Exponential families and Gibbs measures? Any relation? The title says it all. The formulas for exponential families and Gibbs measures seems very similar. Is there any relationship, or some kind of translation table?
 A: Yes, there are close parallels. Let us write an exponential family in the form
$$ \tag{expfam}\label{expfam}
  g(x)\cdot\exp\left( b(\theta)^T T(x) - A(\theta)  \right)
$$
while Gibbs measures are typically written 
$$ \tag{Gibbs}\label{Gibbs}
  \frac1{Z(\eta)}\exp\left( -\eta \cdot E(x)  \right)
$$
$g(x)$ is the base measure, and have no corresponding part in the Gibbs measure formula, but it could have been included. Often we write the exponential family with a canonical parameter by defining $\eta=b(\theta)$, we can see the Gibbs measure is written then in canonical parameter form.  To see the parallels, first write \eqref{expfam} with the canonical parameter as
$$\tag{*}\label{*}
  g(x)\cdot\exp\left( \eta^T T(x) - A(b^{-1}(\eta))  \right)
$$  and \eqref{Gibbs} more resembling the exponential family form
$$ \tag{**}\label{**}
  \exp\left( \eta \cdot ( -E(x))-\log Z(\eta)  \right)
$$ which makes some differences more clear. In \eqref{*} $\eta$ might be a vector, likewise the canonical statistic $T$ might be a vector, while in \eqref{**} $\eta$ is typically a scalar while the energy function (or potential function) $E$ is a scalar.  So we see that a Gibbs measure is just a one-parameter exponential family, usually with some changes in terminology.
The partition function $Z(\eta)$, note that in \eqref{*} we can write $A(b^{-1}(\eta))=\log Z(\eta)$, is related to the mgf (moment generating function) of the canonical statistic $T$.  
