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Stuart objects to the premisses of my approach to the question of joint and conditional probabilities, so I will use renovated terms in their favor. Then he claims that my theory is self-contradictory; I will show that this is not so.

(1999). A Natural Proof of the Chain Rule. The College Mathematics Journal: Vol. 30, No. 3, pp. 216-218.

In this paper we propose an algorithm for the on-line maintenance of the joint probability distribution of a data stream. The joint probability distribution is modeled by a mixture of low dependence Bayesian networks, and maintained by an on-line EM-algorithm. Modeling the joint probability function by a mixture of low dependence Bayesian networks is motivated by two key observations. First, the probability distribution can be maintained with time cost linear in the number of data points and constant time per data point. Whereas other methods like Bayesian networks have polynomial time complexity. Secondly, looking at the literature there is empirical indication [1] that mixtures of Naive-Bayes structures can model the data as accurate as Bayesian networks. In this paper we relax the constraints of the mixture model of Naive-Bayes structures to that of the mixture models of arbitrary low dependence structures. Furthermore we propose an on-line algorithm for the maintenance of a mixture model of arbitrary Bayesian networks. We empirically show that speed-up is achieved with no decrease in performance.

We propose a Bayesian higher-order probability logic reasoning approach with interval probability parameters to the problem of making inference from conditional knowledge, which combines weak conditional probability and conditional event algebra for approximate inferences. We define the bound-limited weak conditional interval probabilities, the corresponding probabilistic description, and the multiplication rules of weak conditional probabilities for joint probability distribution, and use higher-order conditional event to resolve a discrepancy between logic and probability. By extending normal measurable space with conditional event, we bring logic consistent with probability in denoting conditional knowledge, and then transform a higher-order conditional event to normal events and corresponding logical joint events via conditional event algebra. Based on multiplication rules, we compute the quantitative values of the events with interval parameters, and evaluate the value of higher-order conditional event and finish reasoning process. An illustrative example of application of our method shows how we make inferences from conditional knowledge.

Publisher Summary A Bayesian network is a tool for modeling and reasoning with uncertain beliefs; it comprises two parts: a qualitative component in the form of a directed acyclic graph (DAG) and a quantitative component in the form conditional probabilities. Intuitively, the DAG of a Bayesian network explicates variables of interest (DAG nodes) and the direct influences among them (DAG edges). The conditional probabilities of a Bayesian network quantify the dependencies between variables and their parents in the DAG. Formally though, a Bayesian network is interpreted as specifying a unique probability distribution over its variables. Hence, the network can be viewed as a factored (compact) representation of an exponentially sized probability distribution. The formal syntax and semantics of Bayesian networks are discussed in this chapter. The power of Bayesian networks as a representational tool stems both from the ability to represent large probability distributions compactly and the availability of inference algorithms to answer queries about these distributions without necessarily constructing them explicitly. The chapter also discusses exact inference algorithms and approximate inference algorithms.

The rule states that for two events A and B, the probability that any one of them will occur given that the other has occurred or will occur is equal to the probability that both the events will occur together divided by the probability of the event that has occurred or will occur. (Proof/Derivation)

This note explores the convergence properties of certain sequences of conditional probabilities arising in a journal selection problem, where the probabilities of interest are decreasing in either a deterministic or stochastic fashion. We prove the convergence to a nonextreme value of the probability of an eventual event for any choice of problem parameters within the open unit interval. Computational results illustrate the convergence properties.

In this paper, we propose a new family of density estimates closely related to the nearest neighbor estimates introduced by Loftsgaarden and Quesenberry. An optimal estimator, wit respect to the asymptotic mean square error, is obtained for a given distribution.

A causal network is frequently used as a representation for qualitative medical knowledge, in which conditional probability tables on appropriate sets of variables form the quantitative part of the accumulated experience. For probabilities temporarily assumed known, we describe efficient algorithms for propagating the effects of multiple items of evidence around multiply-connected networks and hence providing precise probabilistic revision of beliefs concerning the current patient. As a database accumulates we also require the quantitative aspects of the model to be updated, as well as to learn about the qualitative structure, and we suggest some formal statistical tools for these problems.

