Lotka-Volterra Bayesian Parameter Estimation Benchmarks
Parameter Estimation of Lotka-Volterra Equation using DiffEqBayes.jl
using DiffEqBayes, StanSample, DynamicHMC, Turingusing Distributions, BenchmarkTools, StaticArrays
using OrdinaryDiffEq, RecursiveArrayTools, ParameterizedFunctions
using Plots, LinearAlgebra"""Display ESS/s (effective samples per second) from a Turing chain."""
function display_ess_per_sec(chain, elapsed)
stats = summarystats(chain)
ess_bulk = stats[:, :ess_bulk]
println("Elapsed time: $(round(elapsed; digits=2)) seconds\n")
println("ESS/s (effective samples per second, bulk):")
for (i, param) in enumerate(stats[:, :parameters])
println(" $param: $(round(ess_bulk[i] / elapsed; digits=1))")
end
println("\nMinimum ESS/s (bulk): $(round(minimum(ess_bulk) / elapsed; digits=1))")
end
"""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
endMain.var"##WeaveSandBox#225".display_stan_timinggr(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 dMain.var"##WeaveSandBox#225".LotkaVolterraTest{Main.var"##WeaveSandBox#225"
.var"###ParameterizedDiffEqFunction#227", Main.var"##WeaveSandBox#225".var"
###ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"###Par
ameterizedJacobianFunction#229", Nothing, Nothing, ModelingToolkit.System}(
Main.var"##WeaveSandBox#225".var"##ParameterizedDiffEqFunction#227", Linear
Algebra.UniformScaling{Bool}(true), nothing, Main.var"##WeaveSandBox#225".v
ar"##ParameterizedTGradFunction#228", Main.var"##WeaveSandBox#225".var"##Pa
rameterizedJacobianFunction#229", nothing, nothing, nothing, nothing, nothi
ng, nothing, nothing, [:x, :y], :t, nothing, Model ##Parameterized#226:
Equations (2):
2 standard: see equations(##Parameterized#226)
Unknowns (2): see unknowns(##Parameterized#226)
x(t)
y(t)
Parameters (4): see parameters(##Parameterized#226)
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
1.0prob = 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, 0.8576684823217128]
[1.1758715885890039, 0.6394595702308831]
[1.419680958026516, 0.45699626144050703]
[1.8767193976262215, 0.3247334288460738]
[2.5882501035146133, 0.26336255403957304]
[3.860709084797009, 0.27944581878759106]
[5.750813064347339, 0.5220073551361045]
[6.814978696356636, 1.917783405671627]
[4.392997771045279, 4.194671543390719]
⋮
[2.6142510825026886, 0.2641695435004172]
[4.241070648057757, 0.30512326533052475]
[6.79112182569163, 1.13452538354883]
[6.265374940295053, 2.7416885955953294]
[3.7807688120520893, 4.431164521488331]
[1.8164214705302744, 4.064057991958618]
[1.146502825635759, 2.791173034823897]
[0.955798652853089, 1.623563316340748]
[1.0337581330572414, 0.9063703732075853]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, 0.8576684823217128]
[1.1758715885890039, 0.6394595702308831]
[1.419680958026516, 0.45699626144050703]
[1.8767193976262215, 0.3247334288460738]
[2.5882501035146133, 0.26336255403957304]
[3.860709084797009, 0.27944581878759106]
[5.750813064347339, 0.5220073551361045]
[6.814978696356636, 1.917783405671627]
[4.392997771045279, 4.194671543390719]
⋮
[2.6142510825026886, 0.2641695435004172]
[4.241070648057757, 0.30512326533052475]
[6.79112182569163, 1.13452538354883]
[6.265374940295053, 2.7416885955953294]
[3.7807688120520893, 4.431164521488331]
[1.8164214705302744, 4.064057991958618]
[1.146502825635759, 2.791173034823897]
[0.955798652853089, 1.623563316340748]
[1.0337581330572414, 0.9063703732075853]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}:
2.5141 7.30662 0.856452 1.896 … 3.03282 3.59552 1.58763
-0.268141 2.16871 1.72749 0.00231962 0.82702 4.03775 1.64031Plots 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
The solution converges for tolerance values lower than 1e-3, lower tolerance leads to better accuracy in result but is accompanied by longer warmup and sampling time, truncated normal priors are used for preventing Stan from stepping into negative values.
