Can I use the Bhattacharyya distance as an acceptance criterion for Approximate Bayesian Computation?

Question

I am researching the spread of a disease through a population and want to capture the behavior of this disease with a model.

I already have a model and patient data. The data is a value per patient that is more or less lognormally distributed. I would like to tune the parameters of the model based on the patient data with MCMC. The acceptance criterion would be P(parameters | data). Unfortunately, calculating this likelihood is computationally unfeasible. This led me to using Approximate Bayesian Computation:

I pick parameters, generate data from the model and compare the generated data to the real data. The acceptance criterion is then based on some metric to compare the two datasets. There exist many ways to calculate the distance between distributions. The most intuitive one to pick for me would be the Bhattacharyya distance, but I can't find any literature about it in the context of Approximate Bayesian Computation.

If anyone could give me references where such a distance is used for approximate Bayesian computation or in combination with MCMC, I would greatly appreciate it.

Answer 1

The Bhattacharyya distance is a distance between distributions while, what you need to conduct ABC, is a distance between summary statistics (this is, vectors of numbers that summarise the information in the sample), unless your summary statistic is a functional approximation to the distribution, which I doubt. For this reason, people typically employ the Euclidean distance (more sophisticated and better distances have also been studied).

Have a look at the following references for more details on the ingredients of ABC:

1. https://darrenjw.wordpress.com/2013/03/31/introduction-to-approximate-bayesian-computation-abc/
2. http://www0.cs.ucl.ac.uk/staff/C.Archambeau/AIS/Talks/rwilkinson_ais08.pdf
3. https://www.youtube.com/watch?v=8TGkrkK6pq4

Answer 2

Since ABC is an approximate method, the simplest answer to your question is that you can use any distance you find to your taste! Provided it is a true distance, the basis justification holds that $$\pi_\epsilon(\theta|y^{\text{obs}})=\pi(\theta|d(y(\theta), y^{\text{obs}})<\epsilon)$$where $y(\theta)\sim f(y|\theta)$ is the pseudo-data, converges to$$\pi(\theta|y^{\text{obs}})$$as $\epsilon$ goes to zero.