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An integral representation for topological pressure in terms of conditional probabilities

Given an equilibrium state $μ$ for a continuous function $f$ on a shift of finite type $X$, the pressure of $f$ is the integral, with respect to $μ$, of the sum of $f$ and the information function of $μ$. We show that under certain assumptions on $f$, $X$ and an invariant measure $ν$, the pressure of $f$ can also be represented as the integral with respect to $ν$ of the same integrand. Under stronger hypotheses we show that this representation holds for all invariant measures $ν$. We establish an algorithmic implication for approximation of pressure, and we relate our results to a result in thermodynamic formalism.