Parameter estimation for Gibbs distributions

We consider Gibbs distributions, which are families of probability distributions over a discrete space $Ω$ with probability mass function of the form $μ^Ω_β(ω) \propto e^{βH(ω)}$ for $β$ in an interval $[β_{\min}, β_{\max}]$ and $H( ω) \in \{0 \} \cup [1, n]$. The partition function is the normalization factor $Z(β)=\sum_{ω\inΩ}e^{βH(ω)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(β_{\max})}{Z(β_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we also develop algorithms to compute the partition function $Z$ using $\tilde O(\frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and using $\tilde O(\frac{n^2}{\varepsilon^2})$ samples for integer-valued distributions.