Hi all

I have a medium sized ODE model with four estimated parameters, and I have problems with chain convergence when using Turing (NUTS) for inference. Depending on the initial draws, some chains converge, and some chains get stuck mostly at the bounds of some of the parameters. Extending the burn-in phase up to 6000 iters does not change anything, the stuck chains just remain stuck. I have observed this also with my previous model. Is there a way to mitigate this behaviour, @yebai @torfjelde ? Could it be related to the Bijector, see also this post?

```
# Household stuff
using DifferentialEquations
using CSV, DataFrames
using StatsPlots
using Turing
using Distributions
import Random
using LabelledArrays
using UnPack
using DelimitedFiles
using Serialization
using BenchmarkTools
using ModelingToolkit
Random.seed!(36541)
# ODE model: simple SIR model with seasonally forced contact rate
# Model structure: (agegroups=4 * level=3 * state=4)
function MSIR_als!(du, u, p, t)
## params
β = p[2]
η = p[3]
φ = p[4]
σ = p[5]
ω = 1.0 / p[6]
μ = 1.0 / p[7]
b = p[8]
nₘ = p[9] # number of women in child-bearing age group
d = p[10:13]
ϵ = p[14:17]
n = p[18:21]
c = reshape(p[22:end], (4,4))
## Create views
# current states
N = @view u[:,:,1:end-1]
M = @view u[:,:,1]
S = @view u[:,:,2]
I = @view u[:,:,3]
R = @view u[:,:,4]
## differentials
dN = @view du[:,:,1:end-1]
dM = @view du[:,:,1]
dS = @view du[:,:,2]
dI = @view du[:,:,3]
dR = @view du[:,:,4]
dC = @view du[:,:,5]
# Transitions
## Ageing (includes births)
ageing_out = zeros(promote_type(eltype(u), eltype(p)), size(N))
@. ageing_out = ϵ * N
ageing_in = zeros(promote_type(eltype(u), eltype(p)), size(N))
ageing_in[2:end,:,:] += ageing_out[1:end-1,:,:] # ageing in from age age groups below
## Births
pₘ = sum(R[3,:]) / nₘ # calculate the proportion of women of child-bearing age who are in the R compartments
ageing_in[1,1,1] = b * pₘ # births into M compartment
ageing_in[1,1,2] = b * (1.0-pₘ) # births into S compartment
## Deaths (calculated from births and transitions in and out, to maintain a stable N per age group)
props_a = zeros(promote_type(eltype(u), eltype(p)), size(N))
@. props_a = N / n # calculate proportions of n in each state
deaths = zeros(promote_type(eltype(u), eltype(p)), size(N)) # distribute the age-specific deaths among the states
@. deaths = d * props_a
## FOI
# effective per contact transmission probability
βeff = β * (1.0 + η * cos(2.0 * π * (t-φ) / 365.0))
# total number of infectious agents by age group
I_tot = sum(I, dims=2)
λ = βeff .* vec(sum(c .* I_tot, dims=1))
## infections
infection = .*(λ, S)
## clearance
clearance = .*(σ, I)
## waning immunity
waning_out_M = .*(μ, M) # waning out of M into S
waning_out_R = zeros(promote_type(eltype(u), eltype(p)), size(R)) # waning out of R
@. waning_out_R = *(ω, R)
waning_from_R = zeros(promote_type(eltype(u), eltype(p)), size(S)) # waning from R into S
waning_from_R[:,2:end] += waning_out_R[:,1:end-1]
waning_from_R[:,end] += waning_out_R[:,end]
# Equations
## births
## transitions between age groups
@. dN = ageing_in - ageing_out - deaths
## transitions between states
@. dM = - waning_out_M
@. dS = waning_out_M + waning_from_R - infection
@. dI = infection - clearance
@. dR = clearance - waning_out_R
## cumulative incidence
@. dC = infection
return nothing
end
# Input
# demographics
pop = 1_000_000
t_y = [10.0, 10.0, 20.0, 30.0] # years spent in each age group
lifespan = sum(t_y)
props = t_y ./ lifespan
n = pop .* props
# Parameters
ψ = 0.05
β = 0.15
η = 0.05
φ = 180
σ = 1.0 / 5.0
ω = 365.0
μ = 180.0
