In this tutorial we will go through the steps for bayesian parameter estimation of the Theophylline model in Pumas.jl. During the tutorial we use the following packages.

using Pumas, MCMCChains, StatsPlots

Important Note: Pumas.jl is a proprietary package. It is free to use for non-commercial academic teaching and research purposes. For commercial users, license fees apply. Further, Pumas may not be used under any license agreement for patient management services in a hospital or clinical settings. Please refer to End User License Agreement (https://juliacomputing.com/eula) for details. Please contact sales@juliacomputing.com for purchase.

The core modeling and estimation functionality is provided by the `Pumas`

package. Postprocessing of the posterior data is handled by the `MCMCChains`

package while the plots are made with the `Plots`

package.

First, we have to set up the pharmacometric model to estimate its parameters. For this tutorial, we will retrict the focus to a simple pharmacokinetics (PK) model. The Bayesian estimation procedure also works for more complicated models but such models might take longer to estimate with an MCMC-based procedure.

The prior distribution of a parameter of the model can be any distribution from the Distributions.jl package and is specified with the tilde (`~`

) symbol.

theopmodel_bayes = @model begin @param begin # Mode at [2.0, 0.2, 0.8, 2.0] θ ~ Constrained( MvNormal( [2.0, 0.2, 0.8, 2.0], Diagonal(ones(4)) ), lower = zeros(4), upper = fill(10.0, 4), init = [2.0, 0.2, 0.8, 2.0]) # Mode at diagm(fill(0.2, 3)) Ω ~ InverseWishart(6, diagm(fill(0.2, 3)) .* (6 + 3 + 1)) # Mean at 0.5 and positive density at 0.0 σ ~ Gamma(1.0, 0.5) end @random begin η ~ MvNormal(Ω) end @pre begin Ka = (SEX == 1 ? θ[1] : θ[4])*exp(η[1]) CL = θ[2]*(WT/70) *exp(η[2]) Vc = θ[3] *exp(η[3]) end @covariates SEX WT @dynamics Depots1Central1 @derived begin # The conditional mean μ := @. Central / Vc # Additive error model dv ~ @. Normal(μ, σ) end end

PumasModel Parameters: θ, Ω, σ Random effects: η Covariates: SEX, WT Dynamical variables: Depot, Central Derived: dv Observed: dv

The joint prior distribution of `θ`

is specified as the normal distribution but constrained to avoid negative and extreme draws. In the Bayesian framework, the prior distribution represents our initial *belief* about the value of the parameter *prior* to observing any data. The covariance matrix of the random effects vector `η`

is given a prior distribution of `InverseWishart`

which is parameterized with degrees of freedom parameter `ν`

and scale matrix `Ψ`

. It is sometimes more intuitive to use the mode to specify priors which is our choice. Since the mode of `InverseWishart`

is `Ψ/(ν + p + 1)`

, the mode of our prior is `diagm(fill(0.2, 3))`

which corresponds to an inter-subject variability peaking at 20%. Finally, the prior additive error component `σ`

is modeled as a `Gamma(1.0, 0.5)`

. Notice that `Gamma`

is parameterized with a shape and a *scale* parameter and *not* a rate. Setting the shape parameter equal to one has the advantage that the density has support at zero, meaning the value zero has a non-zero probablility, so the posterior distribution won't be forced away from zero. The scale parameter is then also equal to the mean which makes the interpretation easy. In the Bayesian framework, the posterior distribution represents our *belief* about the value of the parameter *after* observing some data.

To fit the model, we use the `fit`

function. It requires a model, a population, a named tuple of parameters and an estimation method. Since we want to use Bayesian estimation with Markov Chain Monte Carlo (MCMC), we pass `BayesMCMC`

as the estimation method argument. This will return a sample from the joint posterior distribution of the model's parameters. A sample in the MCMC context is also known as a chain.

