Bayesian efficiency is a statistical concept that refers to the ability to make accurate estimates of parameters or quantities using the smallest possible amount of data. Bayesian efficient estimators are those that have the lowest possible expected error, given a certain set of assumptions.

In finance, bayesian efficiency is often used in reference to portfolio management and investment strategies. For example, a fund manager who is able to accurately predict stock prices using a small amount of data would be considered bayesian efficient.

There are many different ways to measure bayesian efficiency, but one common method is through the use of Bayes factors. Bayes factors compare the evidence for two competing models, and the model with the higher Bayes factor is generally considered to be more efficient.

There are a few different ways to calculate Bayes factors, but one popular method is the Laplace-Jeffreys formula. This formula takes into account both the prior probabilities of the two models and the likelihood of the data under each model.

The Laplace-Jeffreys formula is as follows:

Bayes Factor = (P(M1|D)) / (P(M2|D))

where:

M1 and M2 are the two competing models

D is the data set

P(M1|D) is the posterior probability of model 1 given the data set D

P(M2|D) is the posterior probability of model 2 given the data set D

Bayesian efficiency is a powerful concept that can be used to make more accurate predictions with less data. This is especially useful in finance, where accurate predictions can lead to large profits.