Inference in Epidemic Models without Likelihoods

Abstract

Likelihood-based inference for epidemic models can be challenging, in part due to difficulties in evaluating the likelihood. The problem is particularly acute in models of large-scale outbreaks, and unobserved or partially observed data further complicates this process. Here we investigate the performance of Markov Chain Monte Carlo and Sequential Monte Carlo algorithms for parameter inference, where the routines are based on approximate likelihoods generated from model simulations. We compare our results to a gold-standard data-augmented MCMC for both complete and incomplete data. We illustrate our techniques using simulated epidemics as well as data from a recent outbreak of Ebola Haemorrhagic Fever in the Democratic Republic of Congo and discuss situations in which we think simulation-based inference may be preferable to likelihood-based inference. KEYWORDS:× exp − β N S t (i−1) I t (i−1) + δE t (i−1) + γI t (i−1) t (i) − t (i−1)

A tutorial introduction to Bayesian inference for stochastic epidemic models using Approximate Bayesian Computation

“…On the other hand, simulating realisations from a stochastic epidemic model is relatively straightforward. Therefore, ABC algorithms are very well suited to make inference for the parameters of epidemic models based on partially observed data and this has been illustrated when both temporal (McKinley et al, 2009) and non-temporal data Neal (2012) are available.…”

“…An intuitive distance metric is the sum-of-squared differences between observed and simulated counts (L 2 -norm) or perhaps a sum-of-absolute differences (L 1 -norm). Another option is to use a distance metric based on a chi-squared goodness-of-fit criterion (see, McKinley et al, 2009). This is very similar to an L 2 -norm but with the contribution at each bin scaled by the observed data (number of removals in each bin).…”

“…In the standard M-H algorithm one has the ratio of the likelihoods evaluated at the candidate and the current value of the Markov chain, whist in ABC-MCMC the likelihood terms are each approximated by the proportion of K simulations that "match the data". McKinley et al (2009) demonstrated that repeating simulations of the data given θ does not seem to contribute (much) to an improved approximation of the posterior by the ABC sample. This observation appears to be consistent with the findings of Bornn et al (2016) who proved that K = 1 is usually very close to optimal.…”

An overview on Approximate Bayesian computation

“…Joyce and Marjoram (2008) have proposed an inclusion-exclusion scheme assuming we have at our disposal a large set of univariate summary statistics. This scheme is related to the simulation experiments performed by McKinley et al (2009). A review discussing dimension reduction methods in ABC has been written by Blum et al (2012).…”

Rapid simulation of spatial epidemics: A spectral method

“…Moreover, this problem is compounded whenever insight requires the sweeping of large regions of plausible parameter space, or when the results of simulation are required for parameter inference, e.g. when using an approximate bayesian computation (ABC) method [25,26,27]. The requirement to perform very many independent realisations puts great emphasis on the development of highly e cient algorithms for stochastic simulation, even at the cost of mild inexactness (an example of such an approach is [28]).…”