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Abstract:
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We distinguish two questions (i) how much information does the prior contain? and (ii) what is the effect of the prior? Several measures have been proposed for quantifying effective prior sample size, see for example Clark (1996) and Morita et al. (2008). However, these measures typically ignore the likelihood for the inference currently at hand, and therefore address (i) rather (ii). Since in practice (ii) is of great concern, Reimherr et al. (2014) introduced a new class of effective prior sample size measures based on prior-likelihood discordance. We take this idea further towards its natural Bayesian conclusion by proposing measures of effective prior sample size that not only incorporate the likelihood but also the data at hand. Thus, our measures do not average across all possible datasets, or across all possible likelihoods. Consequently, our measures can be highly variable, but we demonstrate that this is because the impact of a prior can be highly variable. We therefore conclude that high variability is both necessary and desirable for measures that quantify the effect of priors. This work is motivated by an instrument calibration problem in astronomy.
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