API

Summary statistics

PosteriorStats.SummaryStats — Type

struct SummaryStats{D, V<:(AbstractVector)}

A container for a column table of values computed by summarize.

This object implements the Tables and TableTraits column table interfaces. It has a custom show method.

SummaryStats behaves like an OrderedDict of columns, where the columns can be accessed using either Symbols or a 1-based integer index.

name::String: The name of the collection of summary statistics, used as the table title in display.
data::Any: The summary statistics for each parameter. It must implement the Tables interface.
parameter_names::AbstractVector: Names of the parameters

SummaryStats([name::String,] data[, parameter_names])
SummaryStats(data[, parameter_names]; name::String="SummaryStats")

Construct a SummaryStats from tabular data with optional stats name and param_names.

data must not contain a column :parameter, as this is reserved for the parameter names, which are always in the first column.

source

PosteriorStats.default_diagnostics — Function

default_diagnostics(focus=Statistics.mean; kwargs...)

Default diagnostics to be computed with summarize.

The value of focus determines the diagnostics to be returned:

Statistics.mean: mcse_mean, mcse_std, ess_tail, ess_bulk, rhat
Statistics.median: mcse_median, ess_tail, ess_bulk, rhat

source

PosteriorStats.default_stats — Function

default_stats(focus=Statistics.mean; prob_interval=0.94, kwargs...)

Default statistics to be computed with summarize.

The value of focus determines the statistics to be returned:

Statistics.mean: mean, std, hdi_94%
Statistics.median: median, mad, eti_94%

If prob_interval is set to a different value than the default, then different HDI and ETI statistics are computed accordingly. hdi refers to the highest-density interval, while eti refers to the equal-tailed interval.

Credible intervals

PosteriorStats.hdi — Function

hdi(samples::AbstractVecOrMat{<:Real}; [prob, sorted, method]) -> IntervalSets.ClosedInterval
hdi(samples::AbstractArray{<:Real}; [prob, sorted, method]) -> Array{<:IntervalSets.ClosedInterval}

Estimate the highest density interval (HDI) of samples for the probability prob.

The HDI is the minimum width Bayesian credible interval (BCI). That is, it is the smallest possible interval containing (100*prob)% of the probability mass.[1] This implementation uses the algorithm of Chen and Shao [2].

LOO and WAIC

PosteriorStats.AbstractELPDResult — Type

abstract type AbstractELPDResult

An abstract type representing the result of an ELPD computation.

Every subtype stores estimates of both the expected log predictive density (elpd) and the effective number of parameters p, as well as standard errors and pointwise estimates of each, from which other relevant estimates can be computed.

Subtypes implement the following functions:

elpd_estimates
information_criterion

source

PosteriorStats.PSISLOOResult — Type

Results of Pareto-smoothed importance sampling leave-one-out cross-validation (PSIS-LOO).

Model comparison

PosteriorStats.ModelComparisonResult — Type

ModelComparisonResult

Result of model comparison using ELPD.

This struct implements the Tables and TableTraits interfaces.

Each field returns a collection of the corresponding entry for each model:

name: Names of the models, if provided.
rank: Ranks of the models (ordered by decreasing ELPD)
elpd_diff: ELPD of a model subtracted from the largest ELPD of any model
se_elpd_diff: Standard error of the ELPD difference
weight: Model weights computed with weights_method
elpd_result: AbstactELPDResults for each model, which can be used to access useful stats like ELPD estimates, pointwise estimates, and Pareto shape values for PSIS-LOO
weights_method: Method used to compute model weights with model_weights

source

PosteriorStats.compare — Function

compare(models; kwargs...) -> ModelComparisonResult

Compare models based on their expected log pointwise predictive density (ELPD).

The ELPD is estimated either by Pareto smoothed importance sampling leave-one-out cross-validation (LOO) or using the widely applicable information criterion (WAIC). loo is the default and recommended method for computing the ELPD. For more theory, see Spiegelhalter et al. [6].

Arguments

models: a Tuple, NamedTuple, or AbstractVector whose values are either AbstractELPDResult entries or any argument to elpd_method.

Keywords

weights_method::AbstractModelWeightsMethod=Stacking(): the method to be used to weight the models. See model_weights for details
elpd_method=loo: a method that computes an AbstractELPDResult from an argument in models.
sort::Bool=true: Whether to sort models by decreasing ELPD.

Returns

ModelComparisonResult: A container for the model comparison results. The fields contain a similar collection to models.

