Bayesian workflow for disease transmission modeling in Stan
Léo Grinsztajn, Elizaveta Semenova, Charles C. Margossian, Julien Riou
BBayesian workflow for disease transmission modeling in Stan
Léo Grinsztajn ∗ , Elizaveta Semenova † , Charles C. Margossian ‡ , Julien Riou § Abstract
This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and shouldbe useful to researchers interested in modeling the COVID-19 outbreak and doing Bayesian inference.Bayesian modeling provides a principled way to quantify uncertainty and incorporate prior knowledgeinto the model. What is more, Stan’s main inference engine, Hamiltonian Monte Carlo sampling, isamiable to diagnostics, which means we can verify whether our inference is reliable. Stan is an expressiveprobabilistic programing language that abstracts the inference and allows users to focus on the modeling.The resulting code is readable and easily extensible, which makes the modeler’s work more transparentand flexible. In this tutorial, we demonstrate with a simple Susceptible-Infected-Recovered (SIR) modelhow to formulate, fit, and diagnose a compartmental model in Stan. We also introduce more advancedtopics which can help practitioners fit sophisticated models; notably, how to use simulations to probe ourmodel and our priors, and computational techniques to scale ODE-based models.
Contents
Introduction 2
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Mathematical transmission model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Statistical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Coding the Model: Stan Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Complete Stan program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Checking our priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Can our inference algorithm recover the right parameters? . . . . . . . . . . . . . . . . . . . . . . . 18
The Computational cost of Stan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Partitioning the code into blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Propagating derivatives through ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Picking the right ODE integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ∗ École polytechnique, Palaiseau, France, [email protected] † Data Sciences and Quantitative Biology, Discovery Sciences, R&D, AstraZeneca, Cambridge, UK ‡ Department of Statistics, Columbia University, New York, NY, USA § Institute of Social and Preventive Medicine, University of Bern, Bern, Switzerland a r X i v : . [ s t a t . C O ] M a y ntroduction The ongoing pandemic of the Coronavirus COVID-19 has led to an increased interest in statistical diseasemodeling and, amongst other approaches, Bayesian modeling (e.g Flaxman et al. (2020); Riou et al. (2020);see this post on the Stan forum for an updated list of examples).Transmission models of infectious diseases can help answer questions about past and future transmission,including the effects of interventions.
Population-based models subdivide the total population into homogeneousgroups, called compartments . The flows between compartments can be described by a system of differentialequations. Individuals within a compartment are considered to be in the same state (e.g. susceptible, infected,etc.). Moreover, the time-dependent volume of each compartment solves a system of ordinary differentialequations (ODEs). Compartment models are relatively easy to formulate and computationally tractable.In this tutorial, we demonstrate with a simple example how to formulate and fit a typical compartmentalmodel in Stan. Stan is a probabilistic programming framework designed to let the user focus on modeling,while inference happens under the hood (Carpenter et al. 2017). This allows for faster implementation andextension of epidemiological models through a principled modeling workflow, and is thus particularly usefulwhen handling rapidly evolving data and knowledge. Stan is an expressive language that supports manyprobability densities, matrix operations, and numerical ODE solvers. We can combine all these elements tospecify a data generating process. We can also compute useful epidemiological parameters, such as the basicreproduction number , R , and make predictions.Generative models formulated in Stan can be used both for simulation and inference. Stan bolsters severalinference methods: full Bayesian inference using Markov chains Monte Carlo (MCMC), approximate Bayesianinference with variational inference, and penalized maximum likelihood estimation. We focus on Bayesianinference with MCMC. Bayesian inference gives us a principled quantification of uncertainty and the abilityto incorporate domain knowledge in the form of priors, while MCMC is a reliable and flexible algorithm.In addition, Stan provides diagnostic tools to evaluate both the inference (e.g. accuracy of the MCMC,convergence of chains) and the model (e.g. posterior predictive checks).Other tutorials on the subject include the work by Chatzilena et al. (2019) and Mihaljevic (2016) ontransmission models, and the case studies by Carpenter (2018), Weber (2018), and Margossian and Gillespie(2017a) on ODE-based models, all of which can serve as complementary reading. Outline
Through the lens of the Susceptible-Infected-Recovered (SIR) model, we show how to put together a principledBayesian workflow in Stan, allowing a faster development of reliable epidemiological models. In Section 1, weintroduce how to build, fit, and diagnose compartment models in Stan. The next sections discuss topics thatcan help practitioners fit more complex models. Section 2 reviews how we can use simulated data to examineour model and our priors, and provides an introduction to inference calibration. Section 3 offers a pragmaticdiscussion on how to efficiently implement and scale up ODEs in Stan. Throughout the tutorial, we useR as a scripting language , and, while we review some elementary concepts, assume the reader has basicfamiliarity with Bayesian inference and Stan. The source code of this case study can be found on Github ,and a HTML version is available . Note that Stan can also be used with other langages such as Python or Julia, see here for the list of Stan interfaces https://github.com/stan-dev/example-models/tree/master/knitr/disease_transmission https://mc-stan.org/users/documentation/case-studies/boarding_school_case_study.html Simple SIR
