simulated summary statistics [in the sky]

Thinking it was related with ABC, although in the end it is not!, I recently read a baffling cosmology paper by Jeffrey and Abdalla. The data d there means an observed (summary) statistic, while the summary statistic is a transform of the parameter, μ(θ), which calibrates the distribution of the data. With nuisance parameters. More intriguing to me is the sentence that the correct likelihood of d is indexed by a simulated version of μ(θ), μ'(θ), rather than by μ(θ). Which seems to assume that the pseudo- or simulated data can be produced for the same value of the parameter as the observed data. The rest of the paper remains incomprehensible for I do not understand how the simulated versions are simulated.

“…the corrected likelihood is more than a factor of exp(30) more probable than the uncorrected. This is further validation of the corrected likelihood; the model (i.e. the corrected likelihood) shows a better goodness-of-fit.”

The authors further ressort to Bayes factors to compare corrected and uncorrected versions of the likelihoods, which leads (see quote) to picking the corrected version. But are they comparable as such, given that the corrected version involves simulations that are treated as supplementary data? As noted by the authors, the Bayes factor  unsurprisingly goes to one as the number M of simulations grows to infinity, as supported by the graph below.

One Response to “simulated summary statistics [in the sky]”

  1. Regarding the last point on the comparison of the corrected vs non-corrected versions of the likelihood, my understanding is that the non-corrected version is also conditional on the simulations, it just ignores the variance across different simulations.

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