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)
>>> 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.JointDistribution([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

`asComposedEvent`()	If the random vector can be viewed as the composition of several `ThresholdEvent` objects, this method builds and returns the composition.
`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 adaptation expansion factor.
`getAdaptationFactor`()	Get the adaptation factor.
`getAdaptationPeriod`()	Get the adaptation step.
`getAdaptationRange`()	Get the expected range for the acceptance rate.
`getAdaptationShrinkFactor`()	Get the adaptation 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.
`getFrozenRealization`(fixedPoint)	Compute realizations of the RandomVector.
`getFrozenSample`(fixedSample)	Compute realizations of the RandomVector.
`getFunction`()	Accessor to the Function in case of a composite RandomVector.
`getHistory`()	Get the history storage.
`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.
`getTargetDistribution`()	Get the target distribution.
`getTargetLogPDF`()	Get the target log-pdf.
`getTargetLogPDFSupport`()	Get the target log-pdf support.
`getThreshold`()	Accessor to the threshold of the Event.
`hasName`()	Test if the object is named.
`isComposite`()	Accessor to know if the RandomVector is a composite one.
`isEvent`()	Whether the random vector is an event.
`setAdaptationExpansionFactor`(expansionFactor)	Set the adaptation expansion factor.
`setAdaptationPeriod`(period)	Set the adaptation period.
`setAdaptationRange`(range)	Set the expected range for the acceptance rate.
`setAdaptationShrinkFactor`(shrinkFactor)	Set the adaptation 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.

__init__(*args)¶

asComposedEvent()¶

If the random vector can be viewed as the composition of several ThresholdEvent objects, this method builds and returns the composition. Otherwise throws.

Returns:

composedRandomVector: Composed event.

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 adaptation expansion factor.

Returns:

expansionFactorfloat: Expansion factor $e$ of the adaptation factor.

getAdaptationFactor()¶

Get the adaptation factor.

Returns:

factorfloat: Current adaptation factor, for inspection.

getAdaptationPeriod()¶

Get the adaptation step.

Returns:

periodpositive int: During burn-in, adaptation is performed once every period iterations.

getAdaptationRange()¶

Get the expected range for the acceptance rate. During burn-in, at the end of every adaptation period, if the acceptance rate does not belong to this range, the adaptation factor is multiplied either by the expansion factor (if the acceptance rate is larger than the upperBound) or by the shrink factor (if the acceptance rate is smaller than the lowerBound).

Returns:

rangeInterval of dimension 1: Range [lowerBound, upperBound] of the expected acceptance rate.

getAdaptationShrinkFactor()¶

Get the adaptation shrink factor.

Returns:

shrinkFactorfloat: Shrink factor $s$ of the adaptation factor.

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: this is the number of iterations after which adaptation stops.

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.

getFrozenRealization(fixedPoint)¶

Compute realizations of the RandomVector.

In the case of a CompositeRandomVector or an event of some kind, this method returns the value taken by the random vector if the root cause takes the value given as argument.

Parameters:

fixedPointPoint: Point chosen as the root cause of the random vector.

Returns:

realizationPoint: The realization corresponding to the chosen root cause.

See also

openturns.RandomVector.getRealization
openturns.RandomVector.getFrozenSample

Examples

>>> import openturns as ot
>>> distribution = ot.Normal()
>>> randomVector = ot.RandomVector(distribution)
>>> f = ot.SymbolicFunction('x', 'x')
>>> compositeRandomVector = ot.CompositeRandomVector(f, randomVector)
>>> event = ot.ThresholdEvent(compositeRandomVector, ot.Less(), 0.0)
>>> print(event.getFrozenRealization([0.2]))
[0]
>>> print(event.getFrozenRealization([-0.1]))
[1]

getFrozenSample(fixedSample)¶

Compute realizations of the RandomVector.

In the case of a CompositeRandomVector or an event of some kind, this method returns the different values taken by the random vector when the root cause takes the values given as argument.

Parameters:

fixedSampleSample: Sample of root causes of the random vector.

Returns:

sampleSample: Sample of the realizations corresponding to the chosen root causes.

See also

openturns.RandomVector.getSample
openturns.RandomVector.getFrozenRealization

Examples

>>> import openturns as ot
>>> distribution = ot.Normal()
>>> randomVector = ot.RandomVector(distribution)
>>> f = ot.SymbolicFunction('x', 'x')
>>> compositeRandomVector = ot.CompositeRandomVector(f, randomVector)
>>> event = ot.ThresholdEvent(compositeRandomVector, ot.Less(), 0.0)
>>> print(event.getFrozenSample([[0.2], [-0.1]]))
    [ y0 ]
0 : [ 0  ]
1 : [ 1  ]

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.

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:

realizationPoint: 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

openturns.RandomVector.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]

getSample(size)¶

Compute realizations of the RandomVector.

Parameters:

nint, $n \geq 0$: Number of realizations needed.

Returns:

realizationsSample: n sequences of values randomly determined from the RandomVector definition. In the case of an event: n realizations of the event (considered as a Bernoulli variable) which are boolean values (1 for the realization of the event and 0 else).

See also

openturns.RandomVector.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 ]

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

getThreshold()¶

Accessor to the threshold of the Event.

Returns:

thresholdfloat: Threshold of the RandomVector.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

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 adaptation expansion factor.

Parameters:

expansionFactorfloat, $e > 1$: Expansion factor $e$ of the adaptation factor.

setAdaptationPeriod(period)¶

Set the adaptation period.

Parameters:

periodpositive int: During burn-in, adaptation is performed once every period iterations.

setAdaptationRange(range)¶

Set the expected range for the acceptance rate. During burn-in, at the end of every adaptation period, if the acceptance rate does not belong to this range, the adaptation factor is multiplied either by the expansion factor (if the acceptance rate is larger than the upperBound) or by the shrink factor (if the acceptance rate is smaller than the lowerBound).

Parameters:

rangeInterval of dimension 1: Range [lowerBound, upperBound] of the expected acceptance rate.

setAdaptationShrinkFactor(shrinkFactor)¶

Set the adaptation shrink factor.

Parameters:

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

setBurnIn(burnIn)¶

Set the length of the burn-in period.

Parameters:

burninint: Length of the burn-in period: this is the number of iterations after which adaptation stops.

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:

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