Mathematics

This paper uses a very important measure-theoretic result, the Eadon-Nikodym theorem, to give a rigorous defi­ nition of conditional expectation a concept necessary in defining a sufficient statistic. The Radon-Nikodym theorem is used only in the case that the dominating measure is either the Lebesgue or counting measure, and the space is Euclidean with the o-algebra consisting of all the Borel sets or the space is countable with the o-algebra consis­ ting of all subsets. In chapter 0, preliminary measure-theoretic defi­ nitions and theorems are given. The classical definitions of conditional expectation, conditional probability, and density functions are presented in the discrete and con­ tinuous cases and are shown to be consistent with the definitions in the general case. In chapter 1, we define what is meant by a suffi­ cient statistic and develop some of its important proper­ ties. The proofs are given only in the discrete and con­ tinuous cases, but are constructed in such a m...

A probability forecaster is asked to give a density p of a random variable w. In return he gets a reward (or score} depending on p and on a subsequently observed value of• w. A scoring rule is called proper if the expected score is maximized when the true density is chosen. The present paper uses convex analysis to generalize McCarthy's characterization of proper scoring rules.

Part of this material is taken from two earlier reports. Chapters II and III are based in part on Technical Report No. 141, "Payoffs of probability forecasters and a theorem of McCarthy," by Arlo D. Hendrickson (1970); Chapter IV is based in part on Technical Report No. 133, "Some procedures for determining subjective probabilities by sequential choices,"

A probability forecaster is asked to give a density p of a random variable w. In return he gets a reward (or score} depending on p and on a subsequently observed value of• w. A scoring rule is called proper if the expected score is maximized when the true density is chosen. The present paper uses convex analysis to generalize McCarthy's characterization of proper scoring rules.

We study the generalized graph splines introduced by Gilbert, Tymoczko, and Viel and focus on an attribute known as the Universal Difference Property (UDP). We prove that paths, trees, and cycles satisfy UDP. We explore UDP on graphs pasted at a single vertex and use Prüfer domains to illustrate that not every edge labeled graph satisfies UDP. We show that UDP must hold for any edge labeled graph over a principal ideal domain. Lastly, we prove that UDP is preserved by isomorphisms of edge labeled graphs. 1. Introduction Splines are perhaps best known for their usage in analysis and for their applications in finding approximate solutions to differential equations, but splines also appear in a variety of other contexts including geometry and topology. To unify these various notions of splines, Gilbert, Tymoczko, and Viel defined generalized splines on graphs in [8], and it is these generalized graph splines we consider here. Definition 1.1. Given a graph G = (V, E) and a ring R, an edge labeling of G is a function α : E → I(R), where I(R) is the set of all ideals in R. The pair (G, α) denotes a graph G with edge labeling α. Definition 1.2. A generalized spline on (G, α) over a ring R is a function ρ : V → R such that for each edge uv, the difference ρ(u) − ρ(v) is an element of α(uv). Note that when working with a graph G and a ring R, an edge labeling of G labels the edges of G with ideals in R, while a generalized spline labels the vertices of G with elements of R. Simply put, edges are labeled with ideals, while vertices are labeled with ring elements. The ring R is called the base ring for the edge labeled graph (G, α), and henceforth we shall refer to generalized splines simply as splines. When working with an edge labeled graph (G, α) with |V (G)| = n, we often number the vertices of G as v 1 , v 2 ,. .. , v n and denote a spline ρ on G by ρ = (ρ(v 1), ρ(v 2),. .. , ρ(v n)). Example 1.1. Figure 1 illustrates an edge labeled graph with base ring Z. We see that f = (2, 16, 26) is a spline on the edge labeled 3-cycle.