  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, for example Clarke [1996] and
Morita et al. [2008]. However, these measures typically ignore the likelihood
for the inference currently at hand, and therefore address (i) rather than
(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 general mathematical form of the likelihood but also
the specific data at hand. Thus, our measures do not average across datasets
from the working model, but condition on the current observed data.
Consequently, our measures can be highly variable, but we demonstrate that this
is because the impact of a prior can be highly variable. Our measures are Bayes
estimates of meaningful quantities and well communicate the extent to which
inference is determined by the prior, or framed differently, the amount of
effort saved due to having prior information. We illustrate our ideas through a
number of examples including a Gaussian conjugate model (continuous
observations), a Beta-Binomial model (discrete observations), and a linear
regression model (two unknown parameters). Future work on further developments
of the methodology and an application to astronomy are discussed at the end.