We use adapt_delta = 0.85 (Stan's default) consistently across all 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)))40.383872 seconds (2.66 M allocations: 182.336 MiB, 0.13% gc time, 3.34% c
ompilation time: <1% of which was recompilation)
57.495048 seconds (8.51 M allocations: 599.210 MiB, 0.31% gc time, 7.46% c
ompilation time: 3% of which was recompilation)
Chains MCMC chain (10000×6×1 Array{Float64, 3}):
Iterations = 1:1:10000
Number of chains = 1
Samples per chain = 10000
parameters = sigma1.1, sigma1.2, theta_1, theta_2, theta_3, theta_4
internals =
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
sigma1.1 0.7686 0.2208 0.0035 4636.2106 4739.3212 0.999
9 ⋯
sigma1.2 0.7339 0.1983 0.0032 4204.6698 4233.5638 1.000
5 ⋯
theta_1 1.5887 0.1248 0.0024 2796.6017 3784.6819 1.000
0 ⋯
theta_2 1.1488 0.1542 0.0029 3490.9163 3201.9562 1.000
0 ⋯
theta_3 2.8142 0.3063 0.0056 2955.1401 4124.7130 0.999
9 ⋯
theta_4 0.9244 0.1166 0.0022 2775.2324 3896.1763 1.000
0 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
sigma1.1 0.4647 0.6171 0.7291 0.8699 1.3007
sigma1.2 0.4547 0.5954 0.6984 0.8337 1.2173
theta_1 1.3701 1.5015 1.5799 1.6662 1.8561
theta_2 0.8980 1.0424 1.1306 1.2344 1.5040
theta_3 2.2512 2.6003 2.7994 3.0127 3.4535
theta_4 0.7170 0.8430 0.9166 0.9965 1.1721Stan's internal timing (excluding data serialization and CSV parsing):
display_stan_timing(bayesian_result_stan)Chain 1 timing (from Stan CSV):
Elapsed Time: 3.092 seconds (Warm-up)
35.939 seconds (Sampling)
39.031 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 1:length(predicted)
data[:, i] ~ MvNormal(predicted[i], Diagonal(σ .^ 2))
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)
chainChains MCMC chain (10000×20×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 27.71 seconds
Compute duration = 27.71 seconds
parameters = σ[1], σ[2], α, β, γ, δ
internals = n_steps, is_accept, acceptance_rate, log_density, hamil
tonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree
_depth, numerical_error, step_size, nom_step_size, logprior, loglikelihood,
logjoint
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
σ[1] 0.7755 0.2279 0.0035 4889.2464 4717.5449 1.000
2 ⋯
σ[2] 0.7319 0.1962 0.0030 4872.8083 5096.6204 1.000
0 ⋯
α 1.5850 0.1241 0.0025 2442.5261 3174.9926 1.001
0 ⋯
β 1.1452 0.1509 0.0030 3081.9890 2462.6228 1.000
1 ⋯
γ 2.8211 0.3063 0.0060 2576.3491 3614.0091 1.001
1 ⋯
δ 0.9288 0.1168 0.0023 2548.0786 3624.9055 1.000
6 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ[1] 0.4652 0.6183 0.7337 0.8807 1.3366
σ[2] 0.4477 0.5935 0.6997 0.8337 1.2040
α 1.3673 1.4987 1.5754 1.6615 1.8539
β 0.8958 1.0446 1.1284 1.2281 1.4947
γ 2.2584 2.6122 2.8083 3.0145 3.4707
δ 0.7208 0.8482 0.9214 1.0025 1.1799display_ess_per_sec(chain, elapsed_turing_direct)Elapsed time: 27.96 seconds
ESS/s (effective samples per second, bulk):
σ[1]: 174.8
σ[2]: 174.3
α: 87.3
β: 110.2
γ: 92.1
δ: 91.1
Minimum ESS/s (bulk): 87.3Turing.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) -> MvNormal(u, Diagonal(σ .^ 2)),
likelihood_dist_priors = [InverseGamma(2, 3), InverseGamma(2, 3)])25.224 s (194722988 allocations: 16.70 GiB)
Chains MCMC chain (10000×20×1 Array{Float64, 3}):
Iterations = 1001:1:11000
Number of chains = 1
Samples per chain = 10000
Wall duration = 25.02 seconds
