Credal model averaging for classification: representing prior ignorance and expert opinions

Bayesian model averaging (BMA) is the state of the art approach for overcoming model uncertainty. Yet, especially on small data sets, the results yielded by BMA might be sensitive to the prior over the models. Credal model averaging (CMA) addresses this problem by substituting the single prior over the models by a set of priors (credal set). Such approach solves the problem of how to choose the prior over the models and automates sensitivity analysis. We discuss various CMA algorithms for building an ensemble of logistic regressors characterized by different sets of covariates. We show how CMA can be appropriately tuned to the case in which one is prior-ignorant and to the case in which instead domain knowledge is available. CMA detects prior-dependent instances, namely instances in which a different class is more probable depending on the prior over the models. On such instances CMA suspends the judgment, returning multiple classes. We thoroughly compare different BMA and CMA variants on a real case study, predicting presence of Alpine marmot burrows in an Alpine valley. We find that BMA is almost a random guesser on the instances recognized as prior-dependent by CMA.     

Imprecise expectations for imprecise linear filtering

In most sensor measure based applications, the raw sensor signal has to be processed by an appropriate filter to increase the signal-to-noise ratio or simply to recover the signal to be measured. In both cases, the filter output is obtained by convoluting the sensor signal with a supposedly known appropriate impulse response. However, in many real life situations, this impulse response cannot be precisely specified. The filtered value can thus be considered as biased by this arbitrary choice of one impulse response among all possible impulse responses considered in this specific context. In this paper, we propose a new approach to perform filtering that aims at computing an interval valued signal containing all outputs of filtering processes involving a coherent family of conventional linear filters. This approach is based on a very straightforward extension of the expectation operator involving appropriate concave capacities.     

Computing absorption probabilities for a Markov chain

We present a new matrix construction algorithm for computing absorption probabilities for a finite, reducible Markov chain. The construction algorithm contains two steps: matrix augmentation and matrix reduction. The algorithm requires more memory and less execution time than the calculation of absorption probabilities by the LU decomposition. We apply the algorithm to a Markovian model of a production line.     

Minimizing Maximum Weighted Error for Imprecise Computation Tasks

We consider the problem of preemptively scheduling a set of imprecise computation tasks on m ≥ 1 identical processors, with each task T_i having two weights, w_i and w′_i. Two performance metrics are considered: (1) the maximum w′-weighted error; (2) the total w-weighted error subject to the constraint that the maximum w′-weighted error is minimized. For the problem of minimizing the maximum w′-weighted error, we give an algorithm which runs in O(n³ log²n) time for multiprocessors and O(n²) time for a single processor. For the problem of minimizing the total w-weighted error subject to the constraint that the maximum w′-weighted error is minimized, we give an algorithm which also has the same time complexity.     

The level probabilities for a simple loop ordering

Bartholomew's statistics for testing homogeneity of normal means with ordered alternatives have null distributions which are mixtures of chisquared or beta distributions depending on whether the variances are known or not. The mixing coefficients depend on the sample sizes and the order restriction. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a loop ordering is appropriate. Exact expressions for the mixing coefficients for a loop ordering and arbitrary sample sizes are given for five or fewer populations and approximations are developed for more than five populations. Also, the mixing coefficients for a loop ordering with equal sample sizes are computed. These mixing coefficients also arise in testing the ordering as the null hypothesis, in testing order restrictions in exponential families and in testing order restrictions nonparametrically.This research was supported by the National Institutes of Health under Grant 1 R01 GM42584-01A1     

Estimates for transition probabilities on a compact manifold

The purpose of this note is to describe a procedure for transferring familiar estimates for transition probabilities on RN to transition probabilities on compact manifolds.     

Return probabilities for random walk on a half-line

A random walk with reflecting zone on the nonnegative integers is a Markov chain whose transition probabilitiesq(x, y) are those of a random walk (i.e.,q(x, y)=p(y–x)) outside a finite set {0, 1, 2,...,K}, and such that the distributionq(x,·) stochastically dominatesp(·–x) for everyx{0, 1, 2,..., K}. Under mild hypotheses, it is proved that when xp _x>0, the transition probabilities satisfyq ⁿ(x, y)C_xyR^–nn^–3/2 asn, and when xp _x=0,q ⁿ(x, y)C_xyn^–1/2.Supported by National Science Foundation Grant DMS-9307855.     

Vagueness, ignorance, and margins for error

We argue that the epistemic theory of vagueness cannot adequately justify its key tenet-that vague predicates have precisely bounded extensions, of which we are necessarily ignorant. Nor can the theory adequately account for our ignorance of the truth values of borderline cases. Furthermore, we argue that Williamson’s promising attempt to explicate our understanding of vague language on the model of a certain sort of “inexact knowledge” is at best incomplete, since certain forms of vagueness do not fit Williamson’s model, and in fact fit an alternative model. Finally, we point out that a certain kind of irremediable inexactitude postulated by physics need not be-and is not commonly-interpreted as epistemic. Thus, there are aspects of contemporary science that do not accord well with the epistemicist outlook.     

