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The typical approach in approximate Bayesian computation (ABC) is to propose parameters from the prior, simulate data $\chi'_\text{sim}$ and then accept data that minimises the data misfit $\lambda$ with the observed data set $\chi'_\text{data}$:

$$\lambda = g\{\eta(\chi'_\text{sim}),\eta(\chi'_\text{data})\}$$

where $g(.)$ is some distance function and $\eta(.)$ are summary statistics. The typical approach is to set some threshold acceptance rate e.g. accept $\alpha$ = 1% of the proposals that have the smallest $\lambda$. However of that 1% some have a smaller data misfit $\lambda$ and are 'better' samples of the posterior than samples that have a larger $\lambda$. An alternative approach to selecting samples from the posterior is to assign all the proposals weights which are proportional to $\lambda$. The approximate posterior distribution is then dependent on all of the proposals (albeit with different weights). Within this context the typical approach of selecting some $\alpha$ is equivalent to selecting a shifted flipped heavy-side weighting function.

I am assume that is not a new idea so can you link me to literature that considers this approach? If this is a flawed idea could you explain the issues with this approach?

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This idea is discussed in Wilkinson (2008) and it is noted that 'ABC algorithms can be generalized by replacing the 0-1 cut-off with an acceptance probability that varies with the distance of the simulated data from the observed data'.

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  • $\begingroup$ This appears in many ABC papers, like Blum (2010), François and Blum (2010), and Fearnhead and Prangle (2012). $\endgroup$ – Xi'an Sep 14 '17 at 8:08

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