Working with data in the real world often leads to complex models. This can cause the Bayesian inference to be computationally challenging. This blog will give a short summary of a seminar given by Professor Chris Drovandi, from Queensland University of Technology, titled Statistical Inference for Implicit Models using Bayesian Synthetic Likelihood.

We know from Bayes’ theorem:

Posterior ∝ Likelihood*Prior

Typically, for real world problems the likelihood is incredibly complicated or cannot be derived. Likelihood-free methods are often used when the likelihood is intractable but we can still simulate from the model. Two common approaches are Approximate Bayesian computation (ABC) and Bayesian synthetic likelihood (BSL).

Both ABC and BSL can result in ill-behaved posteriors if the model is misspecified. With the BSL approach you should be able to use the data generating process to simulate summary statistics that capture the observed summary statistics. The model is said to be misspecified when the model and summaries are incompatible, i.e. it cannot simulate summaries which capture the observed summaries. Chris demonstrated work done on making BSL more robust to misspecification.

Two methods were explored in order to improve the robustness of BSL and mitigate the poor performance of BSL under model misspecification. The first method, mean adjustment (R-BSL-M), adjusts the mean of the simulated summaries. The second method, variance inflation (R-BSL-V), inflates the variance of the summary statistics. Interestingly, it was found that the variance inflation method not only made the BSL approach more robust to misspecification but also improved the computational efficiency of the method.

In an example looking at the movement of Fowler’s Toads modelled by Marchand et al (2017), the MCMC output for BSL, R-BSL-M and R-BSL-V gives an acceptance rate of 9%, 7% and 15% respectively. However, the BSL approach required 2000 simulations while the more robust methods only required 500 simulations. When considering both the acceptance rate and number of simulations there is a computational improvement of around one order of magnitude for the variance inflation method.

To finish, using a Bayesian synthetic likelihood approach offers another tool for dealing with intractable likelihoods and the additional work to make it more robust is particularly interesting, especially as it also offers computational improvements over BSL.

Header image credit: Frazier & Drovandi (2020)

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