Compute duration = 25.02 seconds
parameters = theta[1], theta[2], theta[3], theta[4], σ[1], σ[2]
internals = n_steps, is_accept, acceptance_rate, log_density, hamil
tonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree
_depth, numerical_error, step_size, nom_step_size, logprior, loglikelihood,
logjoint
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rha
t ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6
4 ⋯
theta[1] 1.5799 0.1242 0.0023 2884.0998 2721.9660 1.000
1 ⋯
theta[2] 1.1414 0.1502 0.0026 3856.9388 3227.6186 1.000
0 ⋯
theta[3] 2.8348 0.3125 0.0057 3058.0667 3042.9676 1.000
2 ⋯
theta[4] 0.9332 0.1200 0.0023 2896.2737 2868.6105 1.000
0 ⋯
σ[1] 0.7721 0.2211 0.0036 4326.6437 4256.9748 1.000
2 ⋯
σ[2] 0.7370 0.1987 0.0030 4716.1415 5483.3747 1.000
6 ⋯
1 column om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
theta[1] 1.3562 1.4960 1.5698 1.6564 1.8416
theta[2] 0.8915 1.0394 1.1256 1.2271 1.4844
theta[3] 2.2797 2.6192 2.8218 3.0223 3.5136
theta[4] 0.7238 0.8519 0.9253 1.0037 1.1927
σ[1] 0.4672 0.6208 0.7300 0.8768 1.3180
σ[2] 0.4545 0.5954 0.7033 0.8386 1.2092DynamicHMC.jl backend
@btime bayesian_result_dynamichmc = dynamichmc_inference(
prob, Tsit5(), t, data, priors; num_samples = 10_000)10.714 s (60725558 allocations: 7.99 GiB)
(posterior = @NamedTuple{parameters::Vector{Float64}, σ::Vector{Float64}}[(
parameters = [1.5845208654766372, 1.138867169661393, 2.748229579955255, 0.8
976957619794329], σ = [0.47243680619877465, 0.45007802218418796]), (paramet
ers = [1.6541997933294974, 1.1109268346734695, 2.645285743833328, 0.8771936
45244687], σ = [0.48084962014793603, 0.5132807406930439]), (parameters = [1
.6237307905196579, 1.0396013045247003, 2.636384604164632, 0.841762846115922
8], σ = [0.5063169021979996, 0.5591407914352544]), (parameters = [1.5981555
746360403, 1.1422565651328627, 2.784960662421189, 0.8949894419917843], σ =
[0.5100149407473997, 0.562671414544959]), (parameters = [1.603691569277927,
1.1084350299459318, 2.7053038355673613, 0.9004092721060778], σ = [0.720744
7481142819, 0.44328344848308826]), (parameters = [1.6280704301962836, 1.085
7066887983238, 2.701490121555567, 0.9082811736083803], σ = [0.7178311256631
975, 0.43362088588973574]), (parameters = [1.7339542937233505, 1.2169980467
188464, 2.5093571790244718, 0.7904749445863157], σ = [0.5148060890457063, 0
.6975983254242633]), (parameters = [1.7624868067042392, 1.0882637262732546,
2.4159870356878694, 0.7795449134073417], σ = [0.4927596531556686, 0.796088
9329757739]), (parameters = [1.7939692692843658, 1.1819188337202384, 2.3871
314381438693, 0.7427098080042576], σ = [0.5695369987928809, 0.8457428008417
858]), (parameters = [1.5527412053143406, 1.2295328589602208, 2.92428072077
562, 0.9715258179558816], σ = [1.0086149725198377, 0.5517764918089443]) …
(parameters = [1.4190148406262577, 1.0654099271721023, 3.267568056487823,
1.065110087254], σ = [0.9292091169358979, 0.5437678512514054]), (parameters
= [1.5778035045532566, 1.1258956389872785, 2.89112889819149, 0.85487321395
85024], σ = [0.5963182066315795, 0.7174313618914828]), (parameters = [1.577
385849200119, 1.0072303966716853, 2.6585289970973816, 0.9633438675403629],
σ = [0.6579789139170471, 0.6015760151177838]), (parameters = [1.71188247706
72252, 1.2911330192063164, 2.5598943752826564, 0.8029615161500803], σ = [0.
