We study the parameter estimation problem in mixture models with observational nonidentifiability: the full model (also containing hidden variables) is identifiable, but the marginal (observed) model is not. Hence global maxima of the marginal likelihood are (infinitely) degenerate and predictions of the marginal likelihood are not unique. We show how to generalize the marginal likelihood by introducing an effective temperature, and making it similar to the free energy. This generalization resolves the observational nonidentifiability, since its maximization leads to unique results that are better than a random selection of one degenerate maximum of the marginal likelihood or the averaging over many such maxima. The generalized likelihood inherits many features from the usual likelihood, e.g. it holds the conditionality principle, and its local maximum can be searched for via suitably modified expectation-maximization method. The maximization of the generalized likelihood relates to entropy optimization.

中文翻译：

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观测不可识别性、广义似然和自由能

我们研究了具有观察不可识别性的混合模型中的参数估计问题：完整模型（也包含隐藏变量）是可识别的，但边缘（观察到的）模型不是。因此，边际似然的全局最大值是（无限）退化的，并且边际似然的预测不是唯一的。我们展示了如何通过引入有效温度并使其类似于自由能来概括边际似然。这种概括解决了观察的不可识别性，因为它的最大化会导致独特的结果，这比随机选择一个退化的边际似然最大值或对许多这样的最大值进行平均要好。广义似然继承了通常似然的许多特征，例如它持有条件原则，并且可以通过适当修改的期望最大化方法来搜索其局部最大值。广义似然的最大化与熵优化有关。