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

Notes

Assume we want to sample from a complicated joint distribution. The Gibbs algorithm is a Markov Chain Monte-Carlo algorithm which calls samplers (implemented as MetropolisHastings objects) in a predetermined sequence (by default) or randomly: see setUpdatingMethod() to manually choose the behavior, and note the default behavior can be changed by setting the Gibbs-DefaultUpdatingMethod ResourceMap key. Each sampler samples from 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
>>> prop = 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
>>> coll = [ot.RandomWalkMetropolisHastings(prior, mu0, prop, [i]) for i in range(chainDim)]
>>> for mh in coll: mh.setLikelihood(conditional, y_obs, linkFunction, covariates)
>>> sampler = ot.Gibbs(coll)
>>> # Get a realization
>>> mu = sampler.getRealization()
>>> # Create a Gibbs sampler which updates one randomly chosen component at each step
>>> sampler2 = ot.Gibbs(coll)
>>> sampler2.setUpdatingMethod(ot.Gibbs.RANDOM_UPDATING)
>>> mu2 = sampler2.getRealization()

Methods

asComposedEvent()

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

getAntecedent()

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

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.

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.

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.

getThreshold()

Accessor to the threshold of the Event.

getUpdatingMethod()

Get how the order of the samplers is determined.

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.

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.

setUpdatingMethod(updatingMethod)

Set how the order of the samplers is determined.

__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.

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}).

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.

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.

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.

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.

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:
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).

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]
getRecomputeLogPosterior()

Get the posterior recompute flags.

Returns:
recomputeIndices

Whether the posterior at the state received from the previous sampler is recomputed. For debug use only.

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).

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 ]
getThreshold()

Accessor to the threshold of the Event.

Returns:
thresholdfloat

Threshold of the RandomVector.

getUpdatingMethod()

Get how the order of the samplers is determined.

Available values are:

  • 0 (Gibbs.DETERMINISTIC_UPDATING): the samplers are sequentially called in the given order.

  • 1 (Gibbs.RANDOM_UPDATING): one of the samplers is randomly selected to produce the next realization.

Refer to the setUpdatingMethod() documentation for details.

Returns:
updatingMethodint

See above for the possible values and their meaning.

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}.

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.

setUpdatingMethod(updatingMethod)

Set how the order of the samplers is determined.

Available values are:

  • 0 (Gibbs.DETERMINISTIC_UPDATING): the samplers are sequentially called in the given order.

  • 1 (Gibbs.RANDOM_UPDATING): one of the samplers is randomly selected to produce the next realization.

Note that under DETERMINISTIC_UPDATING, all samplers are called in the specified order every time getRealization() is called. By contrast, under RANDOM_UPDATING, only one sampler is called by getRealization(), and this sampler is randomly chosen among all specified samplers.

Parameters:
updatingMethodint

See above for the possible values and their meaning.

Examples using the class

Gibbs sampling of the posterior distribution

Gibbs sampling of the posterior distribution

Bayesian calibration of a computer code

Bayesian calibration of a computer code

Bayesian calibration of the flooding model

Bayesian calibration of the flooding model

Customize your Metropolis-Hastings algorithm

Customize your Metropolis-Hastings algorithm

Linear Regression with interval-censored observations

Linear Regression with interval-censored observations