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Bayesian Methods

The following methods require DiffEqBayes.jl:

using Pkg
Pkg.add("DiffEqBayes")
using DiffEqBayes

stan_inference

stan_inference(prob::DiffEqBase.DEProblem, alg, t, data, priors = nothing;
    stanmodel = nothing, likelihood = Normal, vars = (StanODEData(), InverseGamma(3, 3)), sample_u0 = false, solve_kwargs = Dict(), diffeq_string = nothing, sample_kwargs = Dict(), output_format = :mcmcchains, print_summary = true, tmpdir = mktempdir())

stan_inference uses Stan.jl to perform the Bayesian inference. The Stan installation process is required to use this function. Currently CmdStan v2.34.1 is supported.

prob can be any DEProblem with a corresponding alg choice. alg is a choice between :rk45 and :bdf, the two internal integrators of Stan. t is the array of time and data is the array where the first dimension (columns) corresponds to the array of system values. priors is an array of prior distributions for each parameter, specified via a Distributions.jl type. likelihood is the likelihood distribution to use with the arguments from vars, and vars is a tuple of priors for the distributions of the likelihood hyperparameters. The special value StanODEData() in this tuple denotes the position that the ODE solution takes in the likelihood's parameter list.

solve_kwargs is a Dict and passed to the stan differential equation solver. solve_kwargs may contain save_idxs, reltol, abstol, and maxiter. save_idxs is documented at DifferentialEquations.jl. sample_kwargs are passed to the stan sampler and accepts num_samples, num_warmups, num_cpp_chains , num_chains, num_threads, delta. Please refer to the stan documentation for more information.

turing_inference

turing_inference(prob::DiffEqBase.DEProblem, alg, t, data, priors;
    likelihood_dist_priors, likelihood, syms, sample_u0 = false, progress = false, solve_kwargs = Dict(), sample_args = NamedTuple(), sample_kwargs= Dict())

turing_inference uses Turing.jl to perform its parameter inference. prob can be any DEProblem with a corresponding alg choice. t is the array of time points and data is the set of observations for the differential equation system at time point t[i] (or higher dimensional). priors is an array of prior distributions for each parameter, specified via a Distributions.jl type.

The turing_inference interacts with SciML.CommonSolve.solve and StatsBase.sample. Both accept many arguments depending on the solver and sampling algorithm. These arguments are supplied to turing_inferene function via solve_kwargs, sample_args, and sample_kwargs arguments. Please refer to the solve documentation for solve_kwargs, e.g. solve_kwargs = Dict(:save_idxs => [1]). The solve keyword arguments default to save_idxs = nothing. Similarly please refer to the sample documentation for sample_args and sample_kwargs. The four positional argument are as following: sampler, the sampling algorithm. Sampling from multiple chains is possible serially or parallelly using parallel_type. Third num_samples, the number of samples per MCMC chain and n_chains, the number of MCMC chains. The positional arguments default to the following values.

sampler = Turing.NUTS(0.65)
parallel_type = MCMCSerial()
num_samples = 1000
n_chains = 1

dynamichmc_inference

dynamichmc_inference(prob::DEProblem, alg, t, data, priors, transformations;
    σ = 0.01, ϵ = 0.001, initial = Float64[])

dynamichmc_inference uses DynamicHMC.jl to perform the Bayesian parameter estimation. prob can be any DEProblem, data is the set of observations for our model which is to be used in the Bayesian Inference process. priors represent the choice of prior distributions for the parameters to be determined, passed as an array of Distributions.jl distributions. t is the array of time points. transformations is an array of Transformations imposed for constraining the parameter values to specific domains. initial values for the parameters can be passed, if not passed the means of the priors are used. ϵ can be used as a kwarg to pass the initial step size for the NUTS algorithm.

abc_inference

abc_inference(prob::DEProblem, alg, t, data, priors; ϵ = 0.001,
    distancefunction = euclidean, ABCalgorithm = ABCSMC, progress = false,
    num_samples = 500, maxiterations = 10^5, kwargs...)

abc_inference uses ApproxBayes.jl, which uses Approximate Bayesian Computation (ABC) to perform its parameter inference. prob can be any DEProblem with a corresponding alg choice. t is the array of time points and data[:,i] is the set of observations for the differential equation system at time point t[i] (or higher dimensional). priors is an array of prior distributions for each parameter, specified via a Distributions.jl type. num_samples is the number of posterior samples. ϵ is the target distance between the data and simulated data. distancefunction is a distance metric specified from the Distances.jl package, the default is euclidean. ABCalgorithm is the ABC algorithm to use, options are ABCSMC or ABCRejection from ApproxBayes.jl, the default is the former which is more efficient. maxiterations is the maximum number of iterations before the algorithm terminates. The extra kwargs are given to the internal differential equation solver.