Fixes to Bayes SDE notebook (#449)

* Updated retcode success check to the current version used by SciMLBase. Introduced new noisy observations that are better suited for the problem, and corrected the model to calculate the likelihood based on multiple trajectories rather than a single trajectory.

* Added more explanation on the likelihood calculation

---------

Co-authored-by: Gabriel Gress <[email protected]>

tutorials/10-bayesian-stochastic-differential-equations/index.qmd

@@ -17,6 +17,7 @@ Pkg.instantiate();
 ```{julia}
 using Turing
 using DifferentialEquations
+using DifferentialEquations.EnsembleAnalysis
 
 # Load StatsPlots for visualizations and diagnostics.
 using StatsPlots
@@ -117,6 +118,10 @@ prob_sde = SDEProblem(lotka_volterra!, multiplicative_noise!, u0, tspan, p)
 ensembleprob = EnsembleProblem(prob_sde)
 data = solve(ensembleprob, SOSRI(); saveat=0.1, trajectories=1000)
 plot(EnsembleSummary(data))
+
+# We generate new noisy observations based on the stochastic model for the parameter estimation tasks in this tutorial.
+# We create our observations by adding random normally distributed noise to the mean of the ensemble simulation. 
+sdedata = reduce(hcat, timeseries_steps_mean(data).u) + 0.8 * randn(size(reduce(hcat, timeseries_steps_mean(data).u)))
 ```
 
 ```{julia}
@@ -132,17 +137,24 @@ plot(EnsembleSummary(data))
 
     # Simulate stochastic Lotka-Volterra model.
     p = [α, β, γ, δ, ϕ1, ϕ2]
-    predicted = solve(prob, SOSRI(); p=p, saveat=0.1)
+    remake(prob, p = p)
+    ensembleprob = EnsembleProblem(prob)
+    predicted = solve(ensembleprob, SOSRI(); saveat=0.1, trajectories = 1000)
 
     # Early exit if simulation could not be computed successfully.
-    if predicted.retcode !== :Success
-        Turing.@addlogprob! -Inf
-        return nothing
+    for i in 1:length(predicted)
+        if !SciMLBase.successful_retcode(predicted[i])
+            Turing.@addlogprob! -Inf
+            return nothing
+        end
     end
 
     # Observations.
-    for i in 1:length(predicted)
-        data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
+    # We compute the likelihood for each trajectory of our simulation in order to better approximate the overall likelihood of our choice of parameters
+    for j in 1:length(predicted)
+        for i in 1:length(predicted[j])
+            data[:, i] ~ MvNormal(predicted[j][i], σ^2 * I)
+        end
     end
 
     return nothing
@@ -154,9 +166,8 @@ Therefore we use NUTS with a low target acceptance rate of `0.25` and specify a
 SGHMC might be a more suitable algorithm to be used here.
 
 ```{julia}
-model_sde = fitlv_sde(odedata, prob_sde)
+model_sde = fitlv_sde(sdedata, prob_sde)
 
-setadbackend(:forwarddiff)
 chain_sde = sample(
     model_sde,
     NUTS(0.25),