9865737017160482, 0.6072554842590152]), (parameters = [1.6773911428212283,
1.3801288597174184, 2.6025595447132157, 0.8500534634096685], σ = [0.7482249
318282505, 0.7538252563879695]), (parameters = [1.7102861140732661, 1.49712
34102343818, 2.548758534813654, 0.8229933447592853], σ = [0.827578500177040
2, 0.7927803773463885]), (parameters = [1.7092573325044462, 1.6322859775365
28, 2.603434584001729, 0.8064596565995802], σ = [0.5426170842548597, 0.9153
159301364666]), (parameters = [1.7198264413354862, 1.675620679859108, 2.616
229996286207, 0.8073343072929491], σ = [0.5824947933905149, 0.7929319224714
372]), (parameters = [1.48904966930348, 0.8702994423951921, 2.9585744357282
96, 1.019102020757928], σ = [0.6330310256795897, 0.8071339590928025]), (par
ameters = [1.478001222771699, 0.918987987184195, 2.936717700027027, 1.01518
81194498644], σ = [0.5926650761735546, 0.881362993955773])], posterior_matr
ix = [0.4602820685548055 0.5033173833388321 … 0.39813811063610755 0.3906906
498407551; 0.1300340575251369 0.10519465310951687 … -0.13891793989100842 -0
.0844822283286471; … ; -0.7498512844874551 -0.7322006977835381 … -0.4572358
443272616 -0.5231258352016717; -0.7983343286152741 -0.666932330686628 … -0.
21426562809054103 -0.12628571298349467], tree_statistics = DynamicHMC.TreeS
tatisticsNUTS[DynamicHMC.TreeStatisticsNUTS(-27.99106343057668, 5, turning
at positions -10:-41, 0.9822486894768828, 63, DynamicHMC.Directions(0x4e27a
5d6)), DynamicHMC.TreeStatisticsNUTS(-26.414811131673254, 5, turning at pos
itions -23:8, 0.6886029083079795, 31, DynamicHMC.Directions(0xeb001528)), D
ynamicHMC.TreeStatisticsNUTS(-25.65113817639523, 4, turning at positions -1
0:5, 0.9843057747585764, 15, DynamicHMC.Directions(0x6dd22875)), DynamicHMC
.TreeStatisticsNUTS(-25.196846543187345, 4, turning at positions -12:3, 0.9
971165284909176, 15, DynamicHMC.Directions(0x2867ec93)), DynamicHMC.TreeSta
tisticsNUTS(-23.033426201328457, 5, turning at positions -6:25, 0.999737254
6367458, 31, DynamicHMC.Directions(0x3026c219)), DynamicHMC.TreeStatisticsN
UTS(-27.960813809496518, 3, turning at positions -5:-8, 0.7522403072713345,
11, DynamicHMC.Directions(0xf9a04b33)), DynamicHMC.TreeStatisticsNUTS(-26.
