We consider Gibbs distributions, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_{\min}, \beta_{\max}]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The partition function is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ 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.

翻译：我们考虑吉布斯分布，这是一类定义在离散空间Ω上的概率分布族，其概率质量函数形式为μ^Ω_β(ω) ∝ e^{β H(ω)}，其中β位于区间[β_min, β_max]内，且H(ω) ∈ {0} ∪ [1, n]。配分函数是归一化因子Z(β)=∑_{ω∈Ω}e^{β H(ω)}。这些分布的两个重要参数是对数配分比q = log Z(β_max)/Z(β_min)以及计数c_x = |H^{-1}(x)|。这些参数与众多物理应用和采样算法中的系统参数相关。我们的第一个主要结果是：对于一般吉布斯分布，使用约Õ(q/ε²)个样本估计计数c_x；对于整值分布，使用约Õ(n²/ε²)个样本（忽略部分二阶项和参数），并证明这在对数因子意义下是最优的。我们通过改进的算法对连通子图、独立集和完美匹配的计数进行了实例说明。作为关键子程序，我们还开发了计算配分函数的算法：对于一般吉布斯分布需用Õ(q/ε²)个样本，对于整值分布需用Õ(n²/ε²)个样本。