Even though the code was copied… it is shown as below:

```
using Turing
using DifferentialEquations
using StatsPlots
using LinearAlgebra
# Set a seed for reproducibility.
using Random
Random.seed!(14);
# Define Lotka-Volterra model.
function lotka_volterra(du, u, p, t)
# Model parameters.
α, β, γ, δ = p
# Current state.
x, y = u
# Evaluate differential equations.
du[1] = (α - β * y) * x # prey
du[2] = (δ * x - γ) * y # predator
return nothing
end
# Define initial-value problem.
u0 = [1.0, 1.0]
p = [1.5, 1.0, 3.0, 1.0]
tspan = (0.0, 10.0)
prob = ODEProblem(lotka_volterra, u0, tspan, p)
# Plot simulation.
plot(solve(prob, Tsit5()))
sol = solve(prob, Tsit5(); saveat=0.1)
odedata = Array(sol) + 0.8 * randn(size(Array(sol)))
# Plot simulation and noisy observations.
plot(sol; alpha=0.3)
scatter!(sol.t, odedata'; color=[1 2], label="")
@model function fitlv(data, prob)
# Prior distributions.
σ ~ InverseGamma(2, 3)
α ~ truncated(Normal(1.5, 0.5), 0.5, 2.5)
β ~ truncated(Normal(1.2, 0.5), 0, 2)
γ ~ truncated(Normal(3.0, 0.5), 1, 4)
δ ~ truncated(Normal(1.0, 0.5), 0, 2)
# Simulate Lotka-Volterra model.
p = [α, β, γ, δ]
predicted = solve(prob, Tsit5(); p=p, saveat=0.1)
# Observations.
for i in 1:length(predicted)
data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
end
return nothing
end
model = fitlv(odedata, prob)
# Sample 3 independent chains with forward-mode automatic differentiation (the default).
chain = sample(model, NUTS(0.65), MCMCSerial(), 1000, 3; progress=true)
plot(chain)
```