002998414426365, 5, turning at positions -31:0, 0.9220553207820794, 31, Dyn
amicHMC.Directions(0x11ed2680)), DynamicHMC.TreeStatisticsNUTS(-27.18863009
773501, 4, turning at positions -10:5, 0.9568581177712628, 15, DynamicHMC.D
irections(0x2217bfa5)), DynamicHMC.TreeStatisticsNUTS(-27.785642538955972,
4, turning at positions 14:29, 1.0, 31, DynamicHMC.Directions(0x2beb84dd)),
DynamicHMC.TreeStatisticsNUTS(-28.8420719461963, 5, turning at positions -
30:-61, 0.9818131816690622, 63, DynamicHMC.Directions(0xf8ca5dc2)) … Dyna
micHMC.TreeStatisticsNUTS(-26.944132797656298, 5, turning at positions -5:2
6, 1.0, 31, DynamicHMC.Directions(0x0c4bae9a)), DynamicHMC.TreeStatisticsNU
TS(-27.48330969174389, 6, turning at positions -51:12, 0.9959568348143363,
63, DynamicHMC.Directions(0x4ce5404c)), DynamicHMC.TreeStatisticsNUTS(-26.9
17024592462234, 4, turning at positions 4:11, 0.9999029242349048, 23, Dynam
icHMC.Directions(0x5e72a413)), DynamicHMC.TreeStatisticsNUTS(-27.6491303680
97268, 5, turning at positions 7:38, 0.9980163875691671, 63, DynamicHMC.Dir
ections(0x71c87866)), DynamicHMC.TreeStatisticsNUTS(-25.941817005137942, 5,
turning at positions -18:13, 0.997739975475867, 31, DynamicHMC.Directions(
0x15d9084d)), DynamicHMC.TreeStatisticsNUTS(-26.299403494433548, 5, turning
at positions -17:14, 0.9930178783091309, 31, DynamicHMC.Directions(0xa4efe
56e)), DynamicHMC.TreeStatisticsNUTS(-29.46806434849075, 5, turning at posi
tions -25:6, 0.881545841080014, 31, DynamicHMC.Directions(0x354023e6)), Dyn
amicHMC.TreeStatisticsNUTS(-28.61207943723124, 4, turning at positions -13:
2, 0.9768622862588074, 15, DynamicHMC.Directions(0x13740002)), DynamicHMC.T
reeStatisticsNUTS(-30.36404550636461, 6, turning at positions 24:39, 0.9328
273323547334, 79, DynamicHMC.Directions(0xb243fad7)), DynamicHMC.TreeStatis
ticsNUTS(-27.023246880445186, 5, turning at positions -17:14, 0.99244595811
58201, 31, DynamicHMC.Directions(0x18c85c6e))], logdensities = [-22.9110648
83912783, -24.163398445682823, -25.0114395909568, -21.58090040157527, -22.3
06582670388913, -23.74418201688408, -23.300147493815825, -25.76886347525031
, -24.830469129352423, -24.93637570817718 … -25.249051821680013, -25.3491
3527099752, -24.751326594659727, -24.299676005932128, -24.0269266414176, -2
5.54831372581869, -26.599831477521203, -26.54771559387234, -24.933742032557
9, -24.668013214535176], κ = Gaussian kinetic energy (Diagonal), √diag(M⁻¹)
: [0.06124548992081966, 0.13195707824684216, 0.0889781260849419, 0.10066074
665562406, 0.2719065589951006, 0.2636760737132186], ϵ = 0.09467262129258684
)Conclusion
Lotka-Volterra Equation is a "predator-prey" model, it models population of two species in which one is the predator (wolf) and the other is the prey (rabbit). It depicts a cyclic behaviour, which is also seen in its Uncertainty Quantification Plots. This behaviour makes it easy to estimate even at very high tolerance values (1e-3).
