Rewrite Turing.jl example

Fixes #118

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
+```