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* Add mcse_sbm
* Update description of `estimator`
* Add specialized estimators for mean, std, and quantile
* Remove vector methods, defaulting to sbm
* Update docstring
* Fix bugs
* Update docstrings
* Update docstring
* Move helper functions to own file
* Rearrange tests
* Update mcse tests
* Export mcse_sbm
* Increment minor version number with DEV suffix
* Increment docs and tests version numbers
* Add additional citation
* Update diagnostics to use new mcse
* Increase tolerance of mcse tests
* Increase tolerance more
* Add mcse_sbm to docs
* Skip high autocorrelation tests for mcse_sbm
* Note underestimation for SBM
* Update src/mcse.jl
* Don't enforce type
* Document kwargs passed to mcse
* Cross-link mcse and ess_rhat docstrings
* Document derivation of mcse for std
* Test type-inferrability of ess_rhat
* Make sure ess_rhat for quantiles not promoted
* Make sure ess_rhat for median type-inferrable
* Implement specific method for median
* Return missing if any are missing
* Add mcse tests
* Decrease the number of checks
* Make ESS/MCSE for median with with Union{Missing,Real}
* Make _fold_around_median type-inferrable
* Increase tolerance for exhaustive tests
* Fix _fold_around_median
* Fix count of checks
* Increase the number of draws
improves the quality of the estimates and reduces random failures
* Apply suggestions from code review
Co-authored-by: David Widmann <[email protected]>
* Make sure heideldiag and gewekediag preserve input type
* Consistently use first and last for ess_rhat
* Copy comment to _fold_around_median
* Make mcse_sbm an internal function
* Update tests
Co-authored-by: David Widmann <[email protected]>- Loading branch information
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,72 +1,129 @@ | ||
| const normcdf1 = 0.8413447460685429 # StatsFuns.normcdf(1) | ||
| const normcdfn1 = 0.15865525393145705 # StatsFuns.normcdf(-1) | ||
|
|
||
| """ | ||
| mcse(x::AbstractVector{<:Real}; method::Symbol=:imse, kwargs...) | ||
| mcse(estimator, samples::AbstractArray{<:Union{Missing,Real}}; kwargs...) | ||
| Estimate the Monte Carlo standard errors (MCSE) of the `estimator` applied to `samples` of | ||
| shape `(draws, chains, parameters)`. | ||
| See also: [`ess_rhat`](@ref) | ||
| ## Estimators | ||
| `estimator` must accept a vector of the same `eltype` as `samples` and return a real estimate. | ||
| Compute the Monte Carlo standard error (MCSE) of samples `x`. | ||
| The optional argument `method` describes how the errors are estimated. Possible options are: | ||
| For the following estimators, the effective sample size [`ess_rhat`](@ref) and an estimate | ||
| of the asymptotic variance are used to compute the MCSE, and `kwargs` are forwarded to | ||
| `ess_rhat`: | ||
| - `Statistics.mean` | ||
| - `Statistics.median` | ||
| - `Statistics.std` | ||
| - `Base.Fix2(Statistics.quantile, p::Real)` | ||
| - `:bm` for batch means [^Glynn1991] | ||
| - `:imse` initial monotone sequence estimator [^Geyer1992] | ||
| - `:ipse` initial positive sequence estimator [^Geyer1992] | ||
| For other estimators, the subsampling bootstrap method (SBM)[^FlegalJones2011][^Flegal2012] | ||
| is used as a fallback, and the only accepted `kwargs` are `batch_size`, which indicates the | ||
| size of the overlapping batches used to estimate the MCSE, defaulting to | ||
