# ABC with non-uniform prior

I had asked some similar questions in the past, but I never got either the answers or the discussion I was hopping for. So I will rephrase the problem to see if I can understand it myself.

I'm trying to fit a complex model to some data that take a large amount of time to run. I'm also unable to write down a Likelihood function to this problem and so I turned to approximate Bayesian computation (ABC).Now, given the slowness of my simulations, I used Sequential ABC (a strategy where the prior are updated at each iteration), as implemented in R EasyABC.

I had realised that the result posterior distribution is highly variable (Sequential ABC get stuck in "local minimum"). This lead me to tun ABC_sequential more than once, and from different prior. In the end I gather all simulation that were done and now I want to analysed this.

I also need to state two other facts:

• I have very little knowledge to build the priors. In fact, contrary to the concept of Bayesian statistic (new knowledge updating old knowledge) I would like to remove all the influence of the priors from my estimates.

• The resulting distributions had several peaks (given from the different runs of ABC_sequential).

I designed also my custom distance function and computed a simple rejection algorithm of the best 5% simulations. However, as is probably obvious by now, this "posterior distribution" is highly influenced by the non-uniform original distributions. How can I remove this influence?

A common suggestion is importance sampling. As far as I understand, this implies computing the ratio between the density in my posterior and my prior and use this as a weight for my simulations. I did this with R package densratio and the results are unsatisfactory... Do you have any suggestions, corrections, etc?

• @Xi'an Can you elaborate? I did not understood what you mean by "reference probability measure". And isn't ABC simulation from prior? Thanks for the quick reply :) – Diogo Santos Apr 23 '18 at 10:35
• So in the context of rejection sampling, it's impossible to remove the influence of the prior?!! Can you elaborate on the reason why the solution I tried is not correct? – Diogo Santos Apr 23 '18 at 10:39
• @Xi'an Thanks for explaining the effect of merging the samples. What I keep wondering if there is really no way to remove the prior in this context... Because in theory, one could estimate or know the prior density and weight each point by it's inverse. Can you point me to some literature or explanation on why is this wrong? – Diogo Santos Apr 26 '18 at 8:05
• Can you expand on that? Maybe on an answer, as that seems to answer my question! :) – Diogo Santos Apr 26 '18 at 13:24
• If you don't like priors and want to go frequentist there's always indirect inference. (I like priors myself.) arxiv.org/abs/0908.0433 – Cyan Apr 29 '18 at 23:02

Given an intractable likelihood $f(x|\theta)$ of a generative model, associated with an observation $x^\text{obs}$, ABC produces simulations from the joint distribution $$\pi^\text{ABC}_{\epsilon}(\theta,z)\propto \pi(\theta)f(z|\theta) \mathbb{I}_{(0,\epsilon)}d(z,x^\text{obs})$$ or in the kernel generalisation (e.g., Fernhead and Prangle, 2012) $$\pi^\text{ABC}_{\epsilon}(\theta,z)\propto \pi(\theta)f(z|\theta) K(d(z,x^\text{obs})/\epsilon)$$ where $z$ denotes the pseudo-observation and $\epsilon$ the tolerance. There exist many ways of implementing this approach, but a core notion is that it always constitutes a simulation methodology on the parameter x pseudo-observation pair and hence that the parameter must be endowed with a probabilistic structure. Changing or updating the prior is of course feasible by different means, but I do not see a way to reinterpret the ABC output away from a Bayesian perspective.
For instance, reweighing each point $\theta_i$ of an ABC sample by the inverse of the prior density $\pi(\theta_i)^{-1}$ means replacing the initial prior $\pi(\cdot)$ with a flat (and possibly improper) prior. But not leaving the Bayesian paradigm.