Regret and Jeffreys Integrals in Exp. Families

where Z is the partition function Z(β) = exp(βx) dQx, and Γ can := {β | Z(β) < ∞} is the canonical parameter space. We let β sup = sup{β | β ∈ Γ can }, and β inf likewise. The elements of the exponential family are also parametrized by their mean value µ. We write µ β for the mean value corresponding to the canonical parameter β and β µ for the canonical parameter corresponding to the mean value µ. For any x the maximum likelihood distribution is P βx . The Shtarkov integral S is defined as

The variance function V is the function that maps µ ∈ M into the variance of P µ . The Fisher information of an exponential family in its canonical parametrization is I β = V (µ β ) and the Fisher information of the exponential family in its mean value parametrization is I µ = (V (µ))

-1 . The Jeffreys integral J is defined as

More on Fisher information can be found in [2].

As first established by [3], if the parameter space is restricted to a compact subset of the interior of the parameter space with non-empty interior (called an ineccsi set in [2]), then the minimax regret is finite and equal to the logarithm of the Shtarkov integral, which in turn is equal to

It thus becomes quite relevant to investigate whether the same thing still holds if the parameter spaces are not restricted to an ineccsi set. Whether or not this is so is discussed at length and posed as an open problem in [2, Chapter 11, Section 11.1].

Theorem 1: For a 1-dimensional left-truncated exponential family, the following statements are all equivalent:

1. The Shtarkov integral is finite.

2. The minimax individual-sequence regret is finite.

3. The minimax expected redundancy is finite.

4. The exponential family has a dominating distribution Q dom in terms of information divergence, i.e. sup β∈Γ can D(P β Q dom ) < ∞. 5) There is distribution P β with β ∈ Γ can that dominates the exponential family in terms of information divergence. 6) The information channel β → P β has finite capacity. 7) There exists β 0 ∈ Γ can such that

Most of the equivalences between ( 1)-( 6) are quite straightforward. The surprising part is the fact that statements (1)-( 6) are also equivalent to (7).

Theorem 2: Let (Γ can 0 , Q) represent a left-truncated exponential family. If the Shtarkov integral is infinite, then the Jeffreys integral is infinite.

The converse does not hold in general. Theorem 3: Let (Γ can 0 , Q) represent a left-truncated exponential family such that β sup = 0 and Q admits a density q either with respect to Lebesgue measure or counting measure. If q(x) = O(1/x 1+α ) for some α > 0, then the Jeffreys integral 0 β I(γ) 1/2 dγ is finite. In most cases finite Shtarkov implies finite Jeffreys. Theorem 4: Let Q be a measure on the real line with support I. Assume that µ inf is the left end point of I. If Q has density f (x) = (x -µ inf ) γ-1 g (x) in an interval just to the right of a where g is an analytic function and g (µ inf ) > 0 then the left end of the interval I gives a finite contribution to Jeffrey’s integral if and only if Q has a point mass in a.

If Y is a Cauchy distributed random variable then X = exp (Y ) has density 1 π

.

A probability measure Q is defined as a 1/2 and 1/2 mixture of a point mass in 0 and an exponentiated Cauchy distribution. The exponential family based on Q has redundancy upper bounded by 1 bit but the Jeffreys integral is infinite. For exponential families in more dimensions the analysis becomes more more involved and one may even have exponential families with finite redundancy and infinite regret.