## a general framework for updating belief functions [reply from the authors]

Thanks to Christian for the comments and feedback on our paper “A General Framework for Updating Belief Distributions“. We agree with Christian that starting with a summary statistic, or statistics, is an anchor for inference or learning, providing direction and guidance for models, avoiding the alternative vague notion of attempting to model a complete data set. The latter idea has dominated the Bayesian methodology for decades, but with the advent of large and complex data sets, this is becoming increasingly challenging, if not impossible.

However, in order to do work with statistics of interest, we need to find a framework in which this direct approach can be supported by a learning strategy when the formal use of Bayes theorem is not applicable. We achieve this in the paper for a general class of loss functions, which connect observations with a target of interest. A point raised by Christian is how arbitrary these loss functions are. We do not see this at all; for if a target has been properly identified then the most primitive construct between observations informing about a target and the target would come in the form of a loss function. One should always be able to assess the loss of ascertaining a value of $\theta$ as an action and providing the loss in the presence of observation x. The question to be discussed is whether loss functions are objective, as in the case of the median loss,

$l(\theta,x)=|\theta-x|$

or subjective, in the case of the choice between loss functions for estimating a location of a distribution; mean, median or mode? But our work is situated in the former position.

Previous work on loss functions, mostly in the classical literature, has spent a lot of space working out what are optimal loss functions for targets of interest. We are not really dealing with novel targets and so we can draw on the classic literature here. The work can be thought of as the Bayesian version of the M-estimator and associated ideas. In this respect we are dealing with two loss functions for updating belief distributions, one for the data, which we have just discussed, and one for the prior information, which, due to coherence principles, must be the Kullback-Leibler divergence. This raises the thorny issue of how to calibrate the two loss functions. We discuss this in the paper.

To then deal with the statistic problem, mentioned at the start of this discussion, we have found a nice way to proceed by using the loss function $l(\theta,x)=-\log f(x|\theta)$. How this loss function, combined with the use of the exponential family, can be used to estimate functionals of the type

$I=\int g(x)\,f_0(x)\, dx$

is provided in the Walker talk at Bayes 250 in London, titled “The Misspecified Bayesian”, since the “model” $f(x|\theta)$ is designed to be misspecified, a tool to estimate and learn about I only. The basic idea is to evaluate I by ensuring that we learn about the $\theta_0$ for which

$I=\int g(x)\,f(x|\theta_0)\, dx.$

This is the story of the background, we would now like to pick up in more detail on three important points that you raise in your post:

1. The arbitrariness in selecting the loss function.
2. The relative weighting of loss-to-data vs. loss-to-prior.
3. The selection of the loss in the M-free case.

In the absence of complete knowledge of the data generating mechanism, i.e. outside of M-closed,

1. We believe the statistician should weigh up the relative arbitrariness in selecting a loss function targeting the statistic of interest versus the arbitrariness of selecting a misspecified model, known not to be true, for the complete data generating mechanism. There is a wealth of literature on how to select optimal loss functions that target specific statistics, e.g. Hüber (2009) provides a comprehensive overview of how this should be done. As far as we are aware, we know of no formal procedures (that do not rely on loss functions) to select a false sampling distribution $f(x|\theta)$ for the whole of x; see Key, Pericchi and Smith (1999).
2. The relative weighting of loss-to-data vs. loss-to-prior. This is an interesting open problem. Our framework shows in the absence of M-closed or the use of self-information loss that the analyst must select this weighting. In our paper we suggest some default procedures. We have nowhere claimed these were “correct”. You raise concerns regards parameterisation and we agree with you that care is needed, but many of these issues equally hold for existing “Objective” or “Default” Bayes procedures, such as unit-information priors.
3. The selection of the loss in M-free. You say “….there is no optimal choice for the substitute to the loss function…”. We disagree. Our approach is to select an established loss function that directly targets the statistic of interest, and elicit prior beliefs directly on the unknown value of this statistic. There is no notion here of a a pseudo-likelihood or where this converges to.

Thank you again to Christian for his critical observations!

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