| `floor(Int, sqrt(draws * chains))`. Note that SBM tends to underestimate the MCSE, | ||
| especially for highly autocorrelated chains. One should verify that autocorrelation is low | ||
| by checking the bulk- and tail-[`ess_rhat`](@ref) values. | ||
| [^Glynn1991]: Glynn, P. W., & Whitt, W. (1991). Estimating the asymptotic variance with batch means. Operations Research Letters, 10(8), 431-435. | ||
| [^FlegalJones2011]: Flegal JM, Jones GL. (2011) Implementing MCMC: estimating with confidence. | ||
| Handbook of Markov Chain Monte Carlo. pp. 175-97. | ||
| [pdf](http://faculty.ucr.edu/~jflegal/EstimatingWithConfidence.pdf) | ||
| [^Flegal2012]: Flegal JM. (2012) Applicability of subsampling bootstrap methods in Markov chain Monte Carlo. | ||
| Monte Carlo and Quasi-Monte Carlo Methods 2010. pp. 363-72. | ||
| doi: [10.1007/978-3-642-27440-4_18](https://doi.org/10.1007/978-3-642-27440-4_18) | ||
| [^Geyer1992]: Geyer, C. J. (1992). Practical Markov Chain Monte Carlo. Statistical Science, 473-483. | ||
| """ | ||
| function mcse(x::AbstractVector{<:Real}; method::Symbol=:imse, kwargs...) | ||
| return if method === :bm | ||
| mcse_bm(x; kwargs...) | ||
| elseif method === :imse | ||
| mcse_imse(x) | ||
| elseif method === :ipse | ||
| mcse_ipse(x) | ||
| else | ||
| throw(ArgumentError("unsupported MCSE method $method")) | ||
| mcse(f, x::AbstractArray{<:Union{Missing,Real},3}; kwargs...) = _mcse_sbm(f, x; kwargs...) | ||
| function mcse( | ||
| ::typeof(Statistics.mean), samples::AbstractArray{<:Union{Missing,Real},3}; kwargs... | ||
| ) | ||
| S = first(ess_rhat(Statistics.mean, samples; kwargs...)) | ||
| return dropdims(Statistics.std(samples; dims=(1, 2)); dims=(1, 2)) ./ sqrt.(S) | ||
| end | ||
| function mcse( | ||
| ::typeof(Statistics.std), samples::AbstractArray{<:Union{Missing,Real},3}; kwargs... | ||
| ) | ||
| x = (samples .- Statistics.mean(samples; dims=(1, 2))) .^ 2 # expectand proxy | ||
| S = first(ess_rhat(Statistics.mean, x; kwargs...)) | ||
| # asymptotic variance of sample variance estimate is Var[var] = E[μ₄] - E[var]², | ||
| # where μ₄ is the 4th central moment | ||
| # by the delta method, Var[std] = Var[var] / 4E[var] = (E[μ₄]/E[var] - E[var])/4, | ||
| # See e.g. Chapter 3 of Van der Vaart, AW. (200) Asymptotic statistics. Vol. 3. | ||
| mean_var = dropdims(Statistics.mean(x; dims=(1, 2)); dims=(1, 2)) | ||
| mean_moment4 = dropdims(Statistics.mean(abs2, x; dims=(1, 2)); dims=(1, 2)) | ||
| return @. sqrt((mean_moment4 / mean_var - mean_var) / S) / 2 | ||
| end | ||
| function mcse( | ||
| f::Base.Fix2{typeof(Statistics.quantile),<:Real}, | ||
| samples::AbstractArray{<:Union{Missing,Real},3}; | ||
| kwargs..., | ||
| ) | ||
| p = f.x | ||
| S = first(ess_rhat(f, samples; kwargs...)) | ||
| T = eltype(S) | ||
| R = promote_type(eltype(samples), typeof(oneunit(eltype(samples)) / sqrt(oneunit(T)))) | ||
| values = similar(S, R) | ||
| for (i, xi, Si) in zip(eachindex(values), eachslice(samples; dims=3), S) | ||
| values[i] = _mcse_quantile(vec(xi), p, Si) | ||
| end | ||
| return values | ||
| end | ||
| function mcse( | ||
| ::typeof(Statistics.median), samples::AbstractArray{<:Union{Missing,Real},3}; kwargs... | ||
| ) | ||
| S = first(ess_rhat(Statistics.median, samples; kwargs...)) | ||
| T = eltype(S) | ||
| R = promote_type(eltype(samples), typeof(oneunit(eltype(samples)) / sqrt(oneunit(T)))) | ||
