Lotka-Volterra Bayesian Parameter Estimation Benchmarks

Parameter Estimation of Lotka-Volterra Equation using DiffEqBayes.jl

using DiffEqBayes, StanSample, DynamicHMC, Turing

using Distributions, BenchmarkTools, StaticArrays
using OrdinaryDiffEq, RecursiveArrayTools, ParameterizedFunctions
using Plots, LinearAlgebra

"""Print elapsed sampling time and the Turing chain."""
function display_ess_per_sec(chain, elapsed)
    println("Elapsed time: $(round(elapsed; digits=2)) seconds")
    display(chain)
end

# Distributions requires a vector mean; sol[:, i] can be a scalar after ODE 7.
_mvn(u, σ) = MvNormal(
    let n = length(σ)
        μ = u isa AbstractVector ? collect(float.(u)) : [float(u)]
        length(μ) == n ? μ : fill(μ[1], n)
    end,
    Diagonal(σ .^ 2),
)

"""Extract and display Stan's internal timing from its CSV output files."""
function display_stan_timing(stan_result)
    sample_files = stan_result.model.sample_file
    for (chain_idx, f) in enumerate(sample_files)
        isfile(f) || continue
        lines = readlines(f)
        println("Chain $chain_idx timing (from Stan CSV):")
        for line in lines
            if startswith(line, "#") && occursin("Elapsed Time", line)
                println("  ", strip(line[2:end]))
            elseif startswith(line, "#") && occursin("seconds", line)
                println("  ", strip(line[2:end]))
            end
        end
    end
end

Main.var"##WeaveSandBox#277".display_stan_timing

gr(fmt = :png)

Plots.GRBackend()

Initializing the problem

f = @ode_def LotkaVolterraTest begin
    dx = a*x - b*x*y
    dy = -c*y + d*x*y
end a b c d

Main.var"##WeaveSandBox#277".LotkaVolterraTest{Main.var"##WeaveSandBox#277"
.var"###ParameterizedDiffEqFunction#279", Main.var"##WeaveSandBox#277".var"
###ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"###Par
ameterizedJacobianFunction#281", Nothing, Nothing, ModelingToolkitBase.Syst
em}(Main.var"##WeaveSandBox#277".var"##ParameterizedDiffEqFunction#279", Li
nearAlgebra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#27
7".var"##ParameterizedTGradFunction#280", Main.var"##WeaveSandBox#277".var"
##ParameterizedJacobianFunction#281", nothing, nothing, nothing, nothing, n
othing, nothing, nothing, [:x, :y], :t, nothing, Model ##Parameterized#278:
Equations (2):
  2 standard: see equations(##Parameterized#278)
Unknowns (2): see unknowns(##Parameterized#278)
  x(t)
  y(t)
Parameters (4): see parameters(##Parameterized#278)
  a
  b
  c
  d, nothing, nothing)

u0 = [1.0, 1.0]
tspan = (0.0, 10.0)
p = [1.5, 1.0, 3.0, 1.0]

4-element Vector{Float64}:
 1.5
 1.0
 3.0
 1.0

prob = ODEProblem(f, u0, tspan, p)
sol = solve(prob, Tsit5())

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 34-element Vector{Float64}:
  0.0
  0.0776084743154256
  0.2326451370670694
  0.42911851563726466
  0.679082199936808
  0.9444046279774128
  1.2674601918628516
  1.61929140093895
  1.9869755481702074
  2.2640903679981617
  ⋮
  7.5848624442719235
  7.978067891667038
  8.483164641366145
  8.719247691882519
  8.949206449510513
  9.200184762926114
  9.438028551201125
  9.711807820573478
 10.0
u: 34-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [1.0454942346944578, 0.8576684823217127]
 [1.1758715885890039, 0.6394595702308833]
 [1.4196809580265157, 0.456996261440507]
 [1.876719397626222, 0.32473342884607376]
 [2.5882501035146124, 0.26336255403957287]
 [3.86070908479701, 0.27944581878759067]
 [5.750813064347344, 0.5220073551361064]
 [6.814978696356639, 1.9177834056716259]
 [4.392997771045281, 4.194671543390715]
 ⋮
 [2.614251082502646, 0.26416954350041094]
 [4.241070648057705, 0.3051232653305123]
 [6.791121825691655, 1.1345253835487776]
 [6.2653749402951835, 2.741688595595249]
 [3.7807688120522114, 4.431164521488329]
 [1.8164214705303103, 4.064057991958692]
 [1.1465028256357626, 2.7911730348239576]
 [0.9557986528530791, 1.6235633163407794]
 [1.0337581330572256, 0.9063703732076024]

