An interesting paper came out on arXiv in early December, written by Michael Brand from Monash. It is about risk-adverse Bayes estimators, which are defined as avoiding the use of loss functions (although why avoiding loss functions is not made very clear in the paper). Close to MAP estimates, they bypass the dependence of said MAPs on parameterisation by maximising instead π(θ|x)/√I(θ), which is invariant by reparameterisation if not by a change of dominating measure. This form of MAP estimate is called the Wallace-Freeman (1987) estimator [of which I never heard].
The formal definition of a risk-adverse estimator is still based on a loss function in order to produce a proper version of the probability to be “wrong” in a continuous environment. The difference between estimator and true value θ, as expressed by the loss, is enlarged by a scale factor k pushed to infinity. Meaning that differences not in the immediate neighbourhood of zero are not relevant. In the case of a countable parameter space, this is essentially producing the MAP estimator. In the continuous case, for “well-defined” and “well-behaved” loss functions and estimators and density, including an invariance to parameterisation as in my own intrinsic losses of old!, which the author calls likelihood-based loss function, mentioning f-divergences, the resulting estimator(s) is a Wallace-Freeman estimator (of which there may be several). I did not get very deep into the study of the convergence proof, which seems to borrow more from real analysis à la Rudin than from functional analysis or measure theory, but keep returning to the apparent dependence of the notion on the dominating measure, which bothers me.
An interesting if presumably hopeless question spotted on X validated: a lower-truncated Normal distribution is parameterised by its location, scale, and truncation values, μ, σ, and α. There exist formulas to derive the mean and variance of the resulting distribution, that is, when α=0,![\Bbb{E}_{\mu,\sigma}[X]= \mu + \frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\sigma](https://s0.wp.com/latex.php?latex=%5CBbb%7BE%7D_%7B%5Cmu%2C%5Csigma%7D%5BX%5D%3D+%5Cmu+%2B+%5Cfrac%7B%5Cvarphi%28%5Cmu%2F%5Csigma%29%7D%7B1-%5CPhi%28-%5Cmu%2F%5Csigma%29%7D%5Csigma&bg=000000&fg=B0B0B0&s=0 "\Bbb{E}_{\mu,\sigma}[X]= \mu + \frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\sigma")
![\text{var}_{\mu,\sigma}(X)=\sigma^2\left[1-\frac{\mu\varphi(\mu/\sigma)/\sigma}{1-\Phi(-\mu/\sigma)} -\left(\frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\right)^2\right]](https://s0.wp.com/latex.php?latex=%5Ctext%7Bvar%7D_%7B%5Cmu%2C%5Csigma%7D%28X%29%3D%5Csigma%5E2%5Cleft%5B1-%5Cfrac%7B%5Cmu%5Cvarphi%28%5Cmu%2F%5Csigma%29%2F%5Csigma%7D%7B1-%5CPhi%28-%5Cmu%2F%5Csigma%29%7D++-%5Cleft%28%5Cfrac%7B%5Cvarphi%28%5Cmu%2F%5Csigma%29%7D%7B1-%5CPhi%28-%5Cmu%2F%5Csigma%29%7D%5Cright%29%5E2%5Cright%5D&bg=000000&fg=B0B0B0&s=0 "\text{var}_{\mu,\sigma}(X)=\sigma^2\left[1-\frac{\mu\varphi(\mu/\sigma)/\sigma}{1-\Phi(-\mu/\sigma)} -\left(\frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\right)^2\right]")