Examples

Compare the centered and non centered models of the eight school problem using the defaults: loo and Stacking weights. A custom myloo method formates the inputs as expected by loo.

julia> using ArviZExampleData

julia> models = (
           centered=load_example_data("centered_eight"),
           non_centered=load_example_data("non_centered_eight"),
       );

julia> function myloo(idata)
           log_like = PermutedDimsArray(idata.log_likelihood.obs, (2, 3, 1))
           return loo(log_like)
       end;

julia> mc = compare(models; elpd_method=myloo)
┌ Warning: 1 parameters had Pareto shape values 0.7 < k ≤ 1. Resulting importance sampling estimates are likely to be unstable.
└ @ PSIS ~/.julia/packages/PSIS/...
ModelComparisonResult with Stacking weights
               rank  elpd  se_elpd  elpd_diff  se_elpd_diff  weight    p  se_p ⋯
 non_centered     1   -31      1.5       0            0.0       1.0  0.9  0.32 ⋯
 centered         2   -31      1.4       0.03         0.061     0.0  0.9  0.33 ⋯
julia> mc.weight |> pairs
pairs(::NamedTuple) with 2 entries:
  :non_centered => 1.0
  :centered     => 3.50546e-31

Compare the same models from pre-computed PSIS-LOO results and computing BootstrappedPseudoBMA weights:

julia> elpd_results = mc.elpd_result;

julia> compare(elpd_results; weights_method=BootstrappedPseudoBMA())
ModelComparisonResult with BootstrappedPseudoBMA weights
               rank  elpd  se_elpd  elpd_diff  se_elpd_diff  weight    p  se_p ⋯
 non_centered     1   -31      1.5       0            0.0      0.51  0.9  0.32 ⋯
 centered         2   -31      1.4       0.03         0.061    0.49  0.9  0.33 ⋯

References

[6] Spiegelhalter et al. J. R. Stat. Soc. B 64 (2002)

source

PosteriorStats.model_weights — Function

model_weights(elpd_results; method=Stacking())
model_weights(method::AbstractModelWeightsMethod, elpd_results)

Compute weights for each model in elpd_results using method.

elpd_results is a Tuple, NamedTuple, or AbstractVector with AbstractELPDResult entries. The weights are returned in the same type of collection.

Stacking is the recommended approach, as it performs well even when the true data generating process is not included among the candidate models. See Yao et al. [7] for details.

Examples

Compute Stacking weights for two models:

julia> using ArviZExampleData

julia> models = (
           centered=load_example_data("centered_eight"),
           non_centered=load_example_data("non_centered_eight"),
       );

julia> elpd_results = map(models) do idata
           log_like = PermutedDimsArray(idata.log_likelihood.obs, (2, 3, 1))
           return loo(log_like)
       end;
┌ Warning: 1 parameters had Pareto shape values 0.7 < k ≤ 1. Resulting importance sampling estimates are likely to be unstable.
└ @ PSIS ~/.julia/packages/PSIS/...

julia> model_weights(elpd_results; method=Stacking()) |> pairs
pairs(::NamedTuple) with 2 entries:
  :centered     => 3.50546e-31
  :non_centered => 1.0

Now we compute BootstrappedPseudoBMA weights for the same models:

julia> model_weights(elpd_results; method=BootstrappedPseudoBMA()) |> pairs
pairs(::NamedTuple) with 2 entries:
  :centered     => 0.492513
  :non_centered => 0.507487

References

[7] Yao et al. Bayesian Analysis 13, 3 (2018)

source

The following model weighting methods are available

PosteriorStats.AbstractModelWeightsMethod — Type

abstract type AbstractModelWeightsMethod

An abstract type representing methods for computing model weights.

Subtypes implement model_weights(method, elpd_results).

source

PosteriorStats.BootstrappedPseudoBMA — Type

struct BootstrappedPseudoBMA{R<:Random.AbstractRNG, T<:Real} <: AbstractModelWeightsMethod

Model weighting method using pseudo Bayesian Model Averaging using Akaike-type weighting with the Bayesian bootstrap (pseudo-BMA+)[7].

The Bayesian bootstrap stabilizes the model weights.

BootstrappedPseudoBMA(; rng=Random.default_rng(), samples=1_000, alpha=1)
BootstrappedPseudoBMA(rng, samples, alpha)

Construct the method.

rng::Random.AbstractRNG: The random number generator to use for the Bayesian bootstrap
samples::Int64: The number of samples to draw for bootstrapping
alpha::Real: The shape parameter in the Dirichlet distribution used for the Bayesian bootstrap. The default (1) corresponds to a uniform distribution on the simplex.

Predictive checks

PosteriorStats.loo_pit — Function

loo_pit(y, y_pred, log_weights; kwargs...) -> Union{Real,AbstractArray}

Compute leave-one-out probability integral transform (LOO-PIT) checks.

Arguments

y: array of observations with shape (params...,)
y_pred: array of posterior predictive samples with shape (draws, chains, params...).
log_weights: array of normalized log LOO importance weights with shape (draws, chains, params...).

Keywords

is_discrete: If not provided, then it is set to true iff elements of y and y_pred are all integer-valued. If true, then data are smoothed using smooth_data to make them non-discrete before estimating LOO-PIT values.
kwargs: Remaining keywords are forwarded to smooth_data if data is discrete.