Data
In this example, we examine an outbreak of influenza A (H1N1) in 1978 at a British boarding school. Thedata consists of the daily number of students in bed, spanning over a time interval of 14 days. There were763 male students who were mostly full boarders and 512 of them became ill. The outbreak lasted from the22nd of January to the 4th of February. It is reported that one infected boy started the epidemic, whichspread rapidly in the relatively closed community of the boarding school. The data are freely available in theR package outbreaks, maintained as part of the R Epidemics Consortium. library (outbreaks) library (tidyverse) theme_set ( theme_bw ()) head (influenza_england_1978_school) ggplot (data = influenza_england_1978_school) +geom_point (mapping = aes (x = date, y = in_bed)) +labs (y = "Number of students in bed") date N u m be r o f s t uden t s i n bed Figure 1: Plot of the raw data.3 athematical transmission model
The Susceptible-Infected-Recovered (SIR) model splits the population in three time-dependent compartments:the susceptible, the infected (and infectious), and the recovered (and not infectious) compartments. When asusceptible individual comes into contact with an infectious individual, the former can become infected forsome time, and then recover and become immune. The dynamics can be summarized graphically:The temporal dynamics of the sizes of each of the compartments are governed by the following system ofODEs: dSdt = − βS INdIdt = βS IN − γIdRdt = γI where• S ( t ) is the number of people susceptible to becoming infected (no immunity),• I ( t ) is the number of people currently infected (and infectious),• R ( t ) is the number of recovered people (we assume they remain immune indefinitely),• β is the constant rate of infectious contact between people,• γ the constant recovery rate of infected individuals.Let’s give some intuition behind these ODEs. The proportion of infected people among the population is IN .At each time step, given uniform contacts, the probability for a susceptible person to become infected is thus β IN , with β the average number of contacts per person per time, multiplied by the probability of diseasetransmission when a susceptible and an infected subject come in contact. Hence, at each time step, βS IN susceptible individuals become infected, meaning βS IN people leave the S compartment and βS IN peopleenter the I compartment. Similarly, the recovery of an infected individual is taking place at rate γ , and thusthe number of infected individuals decreases with speed γI while the number of recovered grows at the samespeed.The above model holds under several assumptions:• births and deaths are not contributing to the dynamics and the total population N = S + I + R remainsconstant,• recovered individuals do not become susceptible again over time,• the infection rate β and recovery rate γ are constant,• the population is homogeneous,• individuals meet any other individual uniformly at random (homogeneous mixing) and recovery timefollows an exponential distribution with mean γ .• replacing the integer number of people in each compartement by a continuous approximation is legitimate(the population is big enough)In case of boarding school data, the spread of the disease has started with one infected individual which leadsto the initial conditions I (0) = 1 , S (0) = N − , R (0) = 0 . tatistical model We now introduce a sampling distribution (also termed likelihood ) p ( Y | θ )which tells us, given model parameters θ , how to generate data Y . Inference reverse-engineers the datagenerating process and asks: “given a model and observations, Y , what are plausible parameter values?” In aBayesian framework, the set of plausible values is characterized by the posterior distribution , p ( θ | Y ) . Bayes’ rule teaches us that p ( θ | Y ) ∝ p ( Y | θ ) p ( θ )where p ( Y | θ ) is the sampling distribution, p ( θ ) the prior distribution, and ∝ stands for “proportional to”.The prior encodes information about the parameters we have before observing the data. To summarize, aBayesian model couples a mathematical model of what we know about the parameters in the form of a priorand a sampling distribution, i.e. a model of the data generating process. Sampling distribution
Given transmission parameters and initial conditions, a compartmental model defines a unique solution foreach of the compartments, including the number of infected students, I ODE ( t ). We want to link this solutionto the observed data, i.e the number of students in bed, I obs ( t ), at each time point. The latter is a noisyestimate of the true number of infected students. We choose to model the number of students in bed with acount distribution – the Negative Binomial. This distribution allows us to use I ODE ( t ) as the expected valueand account for over-dispersion, through the parameter φ : I obs ( t ) ∼ NegBin( I ODE ( t ) , φ )This gives us p ( Y | θ ), with θ = ( β, γ, φ ), the parameters of the model. Prior distribution
We still need p ( θ ). With θ = ( β, γ, φ ), we specify a prior distribution for each of the three parameters.One advantage of the Bayesian approach is that we can formally incorporate prior knowledge about theparameters into the model. For instance, we specify γ ∼ N (0 . , .
