RandomWalkMetropolisHastings

class RandomWalkMetropolisHastings(*args)

Random Walk Metropolis-Hastings method.

Available constructor:

RandomWalkMetropolisHastings(prior, conditional, observations, initialState, proposal)

RandomWalkMetropolisHastings(prior, conditional, model, parameters, observations, initialState, proposal)

Parameters:

prior : Distribution

Prior distribution of the parameters of the underlying Bayesian statistical model.

conditional : Distribution

Required distribution to define the likelihood of the underlying Bayesian statistical model.

model : Function

Function required to define the likelihood.

observations : 2-d sequence of float

Observations required to define the likelihood.

initialState : sequence of float

Initial state of the Monte-Carlo Markov chain on which the Sampler is based.

parameters : 2-d sequence of float

Parameters of the model to be fixed.

proposal : list of Distribution

Distributions from which the transition kernels of the MCMC are defined, as explained hereafter. In the following of this paragraph, \delta \sim p_j means that the realization \delta is obtained according to the j^{th} Distribution of the list proposal of size d. The underlying MCMC algorithm is a Metropolis-Hastings one which draws candidates (for the next state of the chain) using a random walk: from the current state \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 does not depend on \vect{\theta}^k. More precisely, here, during the k^{th} Metropolis-Hastings iteration, only the j^{th} component \delta_j^k of \vect{\delta}^k , with j=k \mod d, is not zero and \delta_j^k = \lambda_j^k \delta^k where \lambda_j^k is a deterministic scalar calibration coefficient and where \delta^k \sim p_j. Moreover, \lambda_j^k = 1 by default, but adaptive strategy based on the acceptance rate of each component can be defined using the method setCalibrationStrategyPerComponent().

Notes

A RandomWalkMetropolisHastings enables to carry out MCMC sampling according to the preceding statements. It is important to note that sampling one new realization comes to carrying out d Metropolis- Hastings iterations (such as described above): all of the components of the new realization can differ from the corresponding components of the previous realization. Besides, the burn-in and thinning parameters do not take into consideration the number of MCMC iterations indeed, but the number of sampled realizations.

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(y, obsDim)
>>> # Parameters
>>> p = ot.Sample(obsSize, chainDim)
>>> for i in range(obsSize):
...     for j in range(chainDim):
...         p[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)
>>> model = ot.ParametricFunction(fullModel, parametersSet, parametersValue)
>>> # Calibration parameters
>>> calibrationColl = [ot.CalibrationStrategy()]*chainDim
>>> # Proposal distribution
>>> proposalColl = [ot.Uniform(-1.0, 1.0)]*chainDim
>>> # 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 metropolis-hastings sampler
>>> # prior =a distribution of dimension chainDim, the a priori distribution of the parameter
>>> # conditional =a distribution of dimension 1, the observation error on the output
>>> # model =the link between the parameters and the output
>>> # y_obs =noisy observations of the output
>>> # mu0 =starting point of the chain
>>> sampler = ot.RandomWalkMetropolisHastings(
...     prior, conditional, model, p, y_obs, mu0, proposalColl)
>>> sampler.setCalibrationStrategyPerComponent(calibrationColl)
>>> sampler.setBurnIn(200)
>>> sampler.setThinning(10)
>>> # Get a realization
>>> print(sampler.getRealization())
[1.22816,1.0049,-1.99008]

Methods

computeLogLikelihood(currentState) Compute the logarithm of the likelihood w.r.t.
getAcceptanceRate() Get acceptance rate.
getBurnIn() Get the length of the burn-in period.
getCalibrationStrategyPerComponent() Get the calibration strategy per component.
getClassName() Accessor to the object’s name.
getConditional() Get the conditional distribution.
getDimension() Get the dimension of the samples generated.
getHistory() Get the history storage.
getId() Accessor to the object’s id.
getModel() Get the model.
getName() Accessor to the object’s name.
getNonRejectedComponents() Get the components to be always accepted.
getObservations() Get the observations.
getParameters() Get the parameters.
getPrior() Get the prior distribution.
getProposal() Get the proposal.
getRealization() Return a realization.
getSample(size) Return several realizations.
getShadowedId() Accessor to the object’s shadowed id.
getThinning() Get the thinning parameter.
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.
setBurnIn(burnIn) Set the length of the burn-in period.
setCalibrationStrategy(calibrationStrategy) Set the calibration strategy.
setCalibrationStrategyPerComponent(...) Set the calibration strategy per component.
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.
setParameters(parameters) Set the parameters.
setPrior(prior) Set the prior distribution.
setProposal(proposal) Set the proposal.
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:

currentState : sequence of float

Current state.

Returns:

logLikelihood : float

Logarithm of the likelihood w.r.t. observations (\vect{y}^1, \dots, \vect{y}^n).

getAcceptanceRate()

Get acceptance rate.