b = 40.0 # daily births
nₘ = n[3]/2.0 # total women of child-bearing age
n_age = 4
n_level = 3
# ageing vector
ϵ = 1.0 ./ (365.0 .* props .* lifespan)
# calculate theoretical daily deaths in each age group from births and transitions in/out to achieve net 0 change
n_out = n .* ϵ
d = vcat(b, n_out[1:end-1]) - n_out
# contacts
c = [1.4e-6 5.6e-6 2.275e-6 5.44444e-7
5.6e-6 1.05e-5 2.8875e-6 1.08889e-6
9.1e-6 1.155e-5 5.25e-6 2.13889e-6
4.9e-6 9.8e-6 4.8125e-6 1.86667e-6]
# parameter vector
p = vcat(ψ, β, η, φ, σ, ω, μ, b, nₘ, d, ϵ, n, vec(c))
# Inits
u0 = [hcat(zeros(4), zeros(4), zeros(4)) ;;; # M
hcat(pop .* props .- [0.0, 0.0, 10.0, 0.0], zeros(4), zeros(4)) ;;; # S
hcat([0.0, 0.0, 10.0, 0.0], zeros(4), zeros(4)) ;;; # I
hcat(zeros(4), zeros(4), zeros(4)) ;;; # R
hcat(zeros(4), zeros(4), zeros(4))]
# Solver settings
tmin = 0.0
tmax = 365.0 #5.0*365.0
tspan = (tmin, tmax)
solvsettings = (abstol = 1.0e-8,
reltol = 1.0e-8,
saveat = 1.0,
solver = Tsit5())
# Initiate ODE problem
problem = ODEProblem(MSIR_als!, u0, tspan, p)
problem_mtk = ODEProblem(modelingtoolkitize(problem), [], tspan, jac=true)
sol = solve(problem,
solvsettings.solver,
abstol=solvsettings.abstol,
reltol=solvsettings.reltol,
isoutofdomain = (u,p,t)->any(<(0),u),
saveat=solvsettings.saveat)
sol_array = Array(sol);
# sum over levels (data cannot distinguish between levels)
inc = dropdims(sum(sol_array, dims=2), dims=2)
inc = diff(inc[:,end,:], dims=2)
#plot(inc')
# observation model
function NegativeBinomial2(ψ, incidence)
p = 1.0/(1.0 + ψ*incidence)
r = 1.0/ψ
return NegativeBinomial(r, p)
end
# Fake some data from model
data = rand.(NegativeBinomial2.(ψ, inc))
#scatter(data',legend = false);
#plot!(inc', legend = false)
# Fit model to fake data
# Set up as Turing model
Turing.setadbackend(:forwarddiff)
@model function prior()
ψ ~ Uniform(1e-6, 0.1) #Beta(1.1, 50.0)
β ~ Uniform(1e-6, 1.0) #Beta(1.5, 5.0) #
η ~ Beta(1.5, 10.0) #Uniform(0.0,1.0)
φ ~ Uniform(0.0,364.0)
return [ψ, β, η, φ]
end
# Define prior and fixed theta
theta_fix = p[5:end]
@model function turingmodel_mtk(prior, theta_fix, problem, n_age, n_level, solvsettings)
issuccess = true
# Sample prior parameters.
theta_est = @submodel prior()
# Update `p`.
#theta_fix=convert.(eltype(theta_est), theta_fix)
#promote_type(eltype(theta_fix), eltype(theta_est))
p = vcat(theta_est, theta_fix)
# Update problem and solve ODEs
problem_new = remake(problem; p=p)
sol = solve(problem_new,
solvsettings.solver,
abstol=solvsettings.abstol,
reltol=solvsettings.reltol,
isoutofdomain = (u,p,t)->any(<(0),u),
saveat=solvsettings.saveat);
# Return early if integration failed
issuccess &= (sol.retcode === :Success)
if !issuccess
Turing.@addlogprob! -Inf
return nothing
end
sol_array = Array(sol);
sol_array = sol_array[end-(n_age*n_level-1):end,:] # the last n_age *n_level rows correspond to the C compartment
sol_array = diff(sol_array, dims=2)
# sum over levels (data cannot distinguish between levels)
incidence = reshape(sol_array, (n_age,n_level,size(sol_array,2)))
incidence = dropdims(sum(incidence, dims=2), dims=2)
# avoid numerical instability issue
incidence = max.(eltype(incidence)(1e-9), incidence)
# likelihood
obs_ts ~ arraydist(@. NegativeBinomial2(theta_est[1], incidence))
return(; sol, incidence, p=p, obs_ts)
end
# Setup and condition model
model = turingmodel_mtk(prior,
theta_fix,
problem_mtk,
n_age,
n_level,
solvsettings) | (obs_ts = data,);
retval, issuccess = model();
# Fit
rng = Random.MersenneTwister(66)
chain = sample(model, NUTS(3000, 0.65), MCMCThreads(), 1000, 6, progress=true)
plot(chain)
```