Estimation of inverse temperature and other Lagrange multipliers: The dual distribution

It is shown that the problem of parameter estimation for distributions of the exponential type, has a unique consistent Bayesian solution: The requirement that Bayes' rule and maximum entropy lead to the same inverse distribution determines the loss function. Similarly, the demand that the best estimate for a random variable, given an observed value of that variable, coincides with the observed value, determines the prior distribution for the corresponding conjugate parameter. Properties of the dual distribution thus determined are investigated. In particular, the symmetrical role of parameter and constraint as a pair of conjugate variables is shown to imply an inherent uncertainty principle. Possible applications to temperature fluctuations and to an imbedding of classical mechanics in a statistical background are indicated.