su0 = SA[1.0, 1.0]
sp = SA[1.5, 1.0, 3.0, 1.0]
sprob = ODEProblem{false, SciMLBase.FullSpecialize}(f, su0, tspan, sp)
sol = solve(sprob, Tsit5())

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 34-element Vector{Float64}:
  0.0
  0.0776084743154256
  0.2326451370670694
  0.42911851563726466
  0.679082199936808
  0.9444046279774128
  1.2674601918628516
  1.61929140093895
  1.9869755481702074
  2.2640903679981617
  ⋮
  7.5848624442719235
  7.978067891667038
  8.483164641366145
  8.719247691882519
  8.949206449510513
  9.200184762926114
  9.438028551201125
  9.711807820573478
 10.0
u: 34-element Vector{StaticArraysCore.SVector{2, Float64}}:
 [1.0, 1.0]
 [1.0454942346944578, 0.8576684823217127]
 [1.1758715885890039, 0.6394595702308833]
 [1.4196809580265157, 0.456996261440507]
 [1.876719397626222, 0.32473342884607376]
 [2.5882501035146124, 0.26336255403957287]
 [3.86070908479701, 0.27944581878759067]
 [5.750813064347344, 0.5220073551361064]
 [6.814978696356639, 1.9177834056716259]
 [4.392997771045281, 4.194671543390715]
 ⋮
 [2.614251082502646, 0.26416954350041094]
 [4.241070648057705, 0.3051232653305123]
 [6.791121825691655, 1.1345253835487776]
 [6.2653749402951835, 2.741688595595249]
 [3.7807688120522114, 4.431164521488329]
 [1.8164214705303103, 4.064057991958692]
 [1.1465028256357626, 2.7911730348239576]
 [0.9557986528530791, 1.6235633163407794]
 [1.0337581330572256, 0.9063703732076024]

We take the solution data obtained and add noise to it to obtain data for using in the Bayesian Inference of the parameters

t = collect(range(1, stop = 10, length = 10))
sig = 0.49
data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(2)) for i in 1:length(t)]))

2×10 Matrix{Float64}:
  2.52709   6.87513  1.21946   2.52779    …  3.96425   3.6102   1.55461
 -0.197236  3.18618  1.53794  -0.0843126     0.271838  5.11646  0.651598

Plots of the actual data and generated data

scatter(t, data[1, :], lab = "#prey (data)")
scatter!(t, data[2, :], lab = "#predator (data)")
plot!(sol)

priors = [truncated(Normal(1.5, 0.5), 0.5, 2.5), truncated(Normal(1.2, 0.5), 0, 2),
    truncated(Normal(3.0, 0.5), 1, 4), truncated(Normal(1.0, 0.5), 0, 2)]

4-element Vector{Distributions.Truncated{Distributions.Normal{Float64}, Dis
tributions.Continuous, Float64, Float64, Float64}}:
 Truncated(Distributions.Normal{Float64}(μ=1.5, σ=0.5); lower=0.5, upper=2.
5)
 Truncated(Distributions.Normal{Float64}(μ=1.2, σ=0.5); lower=0.0, upper=2.
0)
 Truncated(Distributions.Normal{Float64}(μ=3.0, σ=0.5); lower=1.0, upper=4.
0)
 Truncated(Distributions.Normal{Float64}(μ=1.0, σ=0.5); lower=0.0, upper=2.
0)

Stan.jl backend

Truncated normal priors are used to prevent Stan from stepping into negative values.