Returns:

acceptanceRate : Point of dimension d

Sequence whose the j^{th} component corresponds to the acceptance rate of the candidates \vect{c}^k obtained from a state \vect{\theta}^k by only changing its j^{th} component, that is to the acceptance rate only relative to the k^{th} MCMC iterations such that k \mod d=j (see the paragraph dedicated to the constructors of the class above). These are global acceptance rates over all the MCMC iterations performed.

getBurnIn()

Get the length of the burn-in period.

Returns:

lenght : int

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.

getCalibrationStrategyPerComponent()

Get the calibration strategy per component.

Returns:

strategy : list of CalibrationStrategy

A list of CalibrationStrategy strategy, whose j^{th} component strategy[j] defines whether and how the \lambda_j^k (see the paragraph dedicated to the constructors of the class above) are rescaled, on the basis of the last j^{th} component acceptance rate \rho_j^k . The calibration coefficients are rescaled every q\times d MCMC iterations with q = strategy[j].getCalibrationStep(), thus on the basis of the acceptances or refusals of the last q candidates obtained by only changing the j^{th} component of the current state: \lambda_j^k = \Phi_j (\rho_j^k)\lambda_j^{k-qd} where \Phi_j(.) is defined by strategy[j].computeUpdateFactor().

getClassName()

Accessor to the object’s name.

Returns:

class_name : str

The object class name (object.__class__.__name__).

getConditional()

Get the conditional distribution.

Returns:

conditional : Distribution

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.

getDimension()

Get the dimension of the samples generated.

Returns:

dimension : int

Dimension of the samples that the Sampler can generate.

getHistory()

Get the history storage.

Returns:

history : HistoryStrategy

Used to record the chain.

getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.

getModel()

Get the model.

Returns:

model : Function

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:

name : str

The name of the object.

getNonRejectedComponents()

Get the components to be always accepted.

Returns:

nonRejectedComponents : Indices

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:

observations : Sample

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.

getParameters()

Get the parameters.

Returns:

parameters : Point

Fixed parameters of the model g required to define the likelihood.

getPrior()

Get the prior distribution.

Returns:

prior : Distribution

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.

getProposal()

Get the proposal.

Returns:

proposal : list of Distribution

The d-tuple of Distributions p_j (1 \leq j \leq d) from which the transition kernels of the random walk Metropolis-Hastings algorithm are defined; look at the paragraph dedicated to the constructors of the class above.

getRealization()

Return a realization.

Returns:

realization : Point

A new realization.

getSample(size)

Return several realizations.

Parameters:

size : int, size \leq 0

Number of realizations needed.

Returns:

realizations : Sample

Sequence composed of size new realizations.

getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.

getThinning()

Get the thinning parameter.

Returns:

thinning : int

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.

getVerbose()

Tell whether the verbose mode is activated or not.

Returns:

isVerbose : bool

The verbose mode is activated if it is True, desactivated otherwise.

getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.

hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.

hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.

setBurnIn(burnIn)

Set the length of the burn-in period.

Parameters:

lenght : int

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.

setCalibrationStrategy(calibrationStrategy)

Set the calibration strategy.

Parameters:

strategy : CalibrationStrategy

Same strategy applied for each component \lambda_j^k.

setCalibrationStrategyPerComponent(calibrationStrategy)

Set the calibration strategy per component.

Parameters:

strategy : list of CalibrationStrategy

A list of CalibrationStrategy strategy, whose j^{th} component strategy[j] defines whether and how the \lambda_j^k (see the paragraph dedicated to the constructors of the class above) are rescaled, on the basis of the last j^{th} component acceptance rate \rho_j^k . The calibration coefficients are rescaled every q\times d MCMC iterations with q = strategy[j].getCalibrationStep(), thus on the basis of the acceptances or refusals of the last q candidates obtained by only changing the j^{th} component of the current state: \lambda_j^k = \Phi_j (\rho_j^k)\lambda_j^{k-qd} where \Phi_j(.) is defined by strategy[j].computeUpdateFactor().

setHistory(strategy)

Set the history storage.

Parameters:

history : HistoryStrategy

Used to record the chain.

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setNonRejectedComponents(nonRejectedComponents)

Set the components to be always accepted.

Parameters:

nonRejectedComponents : sequence 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:

observations : 2-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.

setParameters(parameters)

Set the parameters.

Parameters:

parameters : sequence of float

Fixed parameters of the model g required to define the likelihood.

setPrior(prior)

Set the prior distribution.

Parameters:

prior : Distribution

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.

setProposal(proposal)

Set the proposal.

Parameters:

proposal : list of Distribution

The d-tuple of Distributions p_j (1 \leq j \leq d) from which the transition kernels of the random walk Metropolis-Hastings algorithm are defined; look at the paragraph dedicated to the constructors of the class above.

setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.

setThinning(thinning)

Set the thinning parameter.

Parameters:

thinning : int, 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:

isVerbose : bool

The verbose mode is activated if it is True, desactivated otherwise.

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.