API

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.

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

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_3%, hdi_97%
• Statistics.median: median, mad, eti_3%, eti_97%

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 (i.e. the credible interval computed from symmetric quantiles).

See also: hdi

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

Combinatiton of default_stats and default_diagnostics to be used with summarize.

summarize(data, stats_funs...; name="SummaryStats", [var_names]) -> SummaryStats

Compute the summary statistics in stats_funs on each param in data.

stats_funs is a collection of functions that reduces a matrix with shape (draws, chains) to a scalar or a collection of scalars. Alternatively, an item in stats_funs may be a Pair of the form name => fun specifying the name to be used for the statistic or of the form (name1, ...) => fun when the function returns a collection. When the function returns a collection, the names in this latter format must be provided.

If no stats functions are provided, then those specified in default_summary_stats are computed.

var_names specifies the names of the parameters in data. If not provided, the names are inferred from data.

To support computing summary statistics from a custom object, overload this method specifying the type of data.

See also SummaryStats, default_summary_stats, default_stats, default_diagnostics.

Examples

Compute mean, std and the Monte Carlo standard error (MCSE) of the mean estimate:

julia> using Statistics, StatsBase

julia> x = randn(1000, 4, 3) .+ reshape(0:10:20, 1, 1, :);

julia> summarize(x, mean, std, :mcse_mean => sem; name="Mean/Std")
Mean/Std
       mean    std  mcse_mean
 1   0.0003  0.990      0.016
 2  10.02    0.988      0.016
 3  19.98    0.988      0.016

Avoid recomputing the mean by using mean_and_std, and provide parameter names:

julia> summarize(x, (:mean, :std) => mean_and_std, mad; var_names=[:a, :b, :c])
SummaryStats
         mean    std    mad
 a   0.000305  0.990  0.978
 b  10.0       0.988  0.995
 c  20.0       0.988  0.979

Note that when an estimator and its MCSE are both computed, the MCSE is used to determine the number of significant digits that will be displayed.

julia> summarize(x; var_names=[:a, :b, :c])
SummaryStats
       mean   std  hdi_3%  hdi_97%  mcse_mean  mcse_std  ess_tail  ess_bulk  r ⋯
 a   0.0003  0.99   -1.92     1.78      0.016     0.012      3567      3663  1 ⋯
 b  10.02    0.99    8.17    11.9       0.016     0.011      3841      3906  1 ⋯
 c  19.98    0.99   18.1     21.9       0.016     0.012      3892      3749  1 ⋯
                                                                1 column omitted

Compute just the statistics with an 89% HDI on all parameters, and provide the parameter names:

julia> summarize(x, default_stats(; prob_interval=0.89)...; var_names=[:a, :b, :c])
SummaryStats
         mean    std  hdi_5.5%  hdi_94.5%
 a   0.000305  0.990     -1.63       1.52
 b  10.0       0.988      8.53      11.6
 c  20.0       0.988     18.5       21.6

Compute the summary stats focusing on Statistics.median:

julia> summarize(x, default_summary_stats(median)...; var_names=[:a, :b, :c])
SummaryStats
    median    mad  eti_3%  eti_97%  mcse_median  ess_tail  ess_median  rhat
 a   0.004  0.978   -1.83     1.89        0.020      3567        3336  1.00
 b  10.02   0.995    8.17    11.9         0.023      3841        3787  1.00
 c  19.99   0.979   18.1     21.9         0.020      3892        3829  1.00

hdi(samples::AbstractArray{<:Real}; prob=0.94) -> (; lower, upper)

Estimate the unimodal 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.^{[Hyndman1996]}

samples is an array of shape (draws[, chains[, params...]]). If multiple parameters are present, then lower and upper are arrays with the shape (params...,), computed separately for each marginal.

This implementation uses the algorithm of ^{[ChenShao1999]}.

Examples

Here we calculate the 83% HDI for a normal random variable:

julia> x = randn(2_000);

julia> hdi(x; prob=0.83) |> pairs
pairs(::NamedTuple) with 2 entries:
  :lower => -1.38266
  :upper => 1.25982

We can also calculate the HDI for a 3-dimensional array of samples:

julia> x = randn(1_000, 1, 1) .+ reshape(0:5:10, 1, 1, :);

julia> hdi(x) |> pairs
pairs(::NamedTuple) with 2 entries:
  :lower => [-1.9674, 3.0326, 8.0326]
  :upper => [1.90028, 6.90028, 11.9003]

hdi!(samples::AbstractArray{<:Real}; prob=0.94) -> (; lower, upper)

A version of hdi that sorts samples in-place while computing the HDI.

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

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

See also: loo, AbstractELPDResult

• estimates: Estimates of the expected log pointwise predictive density (ELPD) and effective number of parameters (p)

• pointwise: Pointwise estimates

• psis_result: Pareto-smoothed importance sampling (PSIS) results

Results of computing the widely applicable information criterion (WAIC).

See also: waic, AbstractELPDResult

• estimates: Estimates of the expected log pointwise predictive density (ELPD) and effective number of parameters (p)

• pointwise: Pointwise estimates

elpd_estimates(result::AbstractELPDResult; pointwise=false) -> (; elpd, elpd_mcse, lpd)

Return the (E)LPD estimates from the result.

information_criterion(elpd, scale::Symbol)

Compute the information criterion for the given scale from the elpd estimate.

scale must be one of (:deviance, :log, :negative_log).

See also: loo, waic

information_criterion(result::AbstractELPDResult, scale::Symbol; pointwise=false)

Compute information criterion for the given scale from the existing ELPD result.

scale must be one of (:deviance, :log, :negative_log).

