测量不变性、熵和概率
Measurement Invariance, Entropy, and Probability
摘要：
我们表明，一个特定的问题的测量的自然缩放定义的最有可能的概率分布的观测从该测量尺度。我们的方法扩展了最大熵的方法，使用测量尺度作为一种信息约束。我们认为，一个非常常见的测量尺度是线性的小幅度分级对数在大幅度，导致观察，往往遵循学生的概率分布，具有高斯形状的小波动的平均值和幂律形状的大波动的平均值。逆标度通常会出现，其中测量从小到大的幅度自然从对数到线性分级，导致观察结果通常遵循伽马概率分布。伽马分布对于小幅度具有幂律形状，对于大幅度具有指数形状。这两个测量尺度是通过拉普拉斯积分变换连接的自然逆。这种反转连接了自然界中常见的两种主要的缩放模式。我们还表明，超统计量是一个特殊的情况下，积分变换，因此可以被理解为一种特殊的方式来改变测量的规模。将度量尺度的信息表示为最大熵，为度量、信息和概率之间的关系提供了一种一般的方法。
英文摘要：
We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint. We argue that a very common measurement scale is linear at small magnitudes grading into logarithmic at large magnitudes, leading to observations that often follow Student's probability distribution which has a Gaussian shape for small fluctuations from the mean and a power law shape for large fluctuations from the mean. An inverse scaling often arises in which measures naturally grade from logarithmic to linear as one moves from small to large magnitudes, leading to observations that often follow a gamma probability distribution. A gamma distribution has a power law shape for small magnitudes and an exponential shape for large magnitudes. The two measurement scales are natural inverses connected by the Laplace integral transform. This inversion connects the two major scaling patterns commonly found in nature. We also show that superstatistics is a special case of an integral transform, and thus can be understood as a particular way in which to change the scale of measurement. Incorporating information about measurement scale into maximum entropy provides a general approach to the relations between measurement, information and probability.
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