At least two, different approaches to define and solve statistical models for the analysis of economic systems exist: the typical, econometric one, interpreting the Gravity Model specification as the expected link weight of an arbitrary probability distribution, and the one rooted into statistical physics, constructing maximum-entropy distributions constrained to satisfy certain network properties. In a couple of recent, companion papers they have been successfully integrated within the framework induced by the constrained minimisation of the Kullback-Leibler divergence: specifically, two, broad classes of models have been devised, i.e. the integrated and the conditional ones, defined by different, probabilistic rules to place links, load them with weights and turn them into proper, econometric prescriptions. Still, the recipes adopted by the two approaches to estimate the parameters entering into the definition of each model differ. In econometrics, a likelihood that decouples the binary and weighted parts of a model, treating a network as deterministic, is typically maximised; to restore its random character, two alternatives exist: either solving the likelihood maximisation on each configuration of the ensemble and taking the average of the parameters afterwards or taking the average of the likelihood function and maximising the latter one. The difference between these approaches lies in the order in which the operations of averaging and maximisation are taken - a difference that is reminiscent of the quenched and annealed ways of averaging out the disorder in spin glasses. The results of the present contribution, devoted to comparing these recipes in the case of continuous, conditional network models, indicate that the annealed estimation recipe represents the best alternative to the deterministic one.

翻译：至少存在两种不同的方法来定义和求解用于分析经济系统的统计模型：一种是典型的经济计量学方法，将引力模型规范解释为任意概率分布的期望连边权重；另一种源于统计物理学，构造满足特定网络性质约束的最大熵分布。在最近的两篇配套论文中，这两种方法已成功整合到由Kullback-Leibler散度约束最小化所诱导的框架内：具体而言，设计了两种宽泛的模型类别，即集成模型和条件模型，它们通过不同的概率规则来放置连边、赋予权重，并将其转化为适当的经济计量学规范。然而，这两种方法在估计各模型定义中涉及的参数时所采用的策略有所不同。在经济计量学中，通常最大化一种将模型的二值部分和加权部分解耦、并将网络视为确定性的似然函数；为恢复其随机性，存在两种替代方案：要么对系综中的每个构型求解似然最大化并取参数的平均值，要么先取似然函数的平均值再最大化后者。这两种方法的区别在于平均和最大化运算的顺序不同——这种差异令人联想到自旋玻璃中针对无序性进行平均的淬火方式和退火方式。本文致力于在连续条件网络模型情形下比较这些策略，结果表明退火估计策略是确定性估计的最佳替代方案。