Returns

pitvals: LOO-PIT values with same size as y. If y is a scalar, then pitvals is a scalar.

LOO-PIT is a marginal posterior predictive check. If $y_{-i}$ is the array $y$ of observations with the $i$th observation left out, and $y_i^*$ is a posterior prediction of the $i$th observation, then the LOO-PIT value for the $i$th observation is defined as

\[P(y_i^* \le y_i \mid y_{-i}) = \int_{-\infty}^{y_i} p(y_i^* \mid y_{-i}) \mathrm{d} y_i^*\]

The LOO posterior predictions and the corresponding observations should have similar distributions, so if conditional predictive distributions are well-calibrated, then all LOO-PIT values should be approximately uniformly distributed on $[0, 1]$. [8]

Examples

Calculate LOO-PIT values using as test quantity the observed values themselves.

julia> using ArviZExampleData

julia> idata = load_example_data("centered_eight");

julia> y = idata.observed_data.obs;

julia> y_pred = PermutedDimsArray(idata.posterior_predictive.obs, (:draw, :chain, :school));

julia> log_like = PermutedDimsArray(idata.log_likelihood.obs, (:draw, :chain, :school));

julia> log_weights = loo(log_like).psis_result.log_weights;

julia> loo_pit(y, y_pred, log_weights)
8-element DimArray{Float64,1} with dimensions:
  Dim{:school} Categorical{String} String[Choate, Deerfield, …, St. Paul's, Mt. Hermon] Unordered
 "Choate"            0.942759
 "Deerfield"         0.641057
 "Phillips Andover"  0.32729
 "Phillips Exeter"   0.581451
 "Hotchkiss"         0.288523
 "Lawrenceville"     0.393741
 "St. Paul's"        0.886175
 "Mt. Hermon"        0.638821

Calculate LOO-PIT values using as test quantity the square of the difference between each observation and mu.

julia> using Statistics

julia> mu = idata.posterior.mu;

julia> T = y .- median(mu);

julia> T_pred = y_pred .- mu;

julia> loo_pit(T .^ 2, T_pred .^ 2, log_weights)
8-element DimArray{Float64,1} with dimensions:
  Dim{:school} Categorical{String} String[Choate, Deerfield, …, St. Paul's, Mt. Hermon] Unordered
 "Choate"            0.868148
 "Deerfield"         0.27421
 "Phillips Andover"  0.321719
 "Phillips Exeter"   0.193169
 "Hotchkiss"         0.370422
 "Lawrenceville"     0.195601
 "St. Paul's"        0.817408
 "Mt. Hermon"        0.326795

References

[8] Gabry et al. J. R. Stat. Soc. Ser. A Stat. Soc. 182 (2019).

source

PosteriorStats.r2_score — Function

r2_score(y_true::AbstractVector, y_pred::AbstractArray) -> (; r2, r2_std)

$R²$ for linear Bayesian regression models.[9]

Arguments

y_true: Observed data of length noutputs
y_pred: Predicted data with size (ndraws[, nchains], noutputs)

Examples

julia> using ArviZExampleData

julia> idata = load_example_data("regression1d");

julia> y_true = idata.observed_data.y;

julia> y_pred = PermutedDimsArray(idata.posterior_predictive.y, (:draw, :chain, :y_dim_0));

julia> r2_score(y_true, y_pred) |> pairs
pairs(::NamedTuple) with 2 entries:
  :r2     => 0.683197
  :r2_std => 0.0368838

References

[9] Gelman et al, The Am. Stat., 73(3) (2019)

source

Utilities

PosteriorStats.kde_reflected — Function

kde_reflected(data::AbstractVector{<:Real}; bounds=extrema(data), kwargs...)

Compute the boundary-corrected kernel density estimate (KDE) of data using reflection.

For $x \in (l, u)$, the reflected KDE has the density

\[\hat{f}_R(x) = \hat{f}(x) + \hat{f}(2l - x) + \hat{f}(2u - x),\]

where $\hat{f}$ is the usual KDE of data. This is equivalent to augmenting the original data with 2 additional copies of the data reflected around each bound, computing the usual KDE, trimming the KDE to the bounds, and renormalizing.

Any non-finite bounds are ignored. Remaining kwargs are passed to KernelDensity.kde. The default bandwidth is estimated using the Improved Sheather-Jones (ISJ) method [10].

References

[10] Botev et al. Ann. Stat., 38: 5 (2010)

source

PosteriorStats.smooth_data — Function

smooth_data(y; dims=:, interp_method=CubicSpline, offset_frac=0.01)

Smooth y along dims using interp_method.

interp_method is a 2-argument callabale that takes the arguments y and x and returns a DataInterpolations.jl interpolation method, defaulting to a cubic spline interpolator.

offset_frac is the fraction of the length of y to use as an offset when interpolating.

source