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Joseph B. Kadane

Publications and source records attributed to Joseph B. Kadane.

13 recordsLinked to original sources
arXiv2026

Posterior Invariance of Multiplicative Contrasts under Margin Constraints in Contingency Tables

Measures of association in contingency tables, such as odds ratios and their generalizations, are often studied under different sampling schemes that either fix or leave random the margins of the table. While classical results show that certain odds ratios are unaffected by constraining the margins, it is less clear when this invariance holds more generally. This paper studies posterior inference for a broad class of multiplicative contrasts of multinomial cell probabilities, which we refer to as generalized odds ratios, and addresses exactly when fixing a margin alters inference about them. We consider Bayesian inference under multinomial sampling and under models in which partition sums of the table are fixed in advance, and assume that the marginal and conditional parameters are independent a priori. Under additional mild assumptions, we show that the posterior distribution of a generalized odds ratio is invariant to fixing a margin if and only if the coefficients defining the contrast sum to zero within the margin.

math.ST↗
arXiv2025

A Variance Decomposition Approach to Inconclusives in Forensic Black Box Studies

In the US, `black box' studies are increasingly being used to estimate the error rate of forensic disciplines. A sample of forensic examiner participants are asked to evaluate a set of items whose source is known to the researchers but not to the participants. Participants are asked to make a source determination (typically an identification, exclusion, or some kind of inconclusive). We study inconclusives in two black box studies, one on fingerprints and one on bullets. Rather than treating all inconclusive responses as functionally correct (as is the practice in reported error rates in the two studies we address), irrelevant to reported error rates (as some would do), or treating them all as potential errors (as others would do), we propose that the overall pattern of inconclusives in a particular black box study can shed light on the proportion of inconclusives that are due to examiner variability. Raw item and examiner variances are computed, and compared with the results of a logistic regression model that takes account of which items were addressed by which examiner. The error rates reported in black box studies are substantially smaller than ``failure rate" analyses that take inconclusives into account. The magnitude of this difference is highly dependent on the particular study at hand.

stat.AP↗
arXiv2019

On the probability that two random integers are coprime

We show that there is a non-empty class of finitely additive probabilities on $\mathbb N^2$ such that for each member of the class, each set with limiting relative frequency $p$ has probability $p$. Hence, in that context the probability that two random integers are coprime is $6/π^2$. We also show that two other interpretations of "random integer," namely residue classes and shift invariance, support any number in $[0, 6/π^2]$ for that probability. Finally, we specify a countably additive probability space that also supports $6/π^2$.

math.PR↗
arXiv2018

Prime-Residue-Class of Uniform Charges on the Integers

There is a probability charge on the power set of the integers that gives probability $1/p$ to every residue class modulo a prime $p$. There exists such a charge that gives probability $w$ to the set of prime numbers iff $w \in [0,1/2]$. Similarly, there is such a charge that gives probability $x$ to a residue class modulo $c$, where $c$ is composite, iff $x \in [0,1/y]$, where $y$ is the largest prime factor of $c$.

math.PR↗
arXiv2014

Compensating for the loss of a chance

Civil liability for a lost chance applies to legal cases in which a tortious action changes the probabilities of the outcomes that can be obtained by the victim. A central point in the application of this type of liability is the valuation of damages. Despite the practical importance of the valuation of lost chances, the legal restrictions that guide it have rarely been discussed explicitly. In order to discuss these restrictions, we propose an abstract description of a lost chance case in which there are multiple possible outcomes and the victim can make a choice that affects these outcomes. Given this description, we propose six conceptual questions to guide the valuation of lost chances. We discuss alternative answers to these questions and present the formulas that derive from them. More specifically, we show that the main formulas that have been proposed for medical misdiagnosis cases are particular instances of the alternatives we discuss.

stat.AP↗
arXiv2014

Sums of Possibly Associated Bernoulli Variables: The Conway-Maxwell-Binomial Distribution

The study of sums of possibly associated Bernoulli random variables has been hampered by an asymmetry between positive correlation and negative correlation. The Conway-Maxwell Binomial (COMB) distribution and its multivariate extension, the Conway-Maxwell Multinomial (COMM) distribution, gracefully model both positive and negative association. Sufficient statistics and a family of proper conjugate distributions are found. The relationship of this distribution to the exchangeable special case is explored, and two applications are discussed.

stat.ME↗
arXiv2013

The number of killings in southern rural Norway, 1300-1569

Three dual systems estimates are employed to study the number of killings in southern rural Norway in a period of slightly over 250 years. The first system is a set of five letters sent to each killer as part of the legal process. The second system is the mention of killings from all other contemporary sources. The posterior distributions derived suggest fewer such killings than rough demographic estimates.

stat.AP↗
arXiv2013

Infinite Previsions and Finitely Additive Expectations

We give an extension of de Finetti's concept of coherence to unbounded (but real-valued) random variables that allows for gambling in the presence of infinite previsions. We present a finitely additive extension of the Daniell integral to unbounded random variables that we believe has advantages over Lebesgue-style integrals in the finitely additive setting. We also give a general version of the Fundamental Theorem of Prevision to deal with conditional previsions and unbounded random variables.

math.ST↗
arXiv2006

The Timing of Bid Placement and Extent of Multiple Bidding: An Empirical Investigation Using eBay Online Auctions

Online auctions are fast gaining popularity in today's electronic commerce. Relative to offline auctions, there is a greater degree of multiple bidding and late bidding in online auctions, an empirical finding by some recent research. These two behaviors (multiple bidding and late bidding) are of ``strategic'' importance to online auctions and hence important to investigate. In this article we empirically measure the distribution of bid timings and the extent of multiple bidding in a large set of online auctions, using bidder experience as a mediating variable. We use data from the popular auction site \url{www.eBay.com} to investigate more than 10,000 auctions from 15 consumer product categories. We estimate the distribution of late bidding and multiple bidding, which allows us to place these product categories along a continuum of these metrics (the extent of late bidding and the extent of multiple bidding). Interestingly, the results of the analysis distinguish most of the product categories from one another with respect to these metrics, implying that product categories, after controlling for bidder experience, differ in the extent of multiple bidding and late bidding observed in them. We also find a nonmonotonic impact of bidder experience on the timing of bid placements. Experienced bidders are ``more'' active either toward the close of auction or toward the start of auction. The impact of experience on the extent of multiple bidding, though, is monotonic across the auction interval; more experienced bidders tend to indulge ``less'' in multiple bidding.

math.ST↗
arXiv1999

Transition Matrix Monte Carlo

Although histogram methods have been extremely effective for analyzing data from Monte Carlo simulations, they do have certain limitations, including the range over which they are valid and the difficulties of combining data from independent simulations. In this paper, we describe an complementary approach to extracting information from Monte Carlo simulations that uses the matrix of transition probabilities. Combining the Transition Matrix with an N-fold way simulation technique produces an extremely flexible and efficient approach to rather general Monte Carlo simulations.

cond-mat.stat-mech↗