Hi everyone,

I’m new to Julia and would like to use the Turing package for Bayesian inference on high dimensional ordinary differential equations. With this in mind, efficiency is key.

To get started I implemented a linear toy model with two variables based on lotka volterra tutorial provided here: Bayesian Estimation of Differential Equations

I also tried to work with with the DataFrames packages in a long dataformat as this is how I will import my own data at a later time point.

Unfortunately, the sampler does not identify the correct parameters and neither does it. Furthermore, the sampling is very slow.

What am I doing wrong / what could be improved? Again, an efficient implementation is key as I’d like to potentially extent this code to a system of up to 30 variables (with say a 100 parameters).

I first define the ODEs and generate fake data:

```
using Turing, Distributions, DifferentialEquations, MCMCChains, Plots, StatsPlots, Random, DataFrames, CSV
using Statistics: mean, std
using Random:seed!
seed!(123)
### Define the ODEs
function linear_sdm(du,u,p,t)
x, y = u
α, β, γ, δ = p
du[1] = α * x + β * y
du[2] = γ * x + δ * y
end
### Define ``real'' parameter values
p_mean = [0.025, -0.0075, -0.05, 0.03]
randomness_size = 0.2
### Create initial conditions for simulated individuals
u0_1 = [-0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1.0];
u0_2 = [1, 0.8, 0.6, 0.4, 0.2, 0.2, 0.4, 0.6, 0.8, 1.0];
t_span = (0.0, 48.0)
N = 10
dt = 3
add_missings = false
df = [] # Allocate memory for this dataframe
prob = ODEProblem(linear_sdm, Array([0,0]), t_span, p_mean)
for i = 1:N
#println(i)
u0_i = [u0_1[i], u0_2[i]]
prob1 = remake(prob, u0=u0_i)
sol = solve(prob1, Tsit5())
#display(plot(sol)) # Plot the set of ODEs
### Generate data from ODEs
sol1 = solve(prob1, Tsit5(), saveat=dt)
odedata = Array(sol1) + randomness_size * randn(size(Array(sol1)))
if add_missings # Generate missings in the simulated data
odedata[1,8] = NaN
odedata[1,3] = NaN
odedata[1,9] = NaN
odedata[1,13] = NaN
odedata[2,1] = NaN
odedata[2,6] = NaN
odedata[2,12] = NaN
odedata[2,16] = NaN
end
# Store in dataframe odedata
df_i = DataFrame(ADAS13 = odedata[1,:],
FDG = odedata[2,:],
Time = sol1.t,
RID = repeat([i],
length(sol1.t)))
if i == 1
df = deepcopy(df_i)
else
df = reduce(vcat, [df, df_i]) # Join dataframe with original dataframe
end
end
```

Then I run the model, which gives very poor results

```
### Run Bayesian model
Turing.setadbackend(:forwarddiff)
using Suppressor
@suppress_err begin # Silence warnings
@model function sdm_model(data, prob, N, t_span, dt)
σ ~ InverseGamma(0.2, 0.05)
α ~ Normal(0.1, 0.05)
β ~ Normal(-0.1, 0.05)
γ ~ Normal(-0.1, 0.05)
δ ~ Normal(0.1, 0.05)
p = [α,β,γ,δ]
for k in 1:N
u0_i = Array(df[in([k]).(df.RID), ["ADAS13", "FDG"]][1,:]) # Set baseline
prob_i = remake(prob, p=p, u0=u0_i)
predicted = solve(prob_i, Tsit5(), saveat=dt)
for i = 2:length(predicted) # where {T <: Real}
data = Array(df[in([k]).(df.RID), ["ADAS13", "FDG"]])[i,:]
data ~ MvNormal(predicted[i], σ) #
end
end
end
model = sdm_model(df, prob, N, t_span, dt)
chain = sample(model, NUTS(.65), MCMCThreads(), 1000, 3, progress=true)
end
```

Any suggestions? Thank you