First, we load the theophylline dataset and specify `SEX`

and `WT`

as covariates.

data = read_pumas(example_data("event_data/THEOPP"), covariates = [:SEX,:WT])

Population Subjects: 12 Covariates: SEX, WT Observables: dv

Next, we extract the initial parameters from `theopmodel_bayes`

. Alternatively, the initial values could be specified directly as a `NamedTuple`

with keys matching the parameter names used in `theopmodel_bayes`

.

param = init_param(theopmodel_bayes)

(θ = [2.0, 0.2, 0.8, 2.0], Ω = [1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0], σ = 0.5)

We can now infer the model's parameters given the data using the No U-Turn Sampler (NUTS). The number of samples and the number of steps used for adaptation are set with the `nsamples`

and `nadapts`

arguments respectively. The adaptation uses the prodecure introduced by Stan and it is currently not possible to adjust the adaption parameters.

result = fit(theopmodel_bayes, data, param, Pumas.BayesMCMC(); nsamples=2000, nadapts=1000)

Chains MCMC chain (2000×11×1 Array{Float64,3}): Iterations = 1:2000 Thinning interval = 1 Chains = 1 Samples per chain = 2000 parameters = θ₁, θ₂, θ₃, θ₄, Ω₁,₁, Ω₂,₁, Ω₃,₁, Ω₂,₂, Ω₃,₂, Ω₃,₃, σ Summary Statistics parameters mean std naive_se mcse ess rhat Symbol Float64 Float64 Float64 Float64 Float64 Float64 θ₁ 1.8721 0.4879 0.0109 0.0140 1058.7378 0.9995 θ₂ 0.0417 0.0080 0.0002 0.0003 750.2298 1.0013 θ₃ 0.4752 0.0613 0.0014 0.0018 909.7713 1.0000 θ₄ 1.8088 0.4659 0.0104 0.0143 850.1282 0.9995 Ω₁,₁ 0.5749 0.2584 0.0058 0.0074 1219.7135 0.9995 Ω₂,₁ -0.0546 0.1098 0.0025 0.0045 965.5898 1.0008 Ω₃,₁ -0.0078 0.0941 0.0021 0.0022 1030.1632 0.9997 Ω₂,₂ 0.2573 0.1093 0.0024 0.0034 1222.8134 1.0009 Ω₃,₂ 0.0296 0.0651 0.0015 0.0024 1074.5796 1.0000 Ω₃,₃ 0.1790 0.0824 0.0018 0.0031 825.4327 1.0028 σ 0.7050 0.0627 0.0014 0.0020 756.6869 1.0005 Quantiles parameters 2.5% 25.0% 50.0% 75.0% 97.5% Symbol Float64 Float64 Float64 Float64 Float64 θ₁ 1.0609 1.5257 1.8051 2.1496 2.9694 θ₂ 0.0301 0.0372 0.0411 0.0452 0.0557 θ₃ 0.3668 0.4334 0.4709 0.5102 0.6072 θ₄ 1.0529 1.4840 1.7697 2.0877 2.8534 Ω₁,₁ 0.2496 0.3956 0.5226 0.6889 1.2473 Ω₂,₁ -0.2847 -0.1079 -0.0510 0.0072 0.1632 Ω₃,₁ -0.2185 -0.0592 -0.0062 0.0474 0.1793 Ω₂,₂ 0.1238 0.1830 0.2345 0.3045 0.5528 Ω₃,₂ -0.0849 -0.0078 0.0242 0.0597 0.1827 Ω₃,₃ 0.0852 0.1269 0.1607 0.2083 0.3883 σ 0.6128 0.6666 0.7009 0.7361 0.8172

The show method for the result will print various summary statistics useful for evaluating the sample/chain. The summary output is based on some postprocessing by the MCMCChains.jl package. Hence, a similar output will be presented if the fitted model is converted to a `Chains`

object from `MCMCChains`

.

chains = Pumas.Chains(result)

However, the `MCMCChains`

package provides many other out of the box diagnostics and plotting functionality for MCMC chains. E.g. a default plotting method to that generates time series and kernel density plots for each of the parmeters.

plot(chains)

The plots are based on the complete chain, including the very unstable initial phase of the sampling. This can distort both the scale of the of the times series plot and the shape of the kernel density. Hence, it is often useful to exlcude the initial burn-in phase. This is easily done simply by slicing the `Chains`

structure, i.e.

plot(chains[200:end])