We introduce a new family of estimators for unnormalized statistical models.
Our family of estimators is parameterized by two nonlinear functions and uses a
single sample from an auxiliary distribution, generalizing Maximum Likelihood
Monte Carlo estimation of Geyer and Thompson (1992). The family is such that we
can estimate the partition function like any other parameter in the model. The
estimation is done by optimizing an algebraically simple, well defined
objective function, which allows for the use of dedicated optimization methods.
We establish consistency of the estimator family and give an expression for the
asymptotic covariance matrix, which enables us to further analyze the influence
of the nonlinearities and the auxiliary density on estimation performance. Some
estimators in our family are particularly stable for a wide range of auxiliary
densities. Interestingly, a specific choice of the nonlinearity establishes a
connection between density estimation and classification by nonlinear logistic
regression. Finally, the optimal amount of auxiliary samples relative to the
given amount of the data is considered from the perspective of computational
efficiency.

本研究提出了一种新的非归一化统计模型估计器族，其参数由两个非线性函数组成，使用来自辅助分布的单个样本，推广了 Geyer 和 Thompson（1992）的最大似然蒙特卡罗估计方法，并且可以像模型中的任何其他参数一样估计分区函数。同时，我们可以通过优化代数简单且定义明确的目标函数来进行估计，从而允许使用专用优化方法。最后，我们从计算效率的角度考虑了相对于给定数据量的最佳辅助样本量。