We use adapt_delta = 0.85 (slightly above Stan's default of 0.8) consistently across the Stan and Turing backends for a fair comparison. Stan infers a separate noise parameter (sigma) per data dimension via the vars specification.

bayesian_result_stan = @time stan_inference(
    prob, :rk45, t, data, priors; print_summary = false,
    sample_kwargs = Dict(:delta => 0.85, :num_samples => 10_000),
    vars = (DiffEqBayes.StanODEData(), InverseGamma(2, 3)))

73.207017 seconds (4.89 M allocations: 239.366 MiB, 0.05% gc time, 4.87% c
ompilation time)
108.849019 seconds (34.44 M allocations: 1.700 GiB, 0.27% gc time, 14.85% c
ompilation time: 2% of which was recompilation)
10000×6 DataFrame
   Row │ sigma1.1  sigma1.2  theta_1  theta_2   theta_3  theta_4
       │ Float64   Float64   Float64  Float64   Float64  Float64
───────┼──────────────────────────────────────────────────────────
     1 │ 0.617167  0.419661  1.49087  0.858174  3.14983  0.994666
     2 │ 0.80826   0.470242  1.48046  0.847884  2.87751  1.07767
     3 │ 0.619952  0.551316  1.49387  0.818614  2.91728  1.03369
     4 │ 0.43553   0.450968  1.51067  0.87034   3.03051  0.972755
     5 │ 0.5046    0.624565  1.53053  0.910454  2.88627  0.981822
     6 │ 0.451101  0.636735  1.53673  0.889494  2.90643  0.935123
     7 │ 0.507483  0.770303  1.48067  0.814267  3.02648  1.00119
     8 │ 0.363054  0.664224  1.47708  0.895267  3.06338  1.00459
   ⋮   │    ⋮         ⋮         ⋮        ⋮         ⋮        ⋮
  9994 │ 0.55499   0.360586  1.49714  0.868756  3.04229  1.007
  9995 │ 0.778506  0.545378  1.47982  0.807272  3.0583   1.02886
  9996 │ 0.752129  0.521911  1.48701  0.806011  3.05323  0.994323
  9997 │ 0.979764  0.629234  1.68043  1.00323   2.61308  0.878632
  9998 │ 0.89589   0.509082  1.69472  0.933742  2.55584  0.895639
  9999 │ 0.564077  0.591288  1.60822  0.996099  2.81094  0.917016
 10000 │ 0.536335  0.567878  1.6089   0.972531  2.80288  0.890477
                                                 9985 rows omitted

Stan's internal timing (excluding data serialization and CSV parsing):

display_stan_timing(bayesian_result_stan)

Chain 1 timing (from Stan CSV):
  Elapsed Time: 5.815 seconds (Warm-up)
  63.816 seconds (Sampling)
  69.631 seconds (Total)

Direct Turing.jl

We use a per-dimension noise model (matching Stan) with InverseGamma(2, 3) priors on each σ.

@model function fitlv(data, prob)
    # Prior distributions.
    σ ~ filldist(InverseGamma(2, 3), 2)
    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 = SA[α, β, γ, δ]
    _prob = remake(prob, p = p)
    predicted = solve(_prob, Tsit5(); saveat = t)

    # Observations.
    for i in axes(data, 2)
        data[:, i] ~ _mvn(predicted(t[i]), σ)
    end

    return nothing
end

model = fitlv(data, sprob)

# Warmup run to compile all code paths before timing
sample(model, Turing.NUTS(0.85), 10; progress = false)

elapsed_turing_direct = @elapsed chain = sample(model, Turing.NUTS(0.85), 10_000; progress = false)
chain

╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter  = 1001:11000                                                     
    │
│ → chain = 1:1                                                            
    │
│                                                                          
    │
│ Parameters (5) ── AbstractPPL.VarName                                    
    │
│  Vector{Float64}  σ (2,)                                                 
    │
│  Float64          α, β, γ, δ                                             
    │
│                                                                          
    │
│ Extras (14)                                                              
    │
│  Int64    n_steps, tree_depth                                            
    │
│  Bool     is_accept, numerical_error                                     
    │
│  Float64  acceptance_rate, log_density, hamiltonian_energy,              
    │
│           hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│           nom_step_size, logprior, loglikelihood, logjoint               
    │
╰──────────────────────────────────────────────────────────────────────────
────╯

display_ess_per_sec(chain, elapsed_turing_direct)

