In this paper we leverage on probability over Riemannian manifolds to rethink the interpretation of priors and posteriors in Bayesian inference. The main mindshift is to move away from the idea that "a prior distribution establishes a probability distribution over the parameters of our model" to the idea that "a prior distribution establishes a probability distribution over probability distributions". To do that we assume that our probabilistic model is a Riemannian manifold with the Fisher metric. Under this mindset, any distribution over probability distributions should be "intrinsic", that is, invariant to the specific parametrization which is selected for the manifold. We exemplify our ideas through a simple analysis of distributions over the manifold of Bernoulli distributions. One of the major shortcomings of maximum a posteriori estimates is that they depend on the parametrization. Based on the understanding developed here, we can define the maximum a posteriori estimate which is independent of the parametrization.
翻译:在本文中,我们利用里曼尼方块的概率来重新思考拜伊西亚推理中前方和后方方方块的诠释。 主要的思维转变是,从“先前的分布决定了我们模型参数的概率分布”的理念转向“先前的分布决定了我们模型参数的概率分布”的理念。 为了实现这一点,我们假设我们的概率模型是一个里曼尼方块,与Fisher 测量值相匹配。 在这种思维模式下,任何概率分布的分布都应该是“内在的 ”, 也就是说, 与为多方块所选择的具体平衡值不同。 我们通过对伯努利分布的方块的分布进行简单分析来展示我们的想法。 后方估计的最大缺点之一是它们取决于对准值的分布。 我们可以根据这里形成的理解, 定义与超光谱化无关的后方位估计值的最大值。