5) (truncated at 0), which expresses ourbelief that γ has to be positive, and that P ( γ ≤
1) = 0 . a priori ). We can change this prior if more information becomes available, constraining our parameterestimation more tightly or, on the contrary, increasing its variance.Section 2 discusses how to check if our priors are consistent with available domain knowledge using priorpredictive checks . Predictions and derived quantities
Once we fit the model and obtain a posterior distribution for θ , we can derive additional quantities of interests.We can simulate predictions, Y pred , and work out the plausible range of new observations. Because thesepredictions are based on the posterior, p ( θ | Y ), our simulations account for uncertainty both in the datagenerating process and in our estimates of θ . Moreover, we compute a posterior distribution of predictions p ( Y pred | Y ) = Z p ( Y pred | θ ) p ( θ |Y ) dθ R . R is defined as the expected number of secondary infections producedfrom one infected individual in a fully susceptible population through the entire duration of the infectiousperiod. R > R < p ( R | Y ) . Coding the Model: Stan Program
We will need the following libraries library (rstan) library (gridExtra) rstan_options (auto_write = TRUE) options (mc.cores = parallel ::detectCores ()) set.seed (3)
Coding the ODE in Stan
An ODE takes the form d y d t = f ( t, y )where y are the states , in our example y = ( S, I, R ), and t is time. We also need an initial condition y at t and the times, τ , at which we evaluate the solution.To specify an ODE in Stan, we first code f in the functions block. This function must observe a strictsignature: real[] f (real time, real[] state, real[] theta,real[] x_r, int[] x_i) with• time , t ;• state , the volumes in each compartment, y ;• theta , variables used to compute f , which depend on the model parameters;• x_r , real variables used to evaluate f , which only depend on fixed data;• x_i , integer values used to evaluate f , which only depend on fixed data.We motivate this signature in our discussion on scaling ODEs (section 3).In our example, the ODEs for the SIR model is defined as follows: functions {real[] sir (real t, real[] y, real[] theta,real[] x_r, int[] x_i) {real S = y[1];real I = y[2];real R = y[3];real N = x_i[1];real beta = theta[1]; eal gamma = theta[2];real dS_dt = - beta * I * S / N;real dI_dt = beta * I * S / N - gamma * I;real dR_dt = gamma * I;return {dS_dt, dI_dt, dR_dt};}}
We evaluate the solution numerically by using one of Stan’s numerical integrators. We opt for the Runge-Kutta4 th / 5 th order but, as we discuss in section 3, the user can consider other options. A call to the integratorlooks as follows y = integrate_ode_rk45(sir, y0, t0, ts, theta, x_r, x_i); with• sir , the name of the function that returns the derivatives, f ;• y0 , the initial condition;• t0 , the time of the initial condition;• ts , the times at which we require the solution to be evaluated;• theta , x_r , x_i , arguments to be passed to f .We now have all the ingredients to solve our ODE.Note that in the given example, when we assume that the total population remains constant, the threederivatives dSdt , dIdt , dRdt sum up to 0: We can use this fact to improve computational efficiency of the sir function by deriving the value of dIdt from dSdt and dRdt : real dS_dt = -beta * I * S / N;real dR_dt = gamma * I;real dI_dt = -(dS_dt + dR_dt); Building the model in Stan
We next code the model in Stan, working through the various coding blocks. The functions block specifies sirfunctions {real[] sir (...) {...} //copy code from above } Fixed data is declared in the data block: data {int
7e code transforms of the data in the transformed data block. In this example, we transform our data tomatch the signature of sir (with x_r being of length 0 because we have nothing to put in it). transformed data {real x_r[0];int x_i[1] = { N };}
We next declare the model parameters. If you want some parameter to be bounded, and it is not alreadyguaranteed by his prior, you need to specify
And then transforms of the parameters transformed parameters{real y[n_days, 3];real phi = 1. / phi_inv;{ real theta[2];theta[1] = beta;theta[2] = gamma;y = integrate_ode_rk45(sir, y0, t0, ts, theta, x_r, x_i);}}
With the ODE solution, y , in hand, the only thing left to do is to code the prior and sampling distribution. model { //priors beta ~ normal(2, 1); //truncated at 0 gamma ~ normal(0.4, 0.5); //truncated at 0 phi_inv ~ exponential(5); //sampling distribution//col(matrix x, int n) - The n-th column of matrix x. Here the number of infected people cases ~ neg_binomial_2(col(to_matrix(y), 2), phi);} Untangled from the inference, we can calculate the basic reproduction number, R , and predictions for thenumber of cases in the generated quantities block: generated quantities {real R0 = beta / gamma;real recovery_time = 1 / gamma;real pred_cases[n_days];pred_cases = neg_binomial_2_rng(col(to_matrix(y), 2) + 1e-5, phi);} itting the model in R We now go to R and collect the data into a list. cases <- influenza_england_1978_school $ in_bed N <- 763; n_days <- length (cases)t <- seq (0, n_days, by = 1)t0 = 0t <- t[ - i0 <- 1s0 <- N - i0r0 <- 0y0 = c (S = s0, I = i0, R = r0) data_sir <- list (n_days = n_days, y0 = y0, t0 = t0, ts = t, N = N, cases = cases) niter <- 2000 Next we compile the model, saved in the file sir_negbin.stan , model <- stan_model ("sir_negbin.stan") and run MCMC. For this problem, it suffices to use Stan’s defaults. Note that, as is standard practice, werun 4 Markov chains. fit_sir_negbin <- sampling (model,data = data_sir,iter = niter,chains = 4) Checking the inference