Elapsed time: 77.95 seconds
╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter  = 1001:11000                                                     
    │
│ → chain = 1:1                                                            
    │
│                                                                          
    │
│ Parameters (5) ── AbstractPPL.VarName                                    
    │
│  Vector{Float64}  σ (2,)                                                 
    │
│  Float64          α, β, γ, δ                                             
    │
│                                                                          
    │
│ Extras (14)                                                              
    │
│  Int64    n_steps, tree_depth                                            
    │
│  Bool     is_accept, numerical_error                                     
    │
│  Float64  acceptance_rate, log_density, hamiltonian_energy,              
    │
│           hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│           nom_step_size, logprior, loglikelihood, logjoint               
    │
╰──────────────────────────────────────────────────────────────────────────
────╯

Turing.jl backend

@btime bayesian_result_turing = turing_inference(
    prob, Tsit5(), t, data, priors;
    sample_args = (sampler = Turing.NUTS(0.85), num_samples = 10_000),
    likelihood = (u, p, t, σ) -> _mvn(u, σ),
    likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3)])

77.971 s (375186186 allocations: 21.92 GiB)
╭─FlexiChain (10000 iterations, 1 chain) ──────────────────────────────────
────╮
│ ↓ iter  = 1001:11000                                                     
    │
│ → chain = 1:1                                                            
    │
│                                                                          
    │
│ Parameters (5) ── AbstractPPL.VarName                                    
    │
│  Float64          theta[1], theta[2], theta[3], theta[4]                 
    │
│  Vector{Float64}  σ (2,)                                                 
    │
│                                                                          
    │
│ Extras (14)                                                              
    │
│  Int64    n_steps, tree_depth                                            
    │
│  Bool     is_accept, numerical_error                                     
    │
│  Float64  acceptance_rate, log_density, hamiltonian_energy,              
    │
│           hamiltonian_energy_error, max_hamiltonian_energy_error, step_si
ze, │
│           nom_step_size, logprior, loglikelihood, logjoint               
    │
╰──────────────────────────────────────────────────────────────────────────
────╯

DynamicHMC.jl backend

@btime bayesian_result_dynamichmc = dynamichmc_inference(
    prob, Tsit5(), t, data, priors; num_samples = 10_000)

