ABConic mean evidence approximation

Following a question on X validated about evidence approximation in ABC settings, i.e., on returning an approximation of the evidence based on the outputs of parallel ABC runs for the models under comparison, I wondered at the relevance of an harmonic mean estimator in that context.

Rather than using the original ABC algorithm that proposes a model, a parameter from that model, and a simulated dataset from that model with that parameter, an alternate, cost-free, solution would be to run an ABC version of [harmonic mean evidence approximation à la Newton & Raftery (1994). Since

$\mathcal Z=1\Big/\int \dfrac{\pi(\theta|D)}{p(D|\theta)}\,\text d\theta$

the evidence can formally be approximated by

$\hat{\mathcal Z} =1\Big/\frac{1}{N}\sum_{i=1}^N\frac{1}{p(D|\theta_i)}\qquad\theta_i\sim\pi(\theta|D)$

and its ABC version is

$\hat{\mathcal Z} =1\Big/\frac{1}{N}\sum_{i=1}^N\frac{1}{K_\epsilon(d(D,D^\star(\theta_i)))}\qquad\theta_i\sim\pi^\epsilon(\theta|D)$

where K_ε(.) is the kernel used for the ABC acceptance/rejection step and d(.,.) is the distance used to measure the discrepancy between samples. Since the kernel values are already computed for evidence, the cost is null. Obviously, an indicator kernel does not return a useful estimate but something like a Cauchy kernel could do.

However, when toying with a normal-normal model and calibrating the Cauchy scale to fit the actual posterior as in the above graph, the estimated evidence 5 10⁻⁵ proved much smaller than the actual one, 8 10⁻².

This entry was posted on March 7, 2023 at 12:23 am and is filed under Statistics with tags ABC, Approximate Bayesian computation, Cauchy distribution, cross validated, harmonic mean estimator, statistical evidence. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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