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For clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t-test and show that it can be non monotone in the interim mean of the data and non monotone in the non-centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t-test and give a numerical example. Additionally, the conditional rejection probability given the interim t-statistic is investigated. It does not depend on the unknown σ and can be used in stochastic curtailment procedures. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Random set theory provides a convenient mechanism for representing uncertain knowledge including probabilistic and set-based information, and extending it through a function. This paper focuses upon the situation when the available information is in terms of coherent lower and upper probabilities, which are encountered, for example, when a probability distribution is specified by interval parameters. We propose an Iterative Rescaling Method (IRM) for constructing a random set with corresponding belief and plausibility measures that are a close outer approximation to the lower and upper probabilities. The approach is compared with the discrete approximation method of Williamson and Downs (sometimes referred to as the p-box), which generates a closer approximation to lower and upper cumulative probability distributions but in most cases a less accurate approximation to the lower and upper probabilities on the remainder of the power set. Four combination methods are compared by application to example random sets generated using the IRM.

We propose an interpretation of fuzzy set theory (both from a semantic and a syntactic point of view) in terms of conditional events and coherent conditional probabilities. During past years, a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty, or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory. In this paper we emphasize the role of conditioning (in a proper framework, i.e. de Finetti's coherence) to get rid of many controversial aspects. Moreover, we introduce suitable operations between fuzzy subsets, looked on as corresponding operations between conditional events endowed with the relevant conditional probability. Finally, we show how the concept of possibility function naturally arises as a coherent conditional probability.

This paper provides an axiomatic characterization of Bayes' Rule that is widely used for updating beliefs. Bayes' Rule is viewed as a revision rule. Consider an agent whose belief about a set of states is characterized by a point in a unit simplex of appropriate dimension. Now new information emerges that rules out the possible occurrence of some of the states. The revision rule then assigns new probabilities over the subset of states that is not ruled out. The paper provides a set of axioms that characterizes Bayes' Rule. The main axiom is Path Independence. A revision rule satisfies Path Independence if the probability distribution over any set of states is unaffected by the order in which new information comes in.

Conditional independence relation of observables on an MV-algebra with product is formulated and basic properties are proven. An analogy with a classical definition of conditional independence of random variables is discussed and the MV-algebraic concept of conditioning is further explored.

In this paper we study conditional independence structures arising from conditional probabilities and lower conditional probabilities. Such models are based on notions of stochastic independence apt to manage also those situations where zero evaluations on possible events are present: this is particularly crucial for lower probability. The "graphoid" properties of such models are investigated, and the representation problem of conditional independence structures is dealt with by generalizing the wellknown classic separation criteria for undirected and directed acyclic graphs. Our graphical models describe the independence statements and the possible logical dependencies among the random variables.

The conditional probabilities of finite conditional probability spaces are considered for points of a smooth manifold of conditional charges. A linear diffeomorphism on the manifold is constructed so that the conditional probabilities map bijectively onto a permutahedron. The facial structure of the permutahedron corresponds to the ways conditional probability spaces decompose. A new global inversion lemma is devised.

While majority cycles may pose a threat to democratic decision making, actual decisions based inadvertently upon an incorrect majority preference relation may be far more expensive to society. We study majority rule both in a statistical sampling and a Bayesian inference framework. Based on any given paired comparison probabilities or ranking probabilities in a population (i.e., culture) of reference, we derive upper and lower bounds on the probability of a correct or incorrect majority social welfare relation in a random sample (with replacement). We also present upper and lower bounds on the probabilities of majority preference relations in the population given a sample, using Bayesian updating. These bounds permit to map quite precisely the entire picture of possible majority preference relations as well as their probabilities. We illustrate our results using survey data.

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of “and”, “or”, “not” and “conditioning” are then defined on these conditional events with variable conditions. The probabilities of the disjunction (or) and of the conjunction (and) of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed.

The paper discusses the problem of sensitivity analysis in Gaussian Bayesian networks. The algebraic structure of the conditional means and variances, as rational functions involving linear and quadratic functions of the parameters, are used to simplify the sensitivity analysis. In particular the probabilities of conditional variables exceeding given values and related probabilities are analyzed. Two examples of application are used to illustrate all the concepts and methods.

This paper presents a method for listing the sample space for a conditional distribution in a discrete generalized linear model. This tabulation is used in conjunction with saddlepoint methods to approximate the associated conditional probabilities. These probabilities are used to calculate conditional p-values.