Let’s specify the parameters of interest. pars= c ('beta', 'gamma', "R0", "recovery_time") We start with a summary table of the results, which displays the posterior mean, standard error, quantiles,and some useful diagnostics. print (fit_sir_negbin, pars = pars) Stan gives us a host of information to evaluate whether the inference is reliable. During sampling, warningscan tell us if something is wrong (here we have no warnings ). In the summary table, several quantities areavailable to check inference. Here we note that b R is close to 1 (< 1.01), indicating the 4 Markov chains arein close agreement with one another. Furthermore the effective samples size, n eff , is large (> 100 ), whichmeans the Markov chains were able to cohesively explore the parameter space. Conversely, large b R and low n eff would indicate that the Markov chains are poorly mixing. Apart from fixing coding errors, improvingthe mixing of the Markov chains almost always requires tweaking the model specification, for example with areparameterization or stronger priors.We can furthermore plot the marginal posterior densities and confirm the Markov chains are in agreementwith one another. stan_dens (fit_sir_negbin, pars = pars, separate_chains = TRUE) R0 recovery_timebeta gamma chain
Figure 2: Posterior densities of model parameters. Note that getting a few warnings is not always concerning, especially if they only occur during the warmup phase. See herefor an overview of Stan’s warnings n eff ≥
100 is considered sufficient to estimate the posterior mean correctly. Note that we may need a bigger n eff to estimateextreme quantiles. hecking the model Now that we trust our inference, let us check the utility of our model. Utility is problem specific and caninclude the precise estimation of a quantity or predicting future behaviors. In general, it is good to check ifour model, once fitted, produces simulations that are consistent with the observed data. This is the ideabehind posterior predictive checks .We sample predictions, Y pred , from p ( Y pred | Y ) and use these samples to construct a fitted curve for studentsin bed, together with the uncertainty (90% interval, meaning observed data is expected to fall outside ofthis interval one in ten times). This posterior predictive check allows us to verify if the model captures thestructure of the data. Here we see that the model gives a satisfying fit to the data, and that the modeluncertainty is able to capture the variation of the data. smr_pred <- cbind ( as.data.frame ( summary (fit_sir_negbin, pars = "pred_cases", probs = c (0.05, 0.5, 0.95)) $ summary), t, cases) colnames (smr_pred) <- make.names ( colnames (smr_pred)) ggplot (smr_pred, mapping = aes (x = t)) +geom_ribbon ( aes (ymin = X5., ymax = X95.), fill = "orange", alpha = 0.6) +geom_line (mapping = aes (x = t, y = X50.)) +geom_point (mapping = aes (y = cases)) +labs (x = "Day", y = "Number of students in bed") Day N u m be r o f s t uden t s i n bed Figure 3: Posterior predictive check.Maybe we also want to access the true number of infected people at each time, and not just the number ofstudents in bed. This is a latent variable for which we have an estimation. params <- lapply (t, function (i){ sprintf ("y[%s,2]", i)}) smr_y <- as.data.frame ( summary (fit_sir_negbin, ars = params, probs = c (0.05, 0.5, 0.95)) $ summary) colnames (smr_y) <- make.names ( colnames (smr_y)) ggplot (smr_y, mapping = aes (x = t)) +geom_ribbon ( aes (ymin = X5., ymax = X95.), fill = "orange", alpha = 0.6) +geom_line (mapping = aes (x = t, y = X50.)) +labs (x = "Day", y = "Number of infected students") Day N u m be r o f i n f e c t ed s t uden t s Figure 4: Estimated number of infected students per day.
Complete Stan program functions {real[] sir(real t, real[] y, real[] theta,real[] x_r, int[] x_i) {real S = y[1];real I = y[2];real R = y[3];real N = x_i[1];real beta = theta[1];real gamma = theta[2];real dS_dt = -beta * I * S / N;real dI_dt = beta * I * S / N - gamma * I;real dR_dt = gamma * I; eturn {dS_dt, dI_dt, dR_dt};}}data {int
We fitted a simple model to a well-understood data set. In practice, we must proceed more cautiously andprobe the behavior of our model and our inference algorithm. To that end, working with fake data can be avery productive endeavour.
