UncertaintyQuantification.jl

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Bayesian Updating ​

Methods for Bayesian updating.

Index ​

Types ​

UncertaintyQuantification.AbstractBayesianMethod Type
julia
AbstractBayesianMethod

Subtypes are used to dispatch to the differenct MCMC methods in bayesianupdating.

Subtypes are:

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UncertaintyQuantification.SingleComponentMetropolisHastings Type
julia
SingleComponentMetropolisHastings(proposal, x0, n, burnin, islog)

Passed to bayesianupdating to run the single-component Metropolis-Hastings algorithm starting from x0 with univariate proposal distibution proposal. Will generate n samples after performing burnin steps of the Markov chain and discarding the samples. The flag islog specifies whether the prior and likelihood functions passed to the bayesianupdating method are already given as logarithms.

Alternative constructor

julia
    SingleComponentMetropolisHastings(proposal, x0, n, burnin)  # `islog` = true

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UncertaintyQuantification.TransitionalMarkovChainMonteCarlo Type
julia
TransitionalMarkovChainMonteCarlo(prior, n, burnin, β, islog)

Passed to [`bayesianupdating`](@ref) to run thetransitional Markov chain Monte Carlo algorithm  with [`RandomVariable'](@ref) vector `prior`. At each transitional level, one sample will be generated from `n` independent Markov chains after `burnin` steps have been discarded. The flag `islog` specifies whether the prior and likelihood functions passed to the  [`bayesianupdating`](@ref) method are already  given as logarithms.

Alternative constructors

julia
    TransitionalMarkovChainMonteCarlo(prior, n, burnin, β)  # `islog` = true
     TransitionalMarkovChainMonteCarlo(prior, n, burnin)    # `β` = 0.2,  `islog` = true

References

[23]

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UncertaintyQuantification.AbstractBayesianPointEstimate Type
julia
AbstractBayesianPointEstimate

Subtypes are used to dispatch to the differenct point estimation methods in bayesianupdating.

Subtypes are:

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UncertaintyQuantification.MaximumAPosterioriBayesian Type
julia
MaximumAPosterioriBayesian(prior, optimmethod, x0; islog, lowerbounds, upperbounds)

Passed to bayesianupdating to estimate one or more maxima of the posterior distribution starting from x0. The optimization uses the method specified in optimmethod. Will calculate one estimation per point in x0. The flag islog specifies whether the prior and likelihood functions passed to the bayesianupdating method are already given as logarithms. lowerbounds and upperbounds specify optimization intervals.

Alternative constructors

julia
    MaximumAPosterioriBayesian(prior, optimmethod, x0; islog) # `lowerbounds` = [-Inf], # `upperbounds` = [Inf]
    MaximumAPosterioriBayesian(prior, optimmethod, x0)  # `islog` = true

See also MaximumLikelihoodBayesian, bayesianupdating, TransitionalMarkovChainMonteCarlo.

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UncertaintyQuantification.MaximumLikelihoodBayesian Type
julia
MaximumLikelihoodBayesian(prior, optimmethod, x0; islog, lowerbounds, upperbounds)

Passed to bayesianupdating to estimate one or more maxima of the likelihood starting from x0. The optimization uses the method specified in optimmethod. Will calculate one estimation per point in x0. The flag islog specifies whether the prior and likelihood functions passed to the bayesianupdating method are already given as logarithms. lowerbounds and upperbounds specify optimization intervals.

Alternative constructors

julia
    MaximumLikelihoodBayesian(prior, optimmethod, x0; islog) # `lowerbounds` = [-Inf], # `upperbounds` = [Inf]
    MaximumLikelihoodBayesian(prior, optimmethod, x0)  # `islog` = true

Notes

The method uses prior only as information on which parameters are supposed to be optimized. The prior itself does not influence the result of the maximum likelihood estimate and can be given as a dummy distribution. For example, if two parameters a and b are supposed to be optimized, the prior could look like this

julia
    prior = RandomVariable.(Uniform(0,1), [:a, :b])

See also MaximumAPosterioriBayesian, bayesianupdating, TransitionalMarkovChainMonteCarlo.

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Methods ​

UncertaintyQuantification.bayesianupdating Function
julia
bayesianupdating(likelihood, models, pointestimate; prior)

Perform bayesian updating using the given likelihood, models and any point estimation method AbstractBayesianPointEstimate.

Notes

Method can be called with an empty Vector of models, i.e.

bayesianupdating(likelihood, [], pointestimate)

If prior is not given, the method will construct a prior distribution from the prior specified in AbstractBayesianPointEstimate.prior.

likelihood is a Julia function which must be defined in terms of a DataFrame of samples, and must evaluate the likelihood for each row of the DataFrame

For example, a loglikelihood based on normal distribution using 'Data':

julia
likelihood(df) = [sum(logpdf.(Normal.(df_i.x, 1), Data)) for df_i in eachrow(df)]

If a model evaluation is required to evaluate the likelihood, a vector of UQModels must be passed to bayesianupdating. For example if the variable x above is the output of a numerical model.

For a general overview of the function, see bayesianupdating.

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julia
bayesianupdating(prior, likelihood, models, mcmc)

Perform bayesian updating using the given prior, likelihood, models and any MCMC sampler AbstractBayesianMethod.

Alternatively the method can be called without models.

bayesianupdating(prior, likelihood, mcmc)

When using TransitionalMarkovChainMonteCarlo the prior can automatically be constructed.

bayesinupdating(likelihood, models, tmcmc)
bayesianupdating(likelihood, tmcmc)

Notes

likelihood is a Julia function which must be defined in terms of a DataFrame of samples, and must evaluate the likelihood for each row of the DataFrame

For example, a loglikelihood based on normal distribution using 'Data':

julia
likelihood(df) = [sum(logpdf.(Normal.(df_i.x, 1), Data)) for df_i in eachrow(df)]

If a model evaluation is required to evaluate the likelihood, a vector of UQModels must be passed to bayesianupdating. For example if the variable x above is the output of a numerical model.

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