The uncertainty in the estimated posterior multinomial probabilities which results from applying Bayes rule using conditional probabilities estimated from a limited data base is investigated. The first order Taylor series estimator for the variance of the estimated posterior probabilities was developed. A simulation study was conducted to ascertain the accuracy of these approximate variance estimates. In addition, the accuracy of both posterior probability estimates and the variance estimates is evaluated for data bases of various sizes. The method is appropriate for any of the usual medical diagnostic problems in which Bayesian methods are applied

We study the usefulness of representing a given joint distribution as a positive linear combination of disjunctions of hypercubes, and generalize the associated results and techniques to Bayesian networks (BNs). The fundamental idea is to pre-compile a given distribution into this form, and employ a host of randomization techniques at runtime to answer various kinds of queries efficiently. Generalizing to BNs, we show that these techniques can be effectively combined with the dynamic programming-based ideas of message-passing and clique-trees to exploit both the topology (conditional independence relationships between the variables) and the numerical structure (structure of the conditional probability tables) of a given BN in efficiently answering queries at runtime.

In practice, the durations of activities are not independent random variables but rather dependent random variables. In this paper, an expert system is proposed to model the dependence of activity durations in project scheduling networks and to update the schedule, in which a Bayesian network is used to describe the interdependence between activities. According to the model, the marginal distribution of activity duration and the concordance coefficient between activities are first estimated by experts; the conditional probability and distribution of the Bayesian Network is calculated consequently; and then the conditional distribution of the durations of all activities and the project duration can be determined. An expert system is constructed, which can update the distribution of the durations of unfinished activities according to the observational durations of finished activities. Results from the numerical example show that the model is promising in reducing the risk of project schedule and predicting the uncertainty of activities.

Interval probabilities have been proposed as one of non-additive measures. The frame of interval probabilities is similar to evidence theory proposed by Dempster and Shafer and they can be regarded as evidences on a finite set. The interval probability is suitable to represent ignorance on the given phenomenon so that it can be used as a kind of subjective probability. We show how to obtain the evidence by a pairwise comparison matrix on a finite set. The pariwise comparisons are usually inconsistent each other since they are given based on human judgements. The interval probabilities from them are determined so as to include such inconsistency. In case of two evidences whose prior and conditional probabilities are obtained as intervals, the marginal and posterior probabilities are also calculated as interval probabilities from the view of possibility. The illustrative numerical example is given in this paper.

This paper discusses military and Homeland Security sensors, sensor systems, and sensor fusion under very general assumptions of statistical performance. In this context, the system performance metrics parameters are analyzed in the form of direct and inverse conditional probabilities, based on so-called signal theory, applied first for automatic target recognition (ATR). In particular, false alarm rate, false positive, false negative rate, accuracy, and probability of detection (or, probability of correct rejection), are discussed as conditional probabilities within classical and Bayesian inference. Several examples from various homeland security areas are also discussed to illustrate the concept. As a result, it is shown that vast majority of sensor systems (in a very general sense) can be discussed in terms of these parameters.

Joint probability density functions (JPDFs) can be used to describe the likelihood of spatial position or events that are dependent on spatial position. Event-based JPDFs are often based on computational outcomes at specific locations, where the number of potential locations may be unlimited. The efficiency and accuracy associated with the estimation of the conditional failure JPDF is dependent on the number of spatial locations and the number of limit state evaluations at each spatial location. An approach is presented for the estimation of a fracture mechanics-based conditional failure JPDF within a finite domain. An approximate JPDF is constructed using probability of fracture values associated with anomalies placed at discrete points along the perimeter of the component and at selected locations within the interior of the component. The discrete points are connected to form a mesh of elements, and anomaly occurrence rates are assigned to elements based on their relative volumes. The final JPDF is obtained from an adaptive refinement of the element mesh where elements are subdivided based on contribution to component risk. The results can be applied to the risk assessment of components that are susceptible to fracture failure.