Shortest Path Problem on a Network with Imprecise Edge Weight

A network with its arc lengths as imprecise number, instead of a real number, namely, interval number and triangular fuzzy number is considered here. Existing ideas on addition and comparison between two imprecise numbers of same type are introduced. To obtain a fuzzy shortest path from a source vertex to all other vertices, a common algorithm is developed which works well on both types of imprecise numbers under consideration. In the proposed algorithm, a decision-maker is to negotiate with the obtained fuzzy shortest paths according to his/her view only when the means are same but the widths are different of the obtained paths. Otherwise, a fuzzy optimal path is obtained to which the decision-maker always satisfies with different grades of satisfaction. All pairs fuzzy shortest paths can be found by repeated use of the proposed algorithm.     

Runs test for a circular distribution and a table of probabilities

Summary A method is suggested for testing whether two samples observed on a circle are drawn from the same distribution. The proposed test is a modification of the well-known Wald-Wolfowitz runs test for a distribution on a straight line. The primary advantage of the proposed test is that it minimizes the number of assumptions on the theoretical distribution. This study was supported in part by the contract NSF-9968, National Science Foundation; in part by the contract Nonr 2249(05), (Nr 301–579), the Office of Naval Research, Dept. of the Navy, with The Catholic University of America. This paper was written while the author was a research associate in the Statistical Laboratory, Department of Mathematics, The Catholic University of America, during the academic year 1962.     

Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

Imprecise parameters for near-optimal control of stochastic SIV epidemic model

The change of parameters may influence the dynamic behaviors of epidemic diseases. Biological system parameters can also be changed due to diverse uncertainties such as lack of data and errors in the statistical approach. The problem of how to define and decide the optimal-control strategies of epidemic diseases with imprecise parameters deserves further researches. The paper presents a stochastic susceptible, infected, and vaccinated (SIV) system that includes imprecise parameters. Firstly, we give the method of parameter estimates of the SIV model. Then, by using Ekeland's principle and Hamiltonian function, we obtain the sufficient conditions and necessary conditions of near-optimal control of the SIV epidemic model with imprecise parameters. At last, numerical examples prove our theoretical results.     

On representing a black box as a dynamical system

Necessary conditions are derived for a black box to be representable as a dynamical system governed by ordinary differential equations. Sufficiency of these conditions depends primarily on the set of inputs. It is shown that the conditions are indeed sufficient for the set of constant inputs and under suitable hypotheses for the set of smooth inputs. These conditions cannot be verified by a finite number of input-output observations.     

Models for representing piecewise linear cost functions

We study formulations of linear programs with piecewise linear objective functions with and without additional binary variables. We show that the two formulations without additional binary variables have the same LP bounds as those of the corresponding formulations with binary variables and therefore are preferable for efficient computation.     

Two inequalities for Poisson probabilities

In this paper two elementary inequalities with respect to the Poisson distribution are proved. (Theorem 1 and 2).     

Bounds for classical ruin probabilities

We derive upper and lower bounds for the ruin probability over infinite time in the classical actuarial risk model (usual independence and equidistribution assumptions; the claim-number process is Poisson). Our starting point is the renewal equation for the ruin probability, but no renewal theory is used, except for the elementary facts proved in the note. Some bounds allow a very simple new proof of an asymptotic result akin to heavy-tailed claim-size distributions.     

Empirical bounds for ruin probabilities

We consider the classical model for an insurance business where the claims occur according to a Poisson process and where the distribution for the cost of each claim fulfills Cramér's tail-condition. Under these conditions Lundberg's constant R is of fundamental importance for ruin calculations.We derive estimates of R, based on an observation of the insurance business and investigate the statistical properties of those estimates. We further derive bounds and confidence intervals for ruin probabilities.     

Projection theorems for hitting probabilities and a theorem of Littlewood

Littlewood (Proc. London Math. Soc. (2), 28 1928, 383–394) showed that a positive superharmonic function u on the unit disc has radial limits a.e. Using techniques due to Doob this result is extended to all rank one symmetric spaces. In addition simplifications are obtained of Doob's (Ann. Inst. Fourier (Grenoble), 15 1965, 113–135) proof of normal convergence a.e. of a positive superharmonic function on a half space. The symmetric space analogue of this half space result is also obtained. The methods used are shown to fail for the potential theory on $\text{R}$ⁿ associated with Δu = αu (α > 4 0). It is an open question as to whether Littlewood's theorem holds in this context.