Gibbs¶

class Gibbs(*args)¶

Gibbs sampling method.

Refer to Bayesian calibration, The Metropolis-Hastings Algorithm.

Parameters

samplerssequence of MetropolisHastings: List of samplers for each bloc of the chain

See also

RandomWalkMetropolisHastings, RandomVectorMetropolisHastings

Notes

Assume we want to sample from a complicated joint distribution. The Gibbs algorithm is a Markov Chain Monte-Carlo algorithm which sequentially calls samplers (implemented as MetropolisHastings objects) of the conditional distributions of one or several components.

Examples

>>> import openturns as ot
>>> ot.RandomGenerator.SetSeed(0)
>>> chainDim = 3
>>> # Observations
>>> obsDim = 1
>>> obsSize = 10
>>> y = [-9.50794871493506, -3.83296694500105, -2.44545713047953,
...      0.0803625289211318, 1.01898069723583, 0.661725805623086,
...      -1.57581204592385, -2.95308465670895, -8.8878164296758,
...      -13.0812290405651]
>>> y_obs = ot.Sample([[yi] for yi in y])
>>> # Parameters
>>> covariates = ot.Sample(obsSize, chainDim)
>>> for i in range(obsSize):
...     for j in range(chainDim):
...         covariates[i, j] = (-2 + 5.0 * i / 9.0) ** j
>>> # Model
>>> fullModel = ot.SymbolicFunction(
...          ['p1', 'p2', 'p3', 'x1', 'x2', 'x3'],
...          ['p1*x1+p2*x2+p3*x3', '1.0'])
>>> parametersSet = range(chainDim)
>>> parametersValue = [0.0] * len(parametersSet) # 0.0 is a placeholder
>>> linkFunction = ot.ParametricFunction(fullModel, parametersSet, parametersValue)
>>> # Calibration parameters
>>> # Proposal distribution
>>> proposal = ot.Uniform(-1.0, 1.0)
>>> # Prior distribution
>>> sigma0 = [10.0]*chainDim
>>> #  Covariance matrix
>>> Q0_inv = ot.CorrelationMatrix(chainDim)
>>> for i in range(chainDim): 
...     Q0_inv[i, i] = sigma0[i] * sigma0[i]
>>> mu0 = [0.0]*chainDim
>>> # x0 ~ N(mu0, sigma0)
>>> prior = ot.Normal(mu0, Q0_inv)
>>> # Conditional distribution y~N(z, 1.0)
>>> conditional = ot.Normal()
>>> # Create a Gibbs sampler
>>> mh_coll = [ot.RandomWalkMetropolisHastings(prior, mu0, proposal, [i]) for i in range(chainDim)]
>>> for mh in mh_coll: mh.setLikelihood(conditional, y_obs, linkFunction, covariates)
>>> sampler = ot.Gibbs(mh_coll)
>>> sampler.setBurnIn(20)
>>> sampler.setThinning(10)
>>> # Get a realization
>>> mu = sampler.getRealization()

Methods

`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.
`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.
`getMetropolisHastingsCollection`()	Get the list of Metropolis-Hastings samplers.
`getName`()	Accessor to the object's name.
`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.
`getRealization`()	Compute one realization of the RandomVector.
`getRecomputeLogPosterior`()	Get the posterior recompute flags.
`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.
`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.
`setParameter`(parameters)	Accessor to the parameter of the distribution.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setThinning`(thinning)	Set the thinning parameter.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

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

lengthint: 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__).

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]

getMetropolisHastingsCollection()¶

Get the list of Metropolis-Hastings samplers.

Returns

samplerssequence of MetropolisHastings: List of all Metropolis-Hastings samplers used in the Gibbs algorithm

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

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.

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.

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

lengthint: 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.

setParameter(parameters)¶

Accessor to the parameter of the distribution.

Parameters

parametersequence of float: Parameter values.

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.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters

visiblebool: Visibility flag.

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