This chapter contains sections titled: Sample Space, Events, and Probabilities Random Variables Cumulative Distribution Functions, Expectation, and Moment Generating Functions Discretely Distributed Random Variables Continuously Distributed Random Variables Some Useful Inequalities Joint Distribution Functions Conditional Expectation Independent Random Variables Conditional Independence A Law of Large Numbers First-Order Autoregressive Estimators Measures of Separation Between Distributions Statistical Confidence Deciding Between Two Alternative Claims Problems

Probabilistic forecasts of variables measured on a categorical or ordinal scale, such as precipitation occurrence or temperatures exceeding a threshold, are typically verified by comparing the relative frequency with which the target event occurs given different levels of forecast confidence. The degree to which this conditional (on the forecast probability) relative frequency of an event corresponds with the actual forecast probabilities is known as reliability, or calibration. Forecast reliability for binary variables can be measured using the Murphy decomposition of the (half) Brier score, and can be presented graphically using reliability and attributes diagrams. For forecasts of variables on continuous scales, however, an alternative measure of reliability is required. The binned probability histogram and the reliability component of the continuous ranked probability score have been proposed as appropriate verification procedures in this context, but are subject to some limitations. A procedure is proposed that is applicable in the context of forecast ensembles and is an extension of the binned probability histogram. Individual ensemble members are treated as estimates of quantiles of the forecast distribution, and the conditional probability that the observed precipitation, for example, exceeds the amount forecast [the conditional exceedance probability (CEP)] is calculated. Generalized linear regression is used to estimate these conditional probabilities. A diagram showing the CEPs for ranked ensemble members is suggested as a useful method for indicating reliability when forecasts are on a continuous scale, and various statistical tests are suggested for quantifying the reliability.

In this study the effectiveness of an arrow diagram which can help students apply the Chain Rule was investigated. Different variations of this diagram were used as mnemonic devices for applying the Chain Rule. For the investigation two instruments were developed, diagnostic test and post-test. The diagnostic test was developed to determine the students' difficulties with the Chain Rule and to create matched groups. It was administered to 76 students taking the Advanced Calculus Course. By matching according to the results of the diagnostic test, the sample of 24 pairs of subjects is obtained. The results of the post-test, administered after the teaching program, indicated that the arrow diagram had positive effects on applying the Chain Rule.

Bayesian belief networks as powerful tools for modeling causes and effects in a wide variety of domains. Using of compact networks of probabilities that capture the probabilistic relationship between variables, as well as historical information about their relationships.

Multiagent Bayesian networks (MABNs) are a powerful new framework for uncertainty management in a distributed environment. In a MABN, a collective joint probability distribution is defined by the conditional probability tables (CPTs) supplied by the individual agents. It is assumed, however, that CPTs supplied by individual agents agree on the variable domains, an assumption that does not necessarily hold in practice. In this paper, we suggest modelling MABNs with inclusion dependencies. Our approach is more flexible, and perhaps realistic, by allowing CPTs supplied by different agents to disagree on variable domains. Our main result is that the input CPTs define a joint probability distribution if and only if certain inclusion dependencies are satisfied. Other advantages, both practical and theoretical, of modelling MABNs with inclusion dependencies are discussed.

This article derives key variables in the analysis of standards of proof in criminal law from basic conditional probabilities. The variables derived are the probability of correct and wrongful conviction, the expected sanction and society’s incarceration costs, while the basic conditional probabilities are the probability of observing (any given) evidence against individual i given that individual j committed the crime (for any j including j equal to i). The variables are derived from the conditional probabilities as a function of the standard of the proof using simple Bayesian updating.

This paper deals with the problems of typicality and conditional typicality of “empirical probabilities” for stochastic process and the estimation of potential functions for Gibbs measures and dynamical systems. The questions of typicality have been studied in [FKT88] for independent sequences, in [BRY98, Ris89] for Markov chains. In order to prove the consistency of estimators of transition probability for Markov chains of unknown order, results on typicality and conditional typicality for some (Ψ)-mixing process where obtained in [CsS, Csi02]. Unfortunately, lots of natural mixing process do not satisfy this Ψ -mixing condition (see [DP05]). We consider a class of mixing process inspired from [DP05]. For this class, we prove strong typicality and strong conditional typicality. In the particular case of Gibbs measures (or complete connexions chains) and for certain dynamical systems, from the typicality results we derive an estimation of the potential as well as a procedure to test the nullity of the asymptotic variance of the process. More formally, we consider X0, ...., Xn, ... a stochastic process taking values on an complete set Σ and a sequence of countable partitions of Σ, (Pk)k∈N such that if P ∈ Pk then there exists a unique P ∈ Pk−1 such that almost surely, Xj ∈ P implies Xj−1 ∈ P . Our aim is to obtain empirical estimates of the probabilities: P(Xj ∈ P ), P ∈ Pk

