The maximum entropy method was originally proposed as a variational technique to determine probability densities from the knowledge of a few expected values. The applications of the method beyond its original role in statistical physics are manifold. An interesting feature of the method is its potential to incorporate errors in the data. Here, we examine two possible ways of doing that. The two approaches have different intuitive interpretations, and one of them allows for error estimation. Our motivating example comes from the field of risk analysis, but the statement of the problem might as well come from any branch of applied sciences. We apply the methodology to a problem consisting of the determination of a probability density from a few values of its numerically-determined Laplace transform. This problem can be mapped onto a problem consisting of the determination of a probability density on [0, 1] from the knowledge of a few of its fractional moments up to some measurement errors stemming from insufficient data.

The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities. For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. In this space, the probabilities on a finite set comprise a submanifold in which exponential families correspond to geodesic surfaces. We shall also obtain a geometric/dynamic interpretation of Jaynes’ method of maximum entropy.

The method of maximum entropy has proven to be a rather powerful way to solve the inverse problem consisting of determining a probability density on from the knowledge of the expected value of a few generalized moments, that is, of functions of the variable S. A version of this problem, of utmost relevance for banking, insurance, engineering and the physical sciences, corresponds to the case in which and the expected values are the values of the Laplace transform of S the points on the real line. Since inverting the Laplace transform is an ill-posed problem, to devise numerical techniques that are efficient is of importance for many applications, especially in cases where all we know is the value of the transform at a few points along the real axis. A simple change of variables transforms the Laplace inversion problem into a fractional moment problem on [0, 1]. In this note, we examine why the maximum entropy method provides a good approximation to the density from a few fractional moments, or correspondingly, why a density can be recovered from a few values of its Laplace transform along the real axis.

In this paper we propose the use of the maximum entropy method to extract pricing densities directly from the weather market prices. The proposed methodology can overcome the data sparsity problem that governs the weather derivatives market and it is model free, non-parametric, robust and computationally fast. We propose a novel method to infer consistent pricing probabilities, and illustrate the method with a motivating example involving market prices of temperature options. The probabilities inferred from a smaller subset of the data are found to consistently reproduce out-of-sample prices, and can be used to value all other possible derivatives in the market sharing the same underlying asset. We examine two sources of the out-of-sample valuation error. First, we use different sets of possible physical state probabilities that correspond to different temperature models. Then, we apply our methodology under three scenarios where the available information in the market is based on historical data, meteorological forecasts or both. Our results indicate that different levels of expertise can affect the accuracy of the valuation. When there is a mix of information available, non-coherent sets of prices are observed in the market.