  In statistical decision theory, a model is said to be Pareto optimal (or
admissible) if no other model carries less risk for at least one state of
nature while presenting no more risk for others. How can you rationally
aggregate/combine a finite set of Pareto optimal models while preserving Pareto
efficiency? This question is nontrivial because weighted model averaging does
not, in general, preserve Pareto efficiency. This paper presents an answer in
four logical steps: (1) A rational aggregation rule should preserve Pareto
efficiency (2) Due to the complete class theorem, Pareto optimal models must be
Bayesian, i.e., they minimize a risk where the true state of nature is averaged
with respect to some prior. Therefore each Pareto optimal model can be
associated with a prior, and Pareto efficiency can be maintained by aggregating
Pareto optimal models through their priors. (3) A prior can be interpreted as a
preference ranking over models: prior $\pi$ prefers model A over model B if the
average risk of A is lower than the average risk of B. (4) A
rational/consistent aggregation rule should preserve this preference ranking:
If both priors $\pi$ and $\pi'$ prefer model A over model B, then the prior
obtained by aggregating $\pi$ and $\pi'$ must also prefer A over B. Under these
four steps, we show that all rational/consistent aggregation rules are as
follows: Give each individual Pareto optimal model a weight, introduce a weak
order/ranking over the set of Pareto optimal models, aggregate a finite set of
models S as the model associated with the prior obtained as the weighted
average of the priors of the highest-ranked models in S. This result shows that
all rational/consistent aggregation rules must follow a generalization of
hierarchical Bayesian modeling. Following our main result, we present
applications to Kernel smoothing, time-depreciating models, and voting
mechanisms.