| values = similar(S, R) | ||
| for (i, xi, Si) in zip(eachindex(values), eachslice(samples; dims=3), S) | ||
| values[i] = _mcse_quantile(vec(xi), 1//2, Si) | ||
| end | ||
| return values | ||
| end | ||
|
|
||
| function mcse_bm(x::AbstractVector{<:Real}; size::Int=floor(Int, sqrt(length(x)))) | ||
| n = length(x) | ||
| m = min(div(n, 2), size) | ||
| m == size || @warn "batch size was reduced to $m" | ||
| mcse = StatsBase.sem(Statistics.mean(@view(x[(i + 1):(i + m)])) for i in 0:m:(n - m)) | ||
| return mcse | ||
| function _mcse_quantile(x, p, Seff) | ||
| Seff === missing && return missing | ||
| S = length(x) | ||
| # quantile error distribution is asymptotically normal; estimate σ (mcse) with 2 | ||
| # quadrature points: xl and xu, chosen as quantiles so that xu - xl = 2σ | ||
| # compute quantiles of error distribution in probability space (i.e. quantiles passed through CDF) | ||
| # Beta(α,β) is the approximate error distribution of quantile estimates | ||
| α = Seff * p + 1 | ||
| β = Seff * (1 - p) + 1 | ||
| prob_x_upper = StatsFuns.betainvcdf(α, β, normcdf1) | ||
| prob_x_lower = StatsFuns.betainvcdf(α, β, normcdfn1) | ||
| # use inverse ECDF to get quantiles in quantile (x) space | ||
| l = max(floor(Int, prob_x_lower * S), 1) | ||
| u = min(ceil(Int, prob_x_upper * S), S) | ||
| iperm = partialsortperm(x, l:u) # sort as little of x as possible | ||
| xl = x[first(iperm)] | ||
| xu = x[last(iperm)] | ||
| # estimate mcse from quantiles | ||
| return (xu - xl) / 2 | ||
| end | ||
|
|
||
| function mcse_imse(x::AbstractVector{<:Real}) | ||
| n = length(x) | ||
| lags = [0, 1] | ||
| ghat = StatsBase.autocov(x, lags) | ||
| Ghat = sum(ghat) | ||
| @inbounds value = Ghat + ghat[2] | ||
| @inbounds for i in 2:2:(n - 2) | ||
| lags[1] = i | ||
| lags[2] = i + 1 | ||
| StatsBase.autocov!(ghat, x, lags) | ||
| Ghat = min(Ghat, sum(ghat)) | ||
| Ghat > 0 || break | ||
| value += 2 * Ghat | ||
| function _mcse_sbm( | ||
| f, | ||
| x::AbstractArray{<:Union{Missing,Real},3}; | ||
| batch_size::Int=floor(Int, sqrt(size(x, 1) * size(x, 2))), | ||
| ) | ||
| T = promote_type(eltype(x), typeof(zero(eltype(x)) / 1)) | ||
| values = similar(x, T, (axes(x, 3),)) | ||
| for (i, xi) in zip(eachindex(values), eachslice(x; dims=3)) | ||
| values[i] = _mcse_sbm(f, vec(xi), batch_size) | ||
| end | ||
|
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||
| mcse = sqrt(value / n) | ||
|
|
||
| return mcse | ||
| return values | ||
| end | ||
|
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||
| function mcse_ipse(x::AbstractVector{<:Real}) | ||
| function _mcse_sbm(f, x, batch_size) | ||
| any(x -> x === missing, x) && return missing | ||
| n = length(x) | ||
| lags = [0, 1] | ||
| ghat = StatsBase.autocov(x, lags) | ||
| @inbounds value = ghat[1] + 2 * ghat[2] | ||
| @inbounds for i in 2:2:(n - 2) | ||
| lags[1] = i | ||
| lags[2] = i + 1 | ||
| StatsBase.autocov!(ghat, x, lags) | ||
| Ghat = sum(ghat) | ||
| Ghat > 0 || break | ||
| value += 2 * Ghat | ||
| end | ||
|
|
||
| mcse = sqrt(value / n) | ||
|
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||
| return mcse | ||
| i1 = firstindex(x) | ||
| v = Statistics.var( | ||
| f(view(x, i:(i + batch_size - 1))) for i in i1:(i1 + n - batch_size); | ||
| corrected=false, | ||
| ) | ||
| return sqrt(v * (batch_size//n)) | ||
| end |
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