Bayesian parameter estimation results in a posterior distribution for model parameters. The user may or may not be interested equally much in all properties of the distribution. Perhaps the user mainly cares for a certain characteristic, e.g. mean, median or a tail quantile.
Equivalently (?), different users may be facing different decision problems and may have different loss or utility functions w.r.t. the estimation result (which in the Bayesian case is a whole distribution).
Question: Do user preferences affect the Bayesian estimation in any way? Or is the estimation unaffected, and only after the posterior is obtained does one consider how to optimally extract information from the posterior to minimize the user's loss function / maximize the utility function?
( I do not immediately see how the prior or the likelihood could be affected by the consideration of the user's preferences. It seems to me the standard practice would be to obtain the posterior without taking this into account and only then derive the properties of the posterior that are of interest (mean, median, tail quantile, ...). I wonder if this "ignorance" at the estimation stage is optimal in any sense. )
P.S. Perhaps the user's preferences affect model formulation / model selection rather than estimation? For example, in the frequentist case, there is the Focused Information Criterion (FIC) which directly reflects the intended use of the model. As Elliott & Timmermann say in Section 1.1.1 of their book "Economic Forecasting" (2016).
Different forecasters approaching the same outcome may well have different loss functions which could result in different choices of forecasting models for the same outcome.