Checking our priors
We can check if our priors are sound by computing the a priori probability of various epidemiologicalparameters of interest. For instance for influenza, we know from domain knowledge that R is typicallybetween 1 and 2, and that the recovery time is approximately 1 week. We want priors that allow for everyreasonable configurations of the data but exclude pattently absurd scenarios, per our domain expertise. Tocheck if our priors fulfill this role, we can do a prior predictive check .To conduct a prior predictive check, we take the same model as before, put the parameters of interestin the generated_quantities code block, and remove the sampling distribution term from the model.Without the sampling distribution, the parameters are not fitted to the data and are thus sampled fromtheir prior distribution. The Stan code is thus the same as the final Stan code, without the cases ~neg_binomial_2(col(to_matrix(y), 2), phi); line. A useful trick to make prior predictive check easy isto add a switch compute_likelihood to the data. Then in the model code block : if (compute_likelihood == 1)cases ~ neg_binomial_2(col(to_matrix(y), 2), phi); This allows to do prior predictive check and inference with the same Stan file.We compile the model without the likelihood term model <- stan_model ("sir_prior.stan") and sample from it. fit_sir_prior <- sampling (model,data = data_sir, seed = 0, chains = 4)
This gives us samples from the a priori distribution of parameters, which we can visualize. Here we show thedistribution of the log of the recovery time, with the red bars showing loose bounds on the recovery time (1/2day and 30 days). We observe that most of the probality mass is between the red bars but we still allowmore extreme values, meaning our posterior can concentrate outside the bars, if the data warrants it. s_prior <- rstan ::extract (fit_sir_prior)df_test <- tibble (r = s_prior $ recovery_time) ggplot (data = df_test) +geom_histogram (mapping = aes (x = r), bins = 30) +labs (x = "log(recovery time)") +geom_vline (xintercept = 0.5, color = "red") +geom_vline (xintercept = 30, color = "red") +scale_x_log10 () log(recovery time) c oun t Figure 5: Prior predictive check of the log-recovery time.We can do the same thing for R (again, on the log-scale), the loose bounds being 0.3 and 30. df_test <- tibble (r = s_prior $ R0) ggplot (data = df_test) +geom_histogram (mapping = aes (x = r), bins = 30) +labs (x = "log(R0)") +geom_vline (xintercept = 0.3, color = "red") +geom_vline (xintercept = 30, color = "red") +scale_x_log10 () log(R0) c oun t Figure 6: Prior predictive check of the log- R .We thus see that these distributions are coherent with domain knowledge. See here for more recommendationson prior choice. We can also plot trajectories of infection according to the prior, that is the number of infected people at eachtime accoring to prior distributions of parameters. n_draws <- 1000draws <- as_tibble ( t (s_prior $ y[,,2][1 : n_draws,])) %>% add_column (t=t)draws <- pivot_longer (draws, c (1 : %>%ggplot () +geom_line (mapping = aes (x = t, y=value, group = draw), alpha = 0.6, size=0.1) +geom_hline (yintercept=763, color="red") +geom_text (x=1.8, y=747, label="Population size", color="red") +labs (x = "Day", y="Number of infected students") Previoulsy, we fitted the data with these priors and found a posteriori R ~ 3 and a recovery time of ~ 2 days. This is quiteunexpected from our basic domain knowledge, but can probably be explained ( R bigger among students? isolated studentscounts for recovered in the model? etc). 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sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation sizePopulation size Day N u m be r o f s t uden t s i n bed Figure 8: Prior predictive check for the number of students in bed per day.It seems that most trajectories are reasonable and quite diverse. Still, some of the curves look a little bitfunky and suggest we could refine our priors and make them more informative, although it may not be neededhere.Typically, we can get away with priors that do not capture all our a priori knowledge, provided the data isinformative enough. However when dealing with complicated models and relatively sparse data, we usuallyneed well constructed priors to regularize our estimates and avoid non-identifiability.
Can our inference algorithm recover the right parameters?
