Geometry of q-Exponential Family of Probability Distributions
Geometry of q-Exponential Family of Probability Distributions
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DOI：
10.3390/e13061170
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发表时间：
2011-06-01
期刊：
影响因子：
2.7
通讯作者：
Ohara, Atsumi
中科院分区：
文献类型：
--
作者：
Amari, Shun-ichi;Ohara, Atsumi
统计物理学的Gibbs分布是概率分布的指数家族，它具有二元性的数学基础，以Legendre转换的形式。对复杂系统的最新研究发现，许多分布遵守了权力法，而不是标准的Gibbs类型分布。 Tsallis Q-entropy是捕获这种现象的典型例子。我们通过将指数函数推广到功率函数的Q家族函数来处理Q-GIBBS分布或Q指数家族，这对于研究各种复杂或非标准的物理现象很有用。我们为与以前给出的Q指数家族提供了新的数学结构。它具有源自Legendre转换的双重几何结构，并且保形几何形状可用于理解它。最大熵定理的Q反相自然是从Q-五头定理诱导的。我们还表明，Q-Escort分布的最大化器是贝叶斯地图（最大a后验概率）估计量。
The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.