Vecer Jan
Department of Probability and Mathematical Statistics, Charles University, Sokolovska 83, 18675 Praha, Czech Republic.
Vysoka Skola Aplikovaneho Prava, Chomutovicka 1443, 14900 Praha, Czech Republic.
Entropy (Basel). 2019 Jan 8;21(1):36. doi: 10.3390/e21010036.
We propose a novel approach of model selection for probability estimates that may be applied in time evolving setting. Specifically, we show that any discrepancy between different probability estimates opens a possibility to compare them by trading on a hypothetical betting market that trades probabilities. We describe the mechanism of such a market, where agents maximize some utility function which determines the optimal trading volume for given odds. This procedure produces supply and demand functions, that determine the size of the bet as a function of a trading probability. These functions are closed form for the choice of logarithmic and exponential utility functions. Having two probability estimates and the corresponding supply and demand functions, the trade matching these estimates happens at the intersection of the supply and demand functions. We show that an agent using correct probabilities will realize a profit in expectation when trading against any other set of probabilities. The expected profit realized by the correct view of the market probabilities can be used as a measure of information in terms of statistical divergence.
我们提出了一种用于概率估计的新型模型选择方法，该方法可应用于随时间演变的情况。具体而言，我们表明不同概率估计之间的任何差异都为通过在交易概率的假设投注市场上进行交易来比较它们提供了可能性。我们描述了这样一个市场的机制，其中参与者最大化某个效用函数，该函数决定了给定赔率下的最优交易量。这个过程产生了供给和需求函数，它们根据交易概率确定赌注的大小。对于对数和指数效用函数的选择，这些函数具有封闭形式。有了两个概率估计值以及相应的供给和需求函数，与这些估计值相匹配的交易发生在供给和需求函数的交点处。我们表明，当与任何其他概率集进行交易时，使用正确概率的参与者预期会实现盈利。由对市场概率的正确看法所实现的预期利润可以用作统计差异方面的信息度量。
