Using Second-Order Probabilities to Make Maximum Entropy Approach to Copulas More Reasonable

Copulas are a general way of describing dependence between two or more random variables. When we only have partial information about the dependence, i.e., when several different copulas are consistent with our knowledge, it is often necessary to select one of these copulas. A frequently used method of selecting this copula is the maximum entropy approach, when we select a copula with the largest entropy. However, in some cases, the maximum entropy approach leads to an unreasonable selection – e.g., even if we know that the two random variables are positively correlated, the maximum entropy approach completely ignores this information. In this paper, we show how to properly modify the maximum entropy approach so that it will lead to more reasonable results: by applying this approach not to the probabilities themselves, but to “second order” probabilities – i.e., probabilities of different probability distributions. 1 Maximum Entropy Approach to Selecting a Copula: Description, Successes, and Limitations Copulas: brief reminder. In many practical situations, we know the cumulative distribution functions F1(x1) = P (X1 ≤ x1) and F2(x2) = P (X2 ≤ x2)