While there exist many theoretical guarantees for MCMC algorithms, modelers should realize that these relyon a set of assumptions which are not always easy to verify and that many of these guarantees are asymptotic .This means they are proven in the limit where we have an infinite number of samples from the posteriordistribution. A very nice, if advanced, review on the subject can be found in Roberts and Rosenthal (2004).As practitioners, we must contend with finite computational resources and assumptions which may or maynot hold. The diagnostics we reviewed earlier, e.g. b R , effective sample sizes, provide necessary conditionsfor the MCMC sampler to work but not sufficient ones. Nevertheless they are potent tools for diagnosingshortcomings in our inference. This section provides further such tools, from both a rigorous and a pragmaticperspective.Fitting the model to simulated data is, if done properly, an effective way to test whether our inference18lgorithm is reliable. If we cannot successfully fit the model in a controlled setting, it is unlikely we can do sowith real data. This of course raises the question of what is meant by “successfully fitting” the model. In aBayesian setting, this means our inference procedure recovers the correct posterior distribution. Unfortunately,even in a controlled setting, the posterior distribution is, but in the simplest cases, not tractable.A powerful method to check the accuracy of our Bayesian inference is simulation-based calibration (SBC)(Talts et al. 2018). SBC exploits a very nice consistency result. The intuition is the following: if we draw ourparameters from our prior distribution θ , ..., θ ∼ p ( θ )and for each θ i simulate a data set Y i , we can by fitting the model multiple times recover the prior distributionfrom the estimated posteriors. This technique is a bit beyond the scope of this tutorial, though we vividlyencourage the reader to consult the original paper, or to see here how this method fits in a principled Bayesianworkflow.For the time being, we focus on a simpler heuristic: fit the model to one simulated data set and check if werecover the correct parameter value. There are serious limitations with this approach: when do we considerthat the estimated posterior distribution covers the correct value? How do we know if the variance of theposterior is properly estimated? etc. But the test is useful: in this controlled setting, do the chains converge?Is the computation of the log density numerically stable (e.g. are we using the right ODE integrator)? Do mypriors prevent the chains from wondering into absurd regions of the parameter space? These are all questionsthis simple test can help us tackle.We take one arbitrary draw from the prior distribution draw <- 12 cases_simu <- s_prior $ pred_cases[draw,] And use it as data which we fit to our model. data_simu <- list (n_days = n_days, y0 = y0, t0 = t0, ts = t, N=N, cases=cases_simu)fit_simu <- sampling (model, data=data_simu, chains=4)
We can then examine the estimated posterior distribution. params = c ("beta", "gamma", "phi") paste ("true beta :", toString (s_prior $ beta[draw])) paste (", true gamma :", toString (s_prior $ gamma[draw])) paste (", true phi :", toString (s_prior $ phi[draw])) print (fit_simu, pars = params) We plot the posterior density (in red) to check if it matches the true value of the parameter (black line).The density is compatible with the true parameters, although not always centered on it. The latter is notalarming, especially if the model parameter θ we sampled, lies on the tail of the prior distribution. We couldrepeat this process a few times to get a better sense of the performance of our inference algorithm. plot_beta <- stan_dens (fit_simu, pars="beta") +geom_vline (xintercept =s_prior $ beta[draw])plot_gamma <- stan_dens (fit_simu, pars="gamma") +geom_vline (xintercept = s_prior $ gamma[draw])plot_phi <- stan_dens (fit_simu, pars="phi_inv") +geom_vline (xintercept = s_prior $ phi_inv[draw]) grid.arrange (plot_beta, plot_gamma, plot_phi, nrow=1) beta gamma phi_inv Figure 9: Estimated posterior distributions.
Doing MCMC on ODE-based models can be computationally intensive, especially as we scale up the numberof observations, parameters, and start using more sophisticated ODEs. If we want to ripe the benefits of afull Bayesian inference, we have to pay the computational cost. But while we cannot get away with a freelunch, we can avoid an overpriced one. Stan is a flexible language, which means there are worse and betterways of coding things. This section develops a few principles to make ODE models in Stan more scalable,drawing on our experience with an SEIR model of Covid-19 (Riou et al. 2020).20 he Computational cost of Stan
Stan abstracts the inference away from the modeling but it’s worth taking a peek inside the black box.Markov chains are constructed using a dynamic Hamiltonian Monte Carlo (HMC) sampler (e.g Betancourt2018; Hoffman and Gelman 2014). To run the sampler, we need to specify the log joint density over theobservations, Y , and the latent variables, θ ,log p ( Y , θ ) = log p ( Y | θ ) + log p ( θ )which conveniently splits between a log likelihood and a log prior. This is ultimately what we specify in the model block. In addition, Hamiltonian Monte Carlo uses the gradient with respect to θ , ∇ θ log p ( Y , θ ) = ∇ θ log p ( θ | Y ) . The gradient contains information about the posterior distribution which, when properly harnessed, allows usto efficiently sample from the posterior. Fortunately, the user does not need to specify the gradient. Instead, automatic differentiation generates the requisite derivatives under the hood, based on computer code toevaluate the log joint density . But, just like evaluating log p ( Y , θ ), differentiation has a computational cost.HMC simulates trajectories across the parameter space by numerically solving Hamilton’s equations of motion.We solve these equations using a numerical leapfrog integrator. At each step the integrator takes, we need to evaluate and differentiate log p ( Y , θ ). Partitioning the code into blocks
