MCMC¶

class MCMC(*args)¶

Monte-Carlo Markov Chain.

Available constructor:

MCMC(prior, conditional, observations, initialState)

MCMC(prior, conditional, model, parameters, observations, initialState)

Parameters

priorDistribution: Prior distribution of the parameters of the underlying Bayesian statistical model.
conditionalDistribution: Required distribution to define the likelihood of the underlying Bayesian statistical model.
modelFunction: Function required to define the likelihood.
observations2-d sequence of float: Observations required to define the likelihood.
initialStatesequence of float: Initial state of the Monte-Carlo Markov chain on which the Sampler is based.
parameters2-d sequence of float: Parameters of the model to be fixed.

Notes

MCMC provides a implementation of the concept of sampler, using a Monte-Carlo Markov Chain (MCMC) algorithm starting from initialState. More precisely, let $t(.)$ be the PDF of its target distribution and $d_{\theta}$ its dimension, $\pi(.)$ be the PDF of the prior distribution, $f(.|\vect{w})$ be the PDF of the conditional distribution when its parameters are set to $\vect{w}$ , $d_w$ be the number of scalar parameters of conditional distribution (which corresponds to the dimension of the above $\vect{w}$ ), $g(.)$ be the function corresponding to model and $(\vect{y}^1, \dots, \vect{y}^n)$ be the sample observations (of size $n$ ):

In the first usage, it creates a sampler based on a MCMC algorithm whose target distribution is defined by:

$t(\vect{\theta}) \quad \propto \quad \underbrace{~\pi(\vect{\theta})~}_{\mbox{prior}} \quad \underbrace{~\prod_{i=1}^n f(\vect{y}^i|\vect{\theta})~}_{\mbox{likelihood}}$

In the second usage, it creates a sampler based on a MCMC algorithm whose target distribution is defined by:

$t(\vect{\theta}) \quad \propto \quad \underbrace{~\pi(\vect{\theta})~}_{\mbox{prior}} \quad \underbrace{~\prod_{i=1}^n f(\vect{y}^i|g^i(\vect{\theta}))~}_{\mbox{likelihood}}$

where the $g^i: \Rset^{d_{\theta}} \rightarrow\Rset^{d_w}$ ( $1\leq{}i\leq{}n$ ) are such that:

$\begin{array}{rcl} g:\Rset^{d_\theta} & \longrightarrow & \Rset^{n\,d_w}\\ \vect{\theta} & \longmapsto & g(\vect{\theta}) = \Tr{(\Tr{g^1(\vect{\theta})}, \cdots, \Tr{g^n(\vect{\theta})})} \end{array}$

In fact, the first usage is a particular case of the second.

The MCMC method implemented is the Random Walk Metropolis-Hastings algorithm. A sample can be generated only through the MCMC’s derived class: RandomWalkMetropolisHastings.

Methods

`computeLogLikelihood`(currentState)	Compute the logarithm of the likelihood w.r.t.
`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.
`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.
`getMarginal`(*args)	Get the random vector corresponding to the $i^{th}$ marginal component(s).
`getMean`()	Accessor to the mean of the RandomVector.
`getModel`()	Get the model.
`getName`()	Accessor to the object's name.
`getNonRejectedComponents`()	Get the components to be always accepted.
`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.
`getParameters`()	Get the parameters.
`getPrior`()	Get the prior distribution.
`getProcess`()	Get the stochastic process.
`getRealization`()	Compute one realization of the RandomVector.
`getSample`(size)	Compute realizations of the RandomVector.
`getShadowedId`()	Accessor to the object's shadowed id.
`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.
`setName`(name)	Accessor to the object's name.
`setNonRejectedComponents`(nonRejectedComponents)	Set the components to be always accepted.
`setObservations`(observations)	Set the observations.
`setParameter`(parameters)	Accessor to the parameter of the distribution.
`setParameters`(parameters)	Set the parameters.
`setPrior`(prior)	Set the prior 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(currentState)¶

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 $(\vect{y}^1, \dots, \vect{y}^n)$ .

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

lenghtint: 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: Distribution taken into account in the definition of the likelihood, whose PDF with parameters $\vect{w}$ corresponds to $f(.|\vect{w})$ in the equations of the target distribution’s PDF.

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 ]]

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.

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)

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]

getModel()¶

Get the model.

Returns

modelFunction: Model take into account in the definition of the likelihood, which corresponds to $g$ , that is the functions $g^i$ ( $1\leq i \leq n$ ) in the equation of the target distribution’s PDF.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getNonRejectedComponents()¶

Get the components to be always accepted.

Returns

nonRejectedComponentsIndices: The indices of the components that are not tuned, and sampled according to the prior distribution in order to take into account the intrinsic uncertainty, as opposed to the epistemic uncertainty corresponding to the tuned variables.

getObservations()¶

Get the observations.

Returns

observationsSample: Sample taken into account in the definition of the likelihood, which corresponds to the $n$ -tuple of the $\vect{y}^i$ ( $1\leq i \leq n$ ) in equations of the target distribution’s PDF.

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.

getParameters()¶

Get the parameters.

Returns

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

getPrior()¶

Get the prior distribution.

Returns

priorDistribution: The prior distribution of the parameter of the underlying Bayesian statistical model, whose PDF corresponds to $\pi$ in the equations of the target distribution’s PDF.

getProcess()¶

Get the stochastic process.

Returns

processProcess: Stochastic process used to define the RandomVector.

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.

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

lenghtint: 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.

setName(name)¶

Accessor to the object’s name.

Parameters

namestr: The name of the object.

setNonRejectedComponents(nonRejectedComponents)¶

Set the components to be always accepted.

Parameters

nonRejectedComponentssequence of int: The indices of the components that are not tuned, and sampled according to the prior distribution in order to take into account the intrinsic uncertainty, as opposed to the epistemic uncertainty corresponding to the tuned variables.

setObservations(observations)¶

Set the observations.

Parameters

observations2-d sequence of float: Sample taken into account in the definition of the likelihood, which corresponds to the $n$ -tuple of the $\vect{y}^i$ ( $1\leq i \leq n$ ) in the equations of the target distribution’s PDF.

setParameter(parameters)¶

Accessor to the parameter of the distribution.

Parameters

parametersequence of float: Parameter values.

setParameters(parameters)¶

Set the parameters.

Parameters

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

setPrior(prior)¶

Set the prior distribution.

Parameters

priorDistribution: The prior distribution of the parameter of the underlying Bayesian statistical model, whose PDF corresponds to $\pi$ in the equations of the target distribution’s PDF.

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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