The article describes probability models that are in the form of a hierarchy; the basic quantities in a situation having their probabilities described in terms of parameters, which themselves are described in terms of other parameters, and so on. This is an arrangement which provides great flexibility in probability modeling. Distinction is made between random effects, to which are attached probabilities, and fixed effects, which have probabilities only in the Bayesian paradigm. Examples of hierarchical models are given, together with illustrations of calculations needed for their resolution. Alternative descriptions in terms of equations are described.

Conditioning for (non-additive) uncertainty measures is still an open problem. This is essentially due to the fact that these measures can be viewed either as lower or upper probabilities or axiomatically defined, independently of probability theory. Focusing on possibility measures on finite domains, the aim is to generalize to the conditional case the interpretation of possibility measures as upper envelopes of extensions of suitable probability measures.

We present a logical system sound and complete with respect to the class of conditional probability spaces. The universal modality has an important role in our language: conditional probabilities have a peculiar behavior when conditionalized over empty events, so we use the universal modality to express nonemptiness of events. We add public announcement to our language in a way that we are able to express sentences like: “the agent believes the probability of p is zero and after the announcement of p he/she believes the probability of p is greater than zero.” Because of the Conditional Probability Language we are able to express reduction axioms for the probability Public Announcement Logic.

In this paper the alternative procedure for updating probabilities (that is, to calculate the posterior distribution from the prior distribution) proposed by Richard Jeffrey is considered, which allows the addition of new information to the prior distribution under more circumstances than with the Bayesian conditioning. A predictivistic approach for the Jeffrey’s rule is introduced and a definition of conjugacy according to this rule (named Jeffrey-conjugacy) is established. Results for Jeffrey-conjugacy in the exponential family are also presented. As a by-product, these results provide full predictivistic characterizations of some predictive distributions. By using both the predictivistic Jeffrey’s rule and Jeffrey-conjugacy, a forecasting procedure which is applied to the Chilean stock market, data is also developed. The Jeffrey’s rule with the Bayesian conditioning according to their capability of incorporating unpredictable information in the forecast is compared.

The assessments for the various conditional probabilities of a Bayesian belief network inevitably are inaccurate, influencing the reliability of its output. By subjecting the network to a i>sensitivity analysis with respect to its conditional probabilities, the reliability of its output can be investigated. Unfortunately, straightforward sensitivity analysis of a belief network is highly time-consuming. In this paper, we show that by qualitative considerations several analyses can be identified as being uninformative as the conditional probabilities under study cannot affect the output. In addition, we show that the analyses that are informative comply with simple mathematical functions. More specifically, we show that a belief network's output can be expressed as a quotient of two functions that are linear in a conditional probability under study. These properties allow for considerably reducing the computational burden of sensitivity analysis of Bayesian belief networks.

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.

Existence of coherent extensions of coherent conditional probabilities is one of the major merits of de Finetti's theory of probability. However, coherent extensions which meet some special property, like σ-additivity or disintegrability, can fail to exist. An example is given where a coherent and σ-additive conditional probability cannot be extended preserving both σ-additivity and coherence. Motivated by such example, conditions are provided in order that a coherent and σ-additive conditional probability admits a coherent and σ-additive extension. Moreover, conditions are given for the existence of disintegrations, possibly σ-additive, of a probability along a partition.

The asymmetry (the tendency to produce different results when the order of presentation of attributes is reversed) of two commonly used measures of conceptual similarity is explored empirically. The observed asymmetry is shown to follow patterns somewhat consistent with conditional probability theory. Implications for theory and measurement development are discussed.

The difficulty involved in the composition of the criteria by weighted average, due to the difficulty of assigning weights to criteria and to sets of criteria, was already signaled in the preceding chapters. The probabilistic approach allows forgo the weighting of the criteria and get overall scores of preference by calculating joint probabilities.

Empirical evidence can sometimes be incorporated in a probabilistic analysis by conditioning with respect to the observations. Usually, the underlying probability distribution and also the conditional distribution are not completely known. The assignment of probabilities will then require a compromise. The making of such a compromise goes beyond mathematical theory: a statistical discussion is needed. It depends on the context whether the result of such discussion is almost compelling, reasonable, or not really agreeable. This is illustrated by means of a simple example from the area of predictive distributional inference.

