

setAdaptationPeriod(period)¶

Set the calibration step.

Parameters

periodpositive int: Number of samples before the adaptation occurs

setAdaptationRange(range)¶

Set the range.

Parameters

rangeInterval of dimension 1: Range $[m,M]$ for the adaptation

setAdaptationShrinkFactor(shrinkFactor)¶

Set the shrink factor.

Parameters

shrinkFactorfloat, $0 < s < 1$: Shrink factor $s$ for the adaptation

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.

setProposal(proposal)¶

Set the proposal distribution.

Parameters

proposalDistribution: The distribution from which the transition kernels of the random walk Metropolis-Hastings algorithm is defined

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.

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