remove SDE part (transfered to TuringLang#445)

Author: yebai

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

@@ -364,83 +364,3 @@ sample(model_sensealg, NUTS(;adtype=AutoZygote()), 1000; progress=false)
 ```
 
 For more examples of adjoint usage on large parameter models, consult the [DiffEqFlux documentation](https://diffeqflux.sciml.ai/dev/).
-
-## Inference of a Stochastic Differential Equation
-
-A [Stochastic Differential Equation (SDE)](https://diffeq.sciml.ai/stable/tutorials/sde_example/) is a differential equation that has a stochastic (noise) term in the expression of the derivatives.
-Here we fit a stochastic version of the Lokta-Volterra system.
-
-We use a quasi-likelihood approach in which all trajectories of a solution are compared instead of a reduction such as mean, this increases the robustness of fitting and makes the likelihood more identifiable.
-We use SOSRI to solve the equation.
-
-```{julia}
-u0 = [1.0, 1.0]
-tspan = (0.0, 10.0)
-function multiplicative_noise!(du, u, p, t)
-    x, y = u
-    du[1] = p[5] * x
-    return du[2] = p[6] * y
-end
-p = [1.5, 1.0, 3.0, 1.0, 0.1, 0.1]
-
-function lotka_volterra!(du, u, p, t)
-    x, y = u
-    α, β, γ, δ = p
-    du[1] = dx = α * x - β * x * y
-    return du[2] = dy = δ * x * y - γ * y
-end
-
-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))
-```
-
-```{julia}
-@model function fitlv_sde(data, prob)
-    # Prior distributions.
-    σ ~ InverseGamma(2, 3)
-    α ~ truncated(Normal(1.3, 0.5); lower=0.5, upper=2.5)
-    β ~ truncated(Normal(1.2, 0.25); lower=0.5, upper=2)
-    γ ~ truncated(Normal(3.2, 0.25); lower=2.2, upper=4)
-    δ ~ truncated(Normal(1.2, 0.25); lower=0.5, upper=2)
-    ϕ1 ~ truncated(Normal(0.12, 0.3); lower=0.05, upper=0.25)
-    ϕ2 ~ truncated(Normal(0.12, 0.3); lower=0.05, upper=0.25)
-
-    # Simulate stochastic Lotka-Volterra model.
-    p = [α, β, γ, δ, ϕ1, ϕ2]
-    predicted = solve(prob, SOSRI(); p=p, saveat=0.1)
-
-    # Early exit if simulation could not be computed successfully.
-    if predicted.retcode !== :Success
-        Turing.@addlogprob! -Inf
-        return nothing
-    end
-
-    # Observations.
-    for i in 1:length(predicted)
-        data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
-    end
-
-    return nothing
-end;
-```
-
-The probabilistic nature of the SDE solution makes the likelihood function noisy which poses a challenge for NUTS since the gradient is changing with every calculation.
-Therefore we use NUTS with a low target acceptance rate of `0.25` and specify a set of initial parameters.
-SGHMC might be a more suitable algorithm to be used here.
-
-```{julia}
-model_sde = fitlv_sde(odedata, prob_sde)
-
-setadbackend(:forwarddiff)
-chain_sde = sample(
-    model_sde,
-    NUTS(0.25),
-    5000;
-    initial_params=[1.5, 1.3, 1.2, 2.7, 1.2, 0.12, 0.12],
-    progress=false,
-)
-plot(chain_sde)
-```