A new simulation approach, called 'subset simulation', is proposed to compute small failure probabilities encountered in reliability analysis of engineering systems. The basic idea is to express the failure probability as a product of larger conditional failure probabilities by introducing intermediate failure events. With a proper choice of the conditional events, the conditional failure probabilities can be made sufficiently large so that they can be estimated by means of simulation with a small number of samples. The original problem of calculating a small failure probability, which is computationally demanding, is reduced to calculating a sequence of conditional probabilities, which can be readily and efficiently estimated by means of simulation. The conditional probabilities cannot be estimated efficiently by a standard Monte Carlo procedure, however, and so a Markov chain Monte Carlo simulation (MCS) technique based on the Metropolis algorithm is presented for their estimation. The proposed method is robust to the number of uncertain parameters and efficient in computing small probabilities. The efficiency of the method is demonstrated by calculating the first-excursion probabilities for a linear oscillator subjected to white noise excitation and for a five-story nonlinear hysteretic shear building under uncertain seismic excitation.

Existing data sets of cases can significantly reduce the knowledge engineering effort required to parameterize Bayesian networks. Unfortunately, when a data set is small, many conditioning cases are represented by too few or no data records and they do not offer sufficient basis for learning conditional probability distributions. We propose a method that uses Noisy-OR gates to reduce the data requirements in learning conditional probabilities. We test our method on H epar II , a model for diagnosis of liver disorders, whose parameters are extracted from a real, small set of patient records. Diagnostic accuracy of the multiple-disorder model enhanced with the Noisy-OR parameters was 6.7% better than the accuracy of the plain multiple-disorder model and 14.3% better than a single-disorder diagnosis model.

The paper is concerned with a modification of the data driven version of Neyman's testfor uniformity proposed by Ledwina (1994). The number of components in Ledwina'stest is chosen by Schwan rule. It is known that for small sample sizes the rule has a tendency to underestimate the model dimension. This gave a mdtivation for the present paper in which we examine an alternative criterion for the choice of the number of cornponents in the smooth test statistic. Schwan rule is an approximation of a Bayesian rule for the choice of the dimension of the exponential model. Trying to improve properties of Ledwina's test we turn to the more aocurate approximation of the Bayesian rule derived by Haughton (1988). Siulations ahow that by the appropriate choice of the prior distribution one obtains a rule which is more flexible than Schwan rule and the resulting test performs even better than the original Ledwina's test. We also give some asymptotic results on the null distribution and consistency of the introduced test.

We study the uncertain dichotomous choice model. In this model a group of decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas, such as medicine, management and banking. The decision rule may be the simple majority rule; however, it is also possible to assign more weight to the opinion of members known to be more qualified. The extreme example of such a rule is the expert decision rule. We are concerned with the probability of the expert rule to be optimal. Our purpose is to investigate the behaviour of this probability as a function of the group size for several rather general types of distributions. One such family of distributions is that where the density function of the correctness probability is a polynomial (on the interval [1/2,1]). Our main result is an explicit formula for the probability in question. This contains formerly known results as very special cases.

The entropic character of the bond multiplicity concept of chemistry is explored within the information theory. The probability schemes of finding a single and two electrons on specified atoms are ...

Higher-order chain rule for differential forms is used to obtain the (higher-order) moments of a random matrix from certain cumulants and vice versa. Computer algorithms are written for the above chain rule. In particular, computer algorithms for finding moments and cumulants of normal and Wishart random matrices are obtained.

In this paper a new Monte-Carlo algorithm for the propagation of probabilities in Bayesian networks is proposed. This algorithm has two stages: in the first one an approximate propagation is carried out by means of a deletion sequence of the variables. In the second stage a sample is obtained using as sampling distribution the calculations of the first step. The different configurations of the sample are weighted according to the importance sampling technique. We show how the use of probability trees to store and to approximate probability potentials, and a careful selection of the deletion sequence, make this algorithm able to propagate over large networks with extreme probabilities.

Our aim in this paper is to clarify the notion of independence for imprecise probabilities. Suppose that two marginal experiments are each described by an imprecise probability model, i.e., by a convex set of probability distributions or an equivalent model such as upper and lower probabilities or previsions. Then there are several ways to define independence of the two experiments and to construct an imprecise probability model for the joint experiment. We survey and compare six definitions of independence. To clarify the meaning of the definitions and the relationships between them, we give simple examples which involve drawing balls from urns. For each concept of independence, we give a mathematical definition, an intuitive or behavioural interpretation, assumptions under which the definition is justified, and an example of an urn model to which the definition is applicable. Each of the independence concepts we study appears to be useful in some kinds of application. The concepts of strong independence and epistemic independence appear to be the most frequently applicable.

We shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity “sets in”. We will show how this ‘quasi’ stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of adualchain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity forabsorbingMarkov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing to establish the simultaneous existence and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.

In this paper a concept of probability defined on a Lukasiewicz-Moisil algebra is proposed. We take some steps in developing the theory, including an extension theorem and some results related to conditional probabilities on Lukasiewicz-Moisil algebras.

Attention is drawn to a statistical notion called conditional probabilities. The importance of understanding this concept and the dangers of mis-understanding it is illustrated by means of a few examples.

It has long been known that several popular default and conditional logics exactly describe infinitesimal probability constructs which betoken virtual certainty regarding the truth of one sentence if some other sentence is true. That the rules for default inference also describe a scheme of approximate inference for all positive-valued conditional probabilities has also long been known. More recently, a class of standard probability measures has been found whose expression of more likely than not is exactly described by default rules. This class can be extended by straightforward algebra so that the range of confidence levels expressed by standard probability distributions which are exactly described by the rules is as complete as for the approximate methods.

In this note we give an altemativity to von Neumann Morgenstem Theory. What we propose is a generalization of a preceeding one (Di Bacco and Regoli, 1994) that assumes the bayesian updating and the additivity of probabilities. We give natural axioms to describe the people’s behaviour in the choice of the “optimal” action and in updating probability assesments. Besides we substitute - in composing probabilities - the additivity with the weaker property of associativity. In these conditions we find again, modulus an isomorphism, Expected Utility Rule and Bayes Rule. More precisely our goal may be summarized in the following three points: 1. Justify the use of the expected utility rule without using the independence axiom which is not universally accepted as “natural” property. 2. Justify the Bayes rule in the context of the statistical theory of decision making, that is in direct connexion with the expected utility rule axiom. 3. Weaken a fundamental characteristic of the probability, the additivit...

Real-world problems can often be couched in terms of conditional probability density function estimation. In particular, pattern recognition, signal detection, and financial prediction are among the multitude of applications requiring conditional density estimation. Previous developments in this direction have used neural nets to estimate statistics of the distribution or the marginal or joint distributions of the input-output variables. We have modified the joint distribution estimating sigmoidal neural network to estimate the conditional distribution. Thus, the probability density of the output conditioned on the inputs is estimated using a neural network. We have derived and implemented the learning laws to train the network. We show that this network has computational advantages over a brute force ratio of joint and marginal distributions. We also compare its performance to a kernel conditional density estimator in a larger scale (higher dimensional) problem simulating more realistic conditions.

An adaptive fusion algorithm is proposed for an environment where the observations and local decisions are dependent from one sensor to another. An optimal decision rule, based on the maximum posterior probability (MAP) detection criterion for such an environment, is derived and compared to the adaptive approach. In the algorithm, the log-likelihood ratio function can be expressed as a linear combination of ratios of conditional probabilities and local decisions. The estimations of the conditional probabilities are adapted by reinforcement learning. The error probability at steady state is analyzed theoretically and, in some cases, found to be equal to the error probability obtained by the optimal fusion rule. The effect of the number of sensors and correlation coefficients on error probability in Gaussian noise is also investigated. Simulation results that conform to the theoretical analysis are also presented.

Results concerning the convolution of distributions and distributors are presented in the context of the Renyi theory of conditional probability spaces.

The need to reason with imprecise probabilities arises in a wealth of situations ranging from pooling of knowledge from multiple experts to abstractiondbased probabilistic planning. Researchers have typically represented imprecise probabilities using intervals and have developed a wide array of different techniques to suit their particular requirements. In this paper we provide an analysis of some of the central issues in representing and reasoning with interval probabilities. At the focus of our analysis is the probability crossdproduct operator and its interval generalization, the ccdoperator. We perform an extensive study of these operators relative to manipulation of sets of probability distributions. This study provides insight into the sources of the strengths and weaknesses of various approaches to handling probability intervals. We demonstrate the application of our results to the problems of inference in interval Bayesian networks and projection and evaluation of abstract probabilistic plans.

