When There are Different Types of Smoke in the Sky It Changes How the Sun Light Goes Through the Sky

When the sky gets smoke in it, it can be more or less bright (white) than before. This is because the small smoke bits can change whether sun light gets through the sky to the ground, can change how much white sky water there is, and can change how the sky air moves around. If the smoke is blacker, it can make the sky warmer and if the smoke is whiter, it can make the sky (and the ground) cooler because the sun light doesn't get through. So we wanted to know whether the smoke is blacker or whiter over the big water far away from here, near where there are a lot of fires on the land.We went to the big water place and watched the sky, the air, the sky smoke, and the white sky water. We did this by flying in the sky with our computers. We watched the smoke in a lot of different ways: we can look at the smoke from near the water and see what sun light gets through the sky, or from above and see how the sun light comes back up to space, or go right in the smoke with our flying computer eyes. Then we looked at whether the different ways of seeing showed that the same smoke was different (blacker or whiter) or the same. A lot of time they said it was the same, at least in some ways, but sometimes the smoke looked different and it changed a lot in different places over the big water.We hope other people can put these answers into their computer studies, or into their computers that fly REALLY high, and then we can better understand how the sky gets warmer or cooler from smoke

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set $\mathcal E$ of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if $\mathcal E$ is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyse the Boltzmann operator in the geometric setting from the point of its Maxwell's weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyv\"arinen divergence. This requires to generalise our approach to Orlicz-Sobolev spaces to include derivatives.%Comment: 39 pages, 1 figure. Expanded version of a paper presente at the conference SigmaPhi 2014 Rhodes GR. Under revision for Entrop

Algebraic Bayesian analysis of contingency tables with possibly zero-probability cells

In this paper we consider a Bayesian analysis of contingency tables allowing for the possibility that cells may have probability zero. In this sense we depart from standard log-linear modeling that implicitly assumes a positivity constraint. Our approach leads us to consider mixture models for contingency tables, where the components of the mixture, which we call model-instances, have distinct support. We rely on ideas from polynomial algebra in order to identify the various model instances. We also provide a method to assign prior probabilities to each instance of the model, as well as describing methods for constructing priors on the parameter space of each instance. We illustrate our methodology through a $5 \times 2$ table involving two structural zeros, as well as a zero count. The results we obtain show that our analysis may lead to conclusions that are substantively different from those that would obtain in a standard framework, wherein the possibility of zero-probability cells is not explicitly accounted for