Consider a list of $n$ files whose popularities are random. These files are updated according to the move-to-front rule and we consider the induced Markov chain at equilibrium. We give the exact limiting distribution of the search-cost per item as $n$ tends to infinity. Some examples are supplied., Comment: move-to-front, search cost, random discrete distribution, limiting distribution, size biased permutation

We develop conditional alpha performance measures that are consistent with conditional mean-variance analysis. Commonly employed conditional alpha measures are inconsistent with conditional mean-variance analysis, conditional Sharpe ratio maximization, the magnitude or sign of the true conditional alphas, and reliable time-varying alphas. This conclusion is based on our derivation of the conditional investment opportunity set and its implied conditional alphas for given population parameters. The analysis is not subject to estimation errors and the important issue of whether empirically estimated performance measures violate the bounds on rational stochastic discount factors (Chen and Knez, 1996). An empirical example demonstrates that the differences between existing measures and our proposed measures are substantive for typical parameterizations.

Bayesian network has been shown to be very successful for many computer vision applications, most of which are solved using the generative approaches. We propose a novel weighted Bayesian network which relaxes the conditional independent assumption in traditional Bayesian network by assigning weights to the estimations of conditional probabilities. In the weighted Bayesian network, the hidden variables are estimated generatively as in the traditional graphical models, and the weights of conditional probabilities are adjusted discriminatively from, the training samples. The combined generative/discriminative approach in a loop preserves the advantage of generative model to perform unsupervised learning and handle missing data while improve the model flexibility and performance by the discriminative learning of probability estimation weights. Our experiments show a number of real-time examples in visual tracking where the performances are significantly improved with the weighted Bayesian networks

The Bayesian networks can express the joint probabilistic distribution compactly between variables and can express the conditionally independence conveniently. The joint probabilistic influence from the parents to their child can be got from the Bayesian network structure however parents are not necessarily have common influence to their child, which are called by the name of causal influence independence other than conditional independence. The causal influence independence extension model of Bayesian networks presented can have wider meaning than traditional Bayesian networks, which is more applicable and easier to understand.

The concept of a probability as a measure of the uncertainty of an event is introduced. Venn diagrams are shown to be useful as a means of representing probabilities and the basics of set theory are outlined. Treating relative frequencies as probabilities is justified using the law of large numbers and independence as a simplifying principle is explained with the help of conditional probabilities. Finally, a probability analysis is applied to a plausible real life situation. Keywords: conditional probability; independence; sample space; Venn diagram

This paper states necessary and sufficient conditions for the existence, uniqueness, and updating according to Bayes’ rule, of subjective probabilities representing individuals’ beliefs. The approach is preference based, and the result is an axiomatic subjective expected utility model of Bayesian decision making under uncertainty with state-dependent preferences. The theory provides foundations for the existence of prior probabilities representing decision makers’ beliefs about the likely realization of events and for the updating of these probabilities according to Bayes’ rule.

According to Bayesian decision theory, the maximum a posteriori (MAP) decision rule is used to minimize the speaker recognition error rate. The a posteriori probability is determined if the a priori probability and the likelihood function are known. However, there has been no method to determine the a priori probability, therefore the maximum likelihood (ML) decision rule is used instead. The paper proposes a method to estimate the a priori probability for speakers based on a training data set and speaker models. Speaker identification experiments performed on 138 Gaussian mixture speaker models in the YOHO database using the MAP rule showed lower error rates than using the ML rule.

Given a Bayesian network relative to a set I of discrete random variables, we are interested in computing the probability distribution PA or the conditional probability distribution PA|B, where A and B are two disjoint subsets of I. The general idea of the algorithm of successive restrictions is to manage the succession of summations on all random variables out of the target A in order to keep on it a structure less constraining than the Bayesian network, but which allows saving in memory ; that is the structure of Bayesian Network of Level Two.