25.221 s (89457194 allocations: 7.56 GiB)
(posterior = [(parameters = [1.4476685178849027, 0.8792982668823641, 3.1162
794317706517, 1.1034065500991816], σ = [0.49033477062861824, 0.460795070633
26086]), (parameters = [1.451856060474085, 0.8758832936246487, 3.1167366325
886956, 1.1103860191663237], σ = [0.4918700018868662, 0.4725372916523627]),
 (parameters = [1.4840730655808463, 0.8927999846188753, 3.0403488626735604,
 0.991119545286339], σ = [0.4207690040815519, 0.49821681352439734]), (param
eters = [1.6039641569556364, 0.9398518550607804, 2.7111741795988786, 0.8905
299810958612], σ = [0.42899698755179005, 0.25920002052495067]), (parameters
 = [1.5747571201637145, 0.9192489314631764, 2.7632771853301206, 0.916288210
5000995], σ = [0.3632975842345243, 0.39109888622079886]), (parameters = [1.
5395145707923743, 0.8905562831181824, 2.8631488169274517, 0.989535728931311
4], σ = [0.5646195120964767, 0.2655960593022276]), (parameters = [1.5576888
746819317, 0.9003297159961149, 2.819395124985265, 0.9622054730932554], σ = 
[0.35434760057705295, 0.3379438146519868]), (parameters = [1.47759477804904
42, 0.8546759068408251, 3.160401560631989, 1.0284929985953877], σ = [0.4870
6738468603233, 0.40517319103178195]), (parameters = [1.4436244903460647, 0.
834001320539935, 3.071597785659058, 1.1072322529578023], σ = [0.62638357293
3718, 0.41897951916793486]), (parameters = [1.4402715394868428, 0.845520822
3883052, 3.2664955438347314, 1.044118969788581], σ = [0.4815175012230368, 0
.5302142579348906])  …  (parameters = [1.5460951604885118, 0.92827653788400
85, 2.8214247309032117, 0.9608324617614382], σ = [0.45671633903357095, 0.44
826294889182267]), (parameters = [1.5242101662123861, 0.9587942708393042, 2
.8885880673714572, 0.9785657582814417], σ = [0.5345138475968254, 0.45041123
813041145]), (parameters = [1.5514969674193846, 0.9263790045875465, 2.84206
7037088315, 0.9380280073768954], σ = [0.3477587142769449, 0.550283875899224
6]), (parameters = [1.5247276117153612, 0.9797821066432957, 2.9302599930637
16, 0.968128630890477], σ = [0.33006591491037696, 0.6623429060553955]), (pa
rameters = [1.5194942898779513, 0.9788037642009224, 2.9143959847095875, 0.9
552146906591953], σ = [0.331481114767876, 0.6673074986893519]), (parameters
 = [1.5113863722180383, 0.9649599281006653, 2.9603161688476014, 0.975574010
449598], σ = [0.3412477390863709, 0.4495285485569228]), (parameters = [1.56
50437055968633, 0.9535452753611688, 2.8500648467958434, 0.9314404155170826]
, σ = [0.3447578631603385, 0.3647962599594296]), (parameters = [1.582507894
1592043, 0.9086762384372914, 2.803963290607017, 0.9357464286247262], σ = [0
.5417908689849672, 0.3053350675328467]), (parameters = [1.4965176045938897,
 0.916633356603956, 3.0457621200746763, 0.9668982954156402], σ = [0.3905332
79768808, 0.46382032132226514]), (parameters = [1.4881346423396538, 0.87634
58670085983, 3.095001611395606, 0.9998113952017841], σ = [0.592953889310063
2, 0.30049516009589117])], posterior_matrix = [0.36995434364376406 0.372842
7795838123 … 0.40314081208632097 0.397523417760655; -0.12863111361883858 -0
.13252242337705866 … -0.08704771590010983 -0.1319944406164839; … ; -0.71266
69157914775 -0.7095408212102717 … -0.9402420898540489 -0.5226386413381634; 
-0.7748018670681315 -0.7496386110307005 … -0.7682580402816838 -1.2023236313
066599], tree_statistics = DynamicHMC.TreeStatisticsNUTS[DynamicHMC.TreeSta
tisticsNUTS(-19.73263124898585, 5, turning at positions 20:51, 0.9458414307
342099, 63, DynamicHMC.Directions(0xc5d16833)), DynamicHMC.TreeStatisticsNU
TS(-20.000207988320724, 2, turning at positions -1:2, 0.9517922502573214, 3
, DynamicHMC.Directions(0x24719f3e)), DynamicHMC.TreeStatisticsNUTS(-20.179
741850522475, 4, turning at positions -14:-21, 0.9585131768580837, 31, Dyna
micHMC.Directions(0x93b0000a)), DynamicHMC.TreeStatisticsNUTS(-17.586016277
82383, 5, turning at positions -25:-32, 0.9918303911809301, 39, DynamicHMC.
Directions(0x760ecbc7)), DynamicHMC.TreeStatisticsNUTS(-17.2778139024511, 5
, turning at positions -17:14, 0.9862925717545266, 31, DynamicHMC.Direction
s(0x6651ba4e)), DynamicHMC.TreeStatisticsNUTS(-17.78007569903434, 5, turnin
g at positions -7:24, 0.7582515442734812, 31, DynamicHMC.Directions(0x374fa
df8)), DynamicHMC.TreeStatisticsNUTS(-18.552635448897014, 5, turning at pos