Many real applications of Bayesian networks (BN) concern problems in which several observations are collected over time on a certain number of similar plants. This situation is typical of the context of medical monitoring, in which several measurements of the relevant physiological quantities are available over time on a population of patients under treatment, and the conditional probabilities that describe the model are usually obtained from the available data through a suitable learning algorithm. In situations with small data sets for each plant, it is useful to reinforce the parameter estimation process of the BN by taking into account the observations obtained from other similar plants. On the other hand, a desirable feature to be preserved is the ability to learn individualized conditional probability tables, rather than pooling together all the available data. In this work we apply a Bayesian hierarchical model able to preserve individual parameterization, and, at the same time, to allow the conditionals of each plant to borrow strength from all the experience contained in the data-base. A testing example and an application in the context of diabetes monitoring will be shown.

The definition of conditional probabilities is based upon the existence of a joint probability. However, a reconstruction of the joint probability from given conditional probabilities imposes certain constraints upon the latter, so that if several conditional probabilities are chosen arbitrarily, the corresponding joint probability may not exist. Such an incompleteness in conditional probabilities can be eliminated by introducing complex probabilities. The physical meaning of the new mathematical formalism, as well as its relation to quantum probabilities, is discussed.

In preparing for battle I have always found that plans are useless, but planning is indispensable. General Dwight D Eisenhower, 34th president of US 1953–1961 (1890–1969) The purpose of this chapter In this chapter we present a step-by-step construction of the Bayesian net associated with a reasonably realistic scenario. This is the scenario for which in Chapters 26–29 we will obtain and analyse in detail the allocation results, and for which we will carry out a detailed sensitivity analysis. Rather than presenting a doctored and sanitized ‘authorized version’, we go through all the steps of the construction of the net, including the false starts, the errors and the hard choices that any real-life implementation inevitably entails. For this reason in this chapter elegance of presentation is not our main concern, and we prefer to show the construction in a warts-and-all manner. As the quote that opens this chapter alludes to, we find the effort put into the preparation phase of a Bayesian net invaluable, and the clarity of thought it stimulates useful, even if, much as General Eisenhower's battle plans, the events in our scenario will not ultimately unfold as we tried to model. Step-by-step construction in a realistic case Roots, leaves and transmission channels Let us start from the following stress scenario, provisionally mapped onto the Bayesian net depicted in Figure 16.1.

A brief description of the material discussed in this chapter is as follows. In the first section, two r.v.’s are considered and the concepts of their joint probability distribution, joint d.f., and joint p.d.f. are defined. The basic properties of the joint d.f. are given, and a number of illustrative examples are provided. On the basis of a joint d.f., marginal d.f.’s are defined. Also, through a joint p.d.f., marginal and conditional p.d.f.’s are defined, and illustrative examples are supplied. By means of conditional p.d.f.’s, conditional expectations and conditional variances are defined and are applied to some examples. These things are done in the second section of the chapter.

The expected supremum of a Gaussian process indexed by the image of an index set under a function class is bounded in terms of separate properties of the index set and the function class. The bound is relevant to the estimation of nonlinear transformations or the analysis of learning algorithms whenever hypotheses are chosen from composite classes, as is the case for multi-layer models.

A Bayesian network model depicts interrelationships in the form of conditional distributions for a collection of random variables. The model is described in terms of a directed acyclic graph in which the nodes are random variables and the directed arcs spell out the structure of conditional distributions. In this chapter, a rudimentary introduction is provided to describe network models. The R package “bnlearn” is invoked to fit a network model using the data on the underlying random variables. Best network model is identified using the “hill-climbing algorithm” via Bayesian Information criterion score. A Bayesian network model is akin to a structural equation model.

This chapter is primarily about conditional expectations and conditional probabilities defined in terms of σ σ -fields. The definitions are based on Theorems 2 and 3 of Chapter 7.

In this paper, we study foundations of interval-based possibilistic networks where possibility degrees associated with nodes are no longer singletons but sub-intervals of [0,1]. This extension allows to compactly encode and reason with epistemic uncertainty and imprecise beliefs as well as with multiple expert knowledge. We propose a natural semantics based on compatible possibilistic networks. The last part of the paper shows that computing the uncertainty bounds of any event can be computed in interval-based networks without extra computational cost with respect to standard possibilistic networks.