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f(*AE=<}a21@?87P8thwFzak@{AYLN&+V8&sIGO+0 literal 0 HcmV?d00001 diff --git a/docs/src/exemples/turing.md b/docs/src/exemples/turing.md index 198b807b..a65019b7 100644 --- a/docs/src/exemples/turing.md +++ b/docs/src/exemples/turing.md @@ -1,14 +1,9 @@ # Bayesian inference with `Turing.jl` -The compatibility with the broader `Distributions.jl`'s API allows a lot of interactions with the broader ecosystem. One of the firsts examples that we discovered was the possibility to do Bayesian inference of model parameters with [`Turing.jl`](https://turing.ml/). +The compatibility with `Distributions.jl`'s API allows a lot of interactions with the broader ecosystem. One of the firsts examples that we discovered was the possibility to do Bayesian inference of model parameters (copula included) with [`Turing.jl`](https://turing.ml/). -Consider that we have a model with : - -- A given parametric dependence structure, say a Clayton Copula. -- A few given parametric marginals. - -Then we can use Turing's `@addlogprob!` to compute the loglikelihood of our model and maximize it around the parameters alongside the chain as follows: +Consider that we have a given model with a certain copula and certain marginals, all having parameters to be fitted. Then we can use Turing's `@addlogprob!` to compute the loglikelihood of our model and maximize it around the parameters alongside the chain as follows: ```julia using Copulas @@ -23,7 +18,7 @@ true_θ₁ = 1 true_θ₂ = 3 true_θ₃ = 2 D = SklarDist(ClaytonCopula(3,true_θ), (Exponential(true_θ₁), Pareto(true_θ₂), Exponential(true_θ₃))) -draws = rand(D, 200) +draws = rand(D, 2000) @model function copula(X) # Priors @@ -45,12 +40,51 @@ end sampler = NUTS() # MH() works too chain = sample(copula(draws), sampler, MCMCThreads(), 100, 4) -plot(chain) ``` -This kind of model could be used to do a lot of stuff. here we use it for a parametric estimation of a sampling model, but it could be used for other things such as, e.g., a Bayesian regression with an error structure given by a parametric copula. +Note that we truncated the θ parameter at -1/3 and not 0 as the `ClaytonCopula` can handle negative dependence structures. We only ran 100 steps for efficiency reasons, you can increase this number easily if needed. The upper code outputs summary of the chain : + +Summary Statistics: + +| parameters | true value | mean | std | mcse | ess_bulk | ess_tail | rhat | ess_per_sec | +|-------------|------------|-----------|----------|----------|-----------|-----------|----------|--------------| +| Symbol | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | +| θ | 7.0 | 6.9319 | 0.1150 | 0.0067 | 291.8858 | 267.1353 | 1.0061 | 0.7238 | +| θ₁ | 1.0 | 0.9954 | 0.0209 | 0.0019 | 116.6941 | 94.3070 | 1.0347 | 0.2894 | +| θ₂ | 3.0 | 2.9839 | 0.0639 | 0.0062 | 108.9185 | 105.5284 | 1.0390 | 0.2701 | +| θ₃ | 2.0 | 2.0055 | 0.0418 | 0.0039 | 114.7324 | 109.5396 | 1.0328 | 0.2845 | + +Quantiles: + +| parameters | 2.5% | 25.0% | 50.0% | 75.0% | 97.5% | +|--------------|----------|----------|----------|----------|-------- | +| Symbol | Float64 | Float64 | Float64 | Float64 | Float64 | +| θ | 6.7286 | 6.8438 | 6.9330 | 7.0150 | 7.1436 | +| θ₁ | 0.9555 | 0.9818 | 0.9953 | 1.0093 | 1.0386 | +| θ₂ | 2.8606 | 2.9426 | 2.9859 | 3.0196 | 3.1186 | +| θ₃ | 1.9254 | 1.9758 | 2.0056 | 2.0336 | 2.0923 | -Note that we truncated the θ parameter at -1/3 and not 0 as the `ClaytonCopula` can handle negative dependence structures. +And then `plot(chain)` produces the following plot: + +![Turing results](assets/turing_plot.png) + +Similar approaches could be used to fit many other dependence structures in a Bayesian settings. The upper example showcases a parametric estimation of a sampling model, but a Bayesian regression with an error structure given by a parametric copula is as easy to implement. + +This was run on the following environment: + +```julia +julia> versioninfo() +Julia Version 1.10.0 +Commit 3120989f39 (2023-12-25 18:01 UTC) +[...] + +(env) pkg> status + [ae264745] Copulas v0.1.20 + [31c24e10] Distributions v0.25.107 + [f3b207a7] StatsPlots v0.15.6 + [fce5fe82] Turing v0.30.3 + [9a3f8284] Random +```