If pointwise=true, then pointwise estimates are returned.

loo(log_likelihood; reff=nothing, kwargs...) -> PSISLOOResult{<:NamedTuple,<:NamedTuple}

Compute the Pareto-smoothed importance sampling leave-one-out cross-validation (PSIS-LOO). ^{[Vehtari2017][LOOFAQ]}

log_likelihood must be an array of log-likelihood values with shape (chains, draws[, params...]).

Keywords

• reff::Union{Real,AbstractArray{<:Real}}: The relative effective sample size(s) of the likelihood values. If an array, it must have the same data dimensions as the corresponding log-likelihood variable. If not provided, then this is estimated using MCMCDiagnosticTools.ess.
• kwargs: Remaining keywords are forwarded to [PSIS.psis].

See also: PSISLOOResult, waic

Examples

Manually compute $R_\mathrm{eff}$ and calculate PSIS-LOO of a model:

julia> using ArviZExampleData, MCMCDiagnosticTools

julia> idata = load_example_data("centered_eight");

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

julia> reff = ess(log_like; kind=:basic, split_chains=1, relative=true);

julia> loo(log_like; reff)
PSISLOOResult with estimates
 elpd  elpd_mcse    p  p_mcse
  -31        1.4  0.9    0.34

and PSISResult with 500 draws, 4 chains, and 8 parameters
Pareto shape (k) diagnostic values:
                    Count      Min. ESS
 (-Inf, 0.5]  good  7 (87.5%)  151
  (0.5, 0.7]  okay  1 (12.5%)  446

waic(log_likelihood::AbstractArray) -> WAICResult{<:NamedTuple,<:NamedTuple}

Compute the widely applicable information criterion (WAIC).^{[Watanabe2010][Vehtari2017][LOOFAQ]}

log_likelihood must be an array of log-likelihood values with shape (chains, draws[, params...]).

See also: WAICResult, loo

Examples

Calculate WAIC of a model:

julia> using ArviZExampleData

julia> idata = load_example_data("centered_eight");

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

julia> waic(log_like)
WAICResult with estimates
 elpd  elpd_mcse    p  p_mcse
  -31        1.4  0.9    0.33

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

• elpd_diff_mcse: Monte Carlo 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

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). We recommend loo. Read more theory here - in a paper by some of the leading authorities on model comparison dx.doi.org/10.1111/1467-9868.00353

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  elpd_mcse  elpd_diff  elpd_diff_mcse  weight    p   ⋯
 non_centered     1   -31        1.4       0              0.0       1.0  0.9   ⋯
 centered         2   -31        1.4       0.06           0.067     0.0  0.9   ⋯
                                                                1 column omitted
julia> mc.weight |> pairs
pairs(::NamedTuple) with 2 entries:
  :non_centered => 1.0
  :centered     => 5.34175e-19

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  elpd_mcse  elpd_diff  elpd_diff_mcse  weight    p   ⋯
 non_centered     1   -31        1.4       0              0.0      0.52  0.9   ⋯
 centered         2   -31        1.4       0.06           0.067    0.48  0.9   ⋯
                                                                1 column omitted

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 ^{[YaoVehtari2018]} for details.

See also: AbstractModelWeightsMethod, compare

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     => 5.34175e-19
  :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     => 0.483723
  :non_centered => 0.516277

abstract type AbstractModelWeightsMethod

An abstract type representing methods for computing model weights.

Subtypes implement model_weights(method, elpd_results).

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+)^{[YaoVehtari2018]}.

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.

See also: Stacking

struct PseudoBMA <: AbstractModelWeightsMethod

Model weighting method using pseudo Bayesian Model Averaging (pseudo-BMA) and Akaike-type weighting.

PseudoBMA(; regularize=false)
PseudoBMA(regularize)

Construct the method with optional regularization of the weights using the standard error of the ELPD estimate.

See also: Stacking

struct Stacking{O<:Optim.AbstractOptimizer} <: AbstractModelWeightsMethod

Model weighting using stacking of predictive distributions^{[YaoVehtari2018]}.

Stacking(; optimizer=Optim.LBFGS(), options=Optim.Options()
Stacking(optimizer[, options])

Construct the method, optionally customizing the optimization.

• optimizer::Optim.AbstractOptimizer: The optimizer to use for the optimization of the weights. The optimizer must support projected gradient optimization via a manifold field.

• options::Optim.Options: The Optim options to use for the optimization of the weights.

See also: BootstrappedPseudoBMA

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 $[0, 1]$.^[Gabry2019]

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} │
├───────────────────────────────┴──────────────────────────────────────── dims ┐
  ↓ school Categorical{String} [Choate, Deerfield, …, St. Paul's, Mt. Hermon] Unordered
└──────────────────────────────────────────────────────────────────────────────┘
 "Choate"            0.943511
 "Deerfield"         0.63797
 "Phillips Andover"  0.316697
 "Phillips Exeter"   0.582252
 "Hotchkiss"         0.295321
 "Lawrenceville"     0.403318
 "St. Paul's"        0.902508
 "Mt. Hermon"        0.655275

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} │
├───────────────────────────────┴──────────────────────────────────────── dims ┐
  ↓ school Categorical{String} [Choate, Deerfield, …, St. Paul's, Mt. Hermon] Unordered
└──────────────────────────────────────────────────────────────────────────────┘
 "Choate"            0.873577
 "Deerfield"         0.243686
 "Phillips Andover"  0.357563
 "Phillips Exeter"   0.149908
 "Hotchkiss"         0.435094
 "Lawrenceville"     0.220627
 "St. Paul's"        0.775086
 "Mt. Hermon"        0.296706

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

$R²$ for linear Bayesian regression models.^{[GelmanGoodrich2019]}

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

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.