This perspective informs how each model block scales. The data and parameters blocks are used to declarevariables. The transformed data block is evaluated once. The transformed parameters and model blocksare evaluated and differentiated at each leapfrog step, which is multiple times per iteration. The generatedquantities block is evaluated once per iteration. Hence operations in the parameters and transformedparameters blocks dominate the computational cost and should only entail operations that depend on θ andare required to compute log p ( Y , θ ).In our boarding school model, the computation of R , for example, is relegated to generated quantities .Even though we want samples from p ( R | Y ) and R depends on the model parameters, R does notcontribute to log p ( Y , θ ). In the model by Riou et al. (2020), we integrate an ODE – denoting the solution y –in transformed parameters from t to τ , the time of the last observation Z τt f ( y, t )d t where d y d t = f ( y, t ) . We then make additional predictions, from time τ to τ , Z τ τ f ( y, t )d t. Because these predicted solutions do not impact our likelihood, we do these calculations in generatedquantities . Compared to the first integral, this second integration is computed a small number of timesand it is not differentiated, making its computational cost marginal. Though heavily used in computational statistics and machine learning, automatic differentiation remains an arcane subjectfor many practitioners; for a review on the subject, we recommend (Margossian 2019). A more comprehensive treatment can befound in (Griewank and Walther 2008). ropagating derivatives through ODEs Our ODE is defined by d y d t = f ( y, t, ϑ, x ) . Here, ϑ contains inputs to f that depend on the model parameters, θ , while x contains inputs which do not depend on θ and therefore remain fixed as the Markov chain moves through the parameter space. Note thatin general, ϑ = θ . In the SIR model for example, ϑ is defined as theta[1] = beta;theta[2] = gamma; and contains two model parameters. Specifically ϑ = ( β, γ ), while θ = ( β, γ, φ − ). To define the integral, weadditionally specify an initial time, t , times of integration, τ , and an initial condition, y , all of which canvary with θ . Hence when propagating derivatives to compute the gradient of the log joint density, we need toworry about how the solution varies with respect to ϑ and potentially y . We say that these elements are varying parameters and denote K the number of such elements. Furthermore, let N be the number of states ,that is the length of y or in a SIR model, the number of compartments.In Stan, we propagate derivatives by solving a coupled system of ODEs. The intuition is the following.Suppose we want to compute d y d ϑ . We do not have an analytical expression for y , so a direct application of automatic differentiation is notfeasible. But we can, assuming the requisite derivatives exist, computed f d ϑ = dd t d y d ϑ and then integrate this quantity numerically. The end result is that, instead of only solving for the N originalstates, we solve an N + N K system to both evaluate and differentiate y .Strictly speaking, we do not need to explicitly compute d y/ d ϑ to propagate derivatives; this is an important,if somewhat counter-intuitive, result of automatic differentiation, and motivates a so-called adjoint method ,which only requires solving 2 N + K ODE states. While this method is very much on our todo list, it is notyet implemented in Stan. But the takeaway is the same for both methods: we should minimize K as much aspossible. Splitting fixed and varying parameters
This sheds light on the function signature of Stan’s numerical integrator. For every element in theta , we addan additional N states to solve for. Hence, components which do not depend on θ should be relegated to x_r and x_i . Fixing the initial conditions (where we can)
Suppose our initial condition, y , are varying parameters, i.e. depend on the model parameters. It is notuncommon for some of the elements in y to not depend on θ . For example, in a compartment model, theinitial condition for the first compartment may depend on model parameters, while it is 0 for the othercompartments. More generally, y may only depend on k < N parameters. The straightforward approach isto pass y as a vector of parameters. Stan interprets this as N additional varying parameters, which meansthe number of ODE we solve increases by N . This is overpriced lunch! While t and τ may depend on model parameters, we here assume they do not to avoid some minute technicalities andsimplify our discussion. For a discussion on adjoint methods, we recommend Hindmarsh and Serban (2020).
22 better, if more intricate, approach is to solve the ODEs for deviations from the baseline and split y between theta and x_r .Concretely, let z = y − y . The initial condition for z is then , an N -vector of 0’s and a fixed quantity. Now,d z d t = d y d t = f ( z + y , t, ϑ, x ) = e f ( z, t, e ϑ, e x )where e f is the same map as f , but applied to z + y instead of z ; e ϑ contains ϑ and the elements of y thatdepend on θ ; and e x contains x and the elements of y that are fixed. With this implementation, K is kept toa minimum. We recover the original y simply by adding y to z .In the SEIR model by (Riou et al. 2020), we have 58 initial conditions but together these depend on a singleparameter. ϑ itself contains 4 elements. Reparameterizing the ODE means we go from K = 62 to K = 5,that is from solving 3596 coupled ODEs to only solving 290. Picking the right ODE integrator
The task of solving and differentiating an ODE boils down to integrating a coupled ODE system. Hence tosure reasonable performance, it is crucial to pick the right solver.