itions -10:21, 0.9012610475326215, 31, DynamicHMC.Directions(0x525193f5)), 
DynamicHMC.TreeStatisticsNUTS(-19.052743213946787, 5, turning at positions 
-15:-18, 0.8207107923369191, 39, DynamicHMC.Directions(0x5781c955)), Dynami
cHMC.TreeStatisticsNUTS(-19.15951569684452, 4, turning at positions -9:-24,
 0.9616764533271717, 31, DynamicHMC.Directions(0x1f311567)), DynamicHMC.Tre
eStatisticsNUTS(-21.234595065689913, 4, turning at positions 12:19, 0.96571
43076680554, 23, DynamicHMC.Directions(0x452a075b))  …  DynamicHMC.TreeStat
isticsNUTS(-17.5325628911112, 5, turning at positions -30:1, 0.865636528113
5845, 31, DynamicHMC.Directions(0x8fb84f01)), DynamicHMC.TreeStatisticsNUTS
(-18.85341410771649, 4, turning at positions 7:22, 0.9080675519037684, 31, 
DynamicHMC.Directions(0xfd8eca76)), DynamicHMC.TreeStatisticsNUTS(-18.32587
5477108966, 5, turning at positions -9:22, 0.9937174741623888, 31, DynamicH
MC.Directions(0x0ef76196)), DynamicHMC.TreeStatisticsNUTS(-18.3834085555629
2, 4, turning at positions 6:21, 0.9473017510997487, 31, DynamicHMC.Directi
ons(0x5ee10a15)), DynamicHMC.TreeStatisticsNUTS(-19.25268569383515, 4, turn
ing at positions 16:23, 0.7384911086290119, 31, DynamicHMC.Directions(0x84c
37597)), DynamicHMC.TreeStatisticsNUTS(-19.551664294280123, 4, turning at p
ositions 13:28, 0.8799347414570727, 31, DynamicHMC.Directions(0x7ae9025c)),
 DynamicHMC.TreeStatisticsNUTS(-19.0108892386747, 4, turning at positions 0
:15, 0.9424887395842007, 15, DynamicHMC.Directions(0xcea652ef)), DynamicHMC
.TreeStatisticsNUTS(-17.999513410564884, 4, turning at positions -7:8, 0.71
61463412351406, 15, DynamicHMC.Directions(0x406031f8)), DynamicHMC.TreeStat
isticsNUTS(-18.2342240812839, 5, turning at positions -31:0, 0.981183017234
2828, 31, DynamicHMC.Directions(0x0bd16660)), DynamicHMC.TreeStatisticsNUTS
(-18.638544434693138, 5, turning at positions -26:5, 0.9572591635995672, 31
, DynamicHMC.Directions(0x8657b225))], logdensities = [-18.29631390767163, 
-19.627948686288104, -15.9073625332707, -16.41386726514752, -15.23062824833
935, -16.151688213415017, -15.93078206034526, -17.301563038246105, -17.4190
16544520968, -17.974287778619747  …  -15.536874732623394, -16.7943818052703
76, -15.690857447251748, -17.167520484506365, -18.79667441427467, -16.10042
878676396, -15.478975004489026, -15.733417149914978, -17.121967286334822, -
17.015232533445566], κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹): [0
.04335825968656034, 0.061563082086944956, 0.060820972358074966, 0.077329634
63850066, 0.28818527850176007, 0.300431029107059], ϵ = 0.12228102613608945)

Conclusion

The Lotka-Volterra equation is a "predator-prey" model: it models the populations of two species, one of which is the predator (wolf) and the other the prey (rabbit). Its solutions are periodic. This page runs each backend at a single ODE solver tolerance, so it does not measure how the estimate depends on the tolerance.

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/BayesianInference","DiffEqBayesLotkaVolterra.jmd")

Computer Information:

Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
  GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/BayesianInference/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
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⌃ [459566f4] DiffEqCallbacks v4.19.4
⌃ [31c24e10] Distributions v0.25.131
⌃ [bbc10e6e] DynamicHMC v3.6.1
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⌃ [65888b18] ParameterizedFunctions v5.27.0
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⌃ [90137ffa] StaticArrays v1.9.20
⌃ [fce5fe82] Turing v0.46.1
  [37e2e46d] LinearAlgebra v1.12.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/BayesianInference/Manifest.toml`
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  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
⌃ [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
⌃ [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
⌃ [35661453] Xorg_xkbcomp_jll v1.4.7+0
⌃ [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
⌃ [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
⌃ [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
⌃ [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.1+2
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.6+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`