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经济系统分析的统计模型：典型的经济计量方法将引力模型规范解释为任意概率分布的期望连接权重，而植根于统计物理学的方法则构建约束于满足某些网络性质的最大熵分布。在近期两篇配套论文中，这两种方法已成功整合到由库尔贝克-莱布勒散度约束最小化所诱导的框架中：具体而言，已设计出两类广义模型——即集成模型与条件模型，它们通过不同的概率规则来放置连接、赋予权重并将其转化为恰当的经济计量规范。然而，这两种方法在估计每个模型定义参数时所采用的策略有所不同。在经济计量学中，通常最大化一个将模型的二元部分与加权部分解耦（将网络视为确定性）的似然函数；为恢复其随机特性，存在两种替代方案：要么在系综的每个配置上求解似然最大化并随后取参数的平均值，要么先取似然函数的平均再最大化后者。这两种方法的核心差异在于平均与最大化操作的执行顺序——这一差异令人联想到自旋玻璃中淬火与退火平均无序的方式。本文致力于比较这些策略在连续条件网络模型中的表现，结果表明退火估计方法代表了确定性估计的最佳替代方案。