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 1.10.11
Commit a2b11907d7b (2026-03-09 14:59 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 64 × AMD EPYC 9354 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
Threads: 64 default, 0 interactive, 32 GC (on 64 virtual cores)
Environment:
JULIA_NUM_THREADS = auto
Package Information:
Status `~/github-runners/demeter3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/BayesianInference/Project.toml`
[6e4b80f9] BenchmarkTools v1.6.3
[ebbdde9d] DiffEqBayes v3.11.0
[459566f4] DiffEqCallbacks v4.12.0
[31c24e10] Distributions v0.25.123
[bbc10e6e] DynamicHMC v3.6.0
[1dea7af3] OrdinaryDiffEq v6.108.0
⌃ [65888b18] ParameterizedFunctions v5.19.0
[91a5bcdd] Plots v1.41.6
[731186ca] RecursiveArrayTools v3.48.0
[31c91b34] SciMLBenchmarks v0.1.3
⌃ [c1514b29] StanSample v7.10.2
⌃ [90137ffa] StaticArrays v1.9.17
⌅ [fce5fe82] Turing v0.42.8
[37e2e46d] LinearAlgebra
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`And the full manifest:
Status `~/github-runners/demeter3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/BayesianInference/Manifest.toml`
[47edcb42] ADTypes v1.21.0
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[1520ce14] AbstractTrees v0.4.5
[7d9f7c33] Accessors v0.1.43
⌃ [79e6a3ab] Adapt v4.4.0
[0bf59076] AdvancedHMC v0.8.3
[5b7e9947] AdvancedMH v0.8.10
[576499cb] AdvancedPS v0.7.2
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⌅ [366bfd00] DynamicPPL v0.39.14
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⌃ [06fc5a27] DynamicQuantities v1.11.0
[cad2338a] EllipticalSliceSampling v2.0.0
⌃ [4e289a0a] EnumX v1.0.6
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[a4df4552] FastPower v1.3.1
[48062228] FilePathsBase v0.9.24
[1a297f60] FillArrays v1.16.0
[64ca27bc] FindFirstFunctions v1.8.0
[6a86dc24] FiniteDiff v2.29.0
[53c48c17] FixedPointNumbers v0.8.5
[1fa38f19] Format v1.3.7
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[22cec73e] InitialValues v0.3.1
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[41ab1584] InvertedIndices v1.3.1
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[f8c6e375] Git_jll v2.53.0+0
[7746bdde] Glib_jll v2.86.3+0
[3b182d85] Graphite2_jll v1.3.15+0
[2e76f6c2] HarfBuzz_jll v8.5.1+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.1.4+0
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.0.1+0
[1d63c593] LLVMOpenMP_jll v18.1.8+0
[dd4b983a] LZO_jll v2.10.3+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.41.3+0
[89763e89] Libtiff_jll v4.7.2+0
[38a345b3] Libuuid_jll v2.41.3+0
[856f044c] MKL_jll v2025.2.0+0
[e7412a2a] Ogg_jll v1.3.6+0
[9bd350c2] OpenSSH_jll v10.2.1+0
[458c3c95] OpenSSL_jll v3.5.5+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.57.0+0
⌅ [30392449] Pixman_jll v0.44.2+0
⌅ [c0090381] Qt6Base_jll v6.8.2+2
⌅ [629bc702] Qt6Declarative_jll v6.8.2+1
⌅ [ce943373] Qt6ShaderTools_jll v6.8.2+1
⌃ [e99dba38] Qt6Wayland_jll v6.8.2+2
[f50d1b31] Rmath_jll v0.5.1+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.2+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[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 v1.8.3+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
[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 v2.44.0+0
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[8f1865be] ZeroMQ_jll v4.3.6+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 v3.13.1+0
[0ac62f75] libass_jll v0.17.4+0
[1183f4f0] libdecor_jll v0.2.2+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 v1.6.55+0
[a9144af2] libsodium_jll v1.0.21+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.0.0+1
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.10.0
[de0858da] Printf
[9abbd945] Profile
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays v1.10.0
[10745b16] Statistics v1.10.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.1.1+0
[deac9b47] LibCURL_jll v8.4.0+0
[e37daf67] LibGit2_jll v1.6.4+0
[29816b5a] LibSSH2_jll v1.11.0+1
[c8ffd9c3] MbedTLS_jll v2.28.2+1
[14a3606d] MozillaCACerts_jll v2023.1.10
[4536629a] OpenBLAS_jll v0.3.23+4
[05823500] OpenLibm_jll v0.8.5+0
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.52.0+1
[3f19e933] p7zip_jll v17.4.0+2
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`