Arsenal of tools
Stan provides a few options (e.g. Margossian and Gillespie 2017a). When available, the user can specifyanalytical solutions. If the ODE is linear, that is it has the formd y d t = Ay for some matrix A , then the solution is e tA y , where e designates the matrix exponential. Stan supports thecomputation and differentiation of the matrix exponential. These methods usually run faster than numericalintegrators and should be used when possible.The majority of the time, we deal with nonlinear ODEs with no analytical solution and must resort tonumerical integrators. In the special case where the ODE can be decoupled, it is possible to combine differentintegration methods. For example, the ODE may have the formd y d t = f ( y , t )d y d t = f ( y , y , t )in which case we can independently solve for y and then solve for y . This scenario arises when y is subjectedto a forcing function, which is itself the solution to a (simple) ODE. We can then solve y analytically and y numerically, and get a decent speed up (e.g. Margossian and Gillespie 2017b). But we may be tradingcomputational time for coding time. Numerical integrators
Stan supports a Runge-Kutta (rk45) method for non-stiff systems, integrate_ode_rk45 and a backward differentiation (bdf) algorithm for stiff systems23 ntegrate_ode_bdf
There is no formal definition of stiffness but the general idea is that the phenomenon occurs when the timestep of the integrator needs to be extremely small – smaller than what is needed to achieve the requiredaccuracy – in order to make the integrator stable. Stiffness can arise when the scale of the solution varieslargely as a function of t . The rk45 integrator is typically faster, so we recommend it as a starting point. Ifhowever the system is stiff, the rk45 integrator will be numerically unstable. It is not uncommon for the bdfmethod, when it is warranted, to produce much more efficient inference.Users should therefore be prepared to adjust their solver. This can mean switching from rk45 to bdf, oradjusting the tuning parameters of the integrator, namely the relative and absolute tolerances, and themaximum number of steps. These features are detailed in the user manual. When an integrator fails to solvean ODE, Stan issues a warning message and rejects the iteration being computed. An excessive number ofsuch failures can indicate the integrator needs to be adjusted.For certain problems, knowing a priori if a system is stiff may not be obvious. What is even less obvious iswhether a coupled system is stiff. And what is yet again less obvious is whether a system is stiff or nonstiffacross the range of parameters the Markov chain explores, both during the warmup and the sampling phases.During warmup the chain can indeed land in odd regions of the parameter space and the ODE can have anunexpected behavior. This is not a fault of the sampler, per se, because we need to explore the parameterspace in order to find regions where the probability mass is high. But it may be a fault of the model if ourpriors allow for the parameters to take absurd values. Constructing more informative priors is then well spenteffort, very much in line with what the folk theorem of statistical computing prescribes. The simple SIR model we have focused on is just one disease progression model that you might fit in Stan.Stan is meant to be flexible because modeling shouldn’t be a cookie-cutter process. Here are a number ofextensions to the model that might be useful to consider in different scenarios.1. One of the assumptions of the current model was that the rate β is constant. Another possibility is β could be modeled as a function of different interventions applied to the system over time.2. The boarding school data describes an isolated population where all of the cases can be observed andrecorded. In a less controlled environment, however, under-reporting might take place. In such cases,models need to be corrected to adjust for the measurement procedure. Here both the selection of thetested population, as well as the test sensitivity might play a role.3. We have treated the number of students in bed on every given day as prevalence data. In many real lifesituations, the reported counts would represent the number of new cases on each given day constitutingincidence data. In the simple model above, incidence can be computed as for (i in 1:n_days-1)incidence[i] = y[i, 1] - y[i + 1, 1]; //S(t) - S(t + 1) Model formulation can then be adjusted as cases ~ neg_binomial_2(incidence, phi);
4. If data for more than one compartment is available, it could be used as well for model fitting in thesame framework. For instance, in the presence of the ‘death due to disease’ compartment, the datablock would have an additional entry int deaths[n_days]; " When you have computational problems, often there’s a problem with your model.", see https://statmodeling.stat.columbia.edu/2008/05/13/the_folk_theore/. deaths ~ neg_binomial_2(col(to_matrix(y),4), phi_death); where y is the 4-entry state of the system, and φ death is the death-specific overdispersion parameter.5. Further extensions could include an adjustment for the natural birth and death and temporal effect ofimmunity.6. Heterogeneity of the population could be reflected in separate compartments for different age groupsand geographical regions.7. Spatial heterogeneity could be modelled either via metapopulation models or models capturing neigh-bouring structure explicitly, such as CAR models. Acknowledgments
The author thank the U.S. National Science Foundation, the Office of Naval Research, and the Institutefor Education Sciences for partial support of this work, as well as Ben Bales and Andrew Gelman for theirhelpful comments.
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