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We define conditional probability and give Bayes Rule for calculating conditional probability in reverse. We give several applications of Bayes Rule to economic problems. We define the concept of independence and conditional independence and give examples illustrating the difference.

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.

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.

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.

This chapter introduces a probabilistic approach to modelling in physiology and medicine: the quantities of interest are modeled as random variables and the focus is on the probabilistic dependencies between these variables. As primary tool in this modelling framework, we present Bayesian networks (BNs), which map the dependencies between a set of random variables to a directed acyclic graph, both increasing human readability and simplifying the representation of the joint probability distribution of the set of variables. The chapter first describes the theoretical foundations of BNs, including a brief review of probability and graph theory, a formal definition of BNs and details on discrete, continuous, and dynamic BNs. Then, a selection of algorithms for inference, conditional probability learning, and structure learning is presented. Finally, several examples of BN applications in biomedicine are reviewed.

This chapter consists of two sections. In the first section, the concept of the conditional probability of an event, given another event, is taken up. Its definition is given and its significance is demonstrated through a number of examples. The section is concluded with three theorems, formulated in terms of conditional probabilities. Through these theorems, conditional probabilities greatly simplify calculation of otherwise complicated probabilities. In the second section, the independence of two events is defined, and we also indicate how it carries over to any finite number of events. A result (Theorem 4) is stated which is often used by many authors without its use even being acknowledged. The section is concluded with an indication of how independence extends to random experiments. The definition of independence of r.v.’s is deferred to another chapter (Chapter 10).

Publisher Summary Statistical inferences rely on probabilities. Bayesian statistics employs probabilities that are relative to evidence. Bayesian statistics studies decisions to illuminate the probabilities it uses to analyze statistical inferences. This chapter explains the Bayesian decision-theoretic approach to statistics. A statistical test of a hypothesis collects statistical data to evaluate the hypothesis. The Bayesian decision-theoretic approach to statistics uses a statistical test along with prior information to evaluate a hypothesis. Bayesians methods generate the hypothesis's probability at the conclusion of the test. This chapter describes Bayesian probabilities' function in representations of decision problems and considers how to use the relationship between Bayesian probabilities and decisions to illuminate these probabilities. It takes them to be rational degrees of belief and does not define them in terms of preferences. The properties of Bayesian probabilities are explored and strategies for meeting objections are presented. Bayesianism advances conditionalization as a rule of probabilistic inference and maximization of expected utility as a decision rule. It compares Bayesian and classical statistics and defends Bayesian statistics against the charge of inordinate subjectivism.

This chapter concerns learning the parameters in a Bayesian network from data. In a Bayesian network the conditional probability distributions are called the parameters. Initially, a direct acyclic graph (DAG) in a Bayesian network was ordinarily hand constructed by a domain expert. Then the conditional probabilities were assessed by the expert, learned from data, or obtained using a combination of both techniques. Researchers developed methods that could learn the DAG from data. Furthermore, they formalized methods for learning the conditional probabilities from data. These methods are discussed in this chapter. It begins by addressing the problem of learning a single parameter. Under this, it presents the method for the mathematical development. After presenting a method for learning the probability of a binomial random variable, it extends the method to multinomial random variables. It then provides guidelines for articulating the prior beliefs concerning probabilities. Following this, it discusses learning all the parameters in a Bayesian network and focuses on learning discrete parameters. Finally, it illustrates a method for learning the parameters in a Gaussian Bayesian network.

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.

Publisher Summary Probability has been applied in the actuarial field for many years. Probability means a measure of the chance of occurring an event, or a set of events. Probabilities can range from 0 to 1, inclusive. A probability of 0 means that the event is impossible; a probability of 1 means the event is certain. Many people think that if a particular outcome does not occur for a long while, the chances of it occurring in the future increase. Probabilities represent a mathematical model of the world. The chapter discusses the use of population. Population refers to the set of subjects using for probability analysis. This could be people or certain sets of people, but it could also be other collections. Sometimes one can develop a set of probabilities by developing a model of an operation or activity of some sort. One can enter numerical inputs into the model using a table of random numbers or a computer random number-generating program to determine numerical outputs. Thus, one can then determine the probabilities of the various outputs.

Several Artificial Intelligence schemes for reasoning under uncertainty explore either explicitly or implicitly asymmetries among probabilities of various states of their uncertain domain models. Even though the correct working of these schemes is practically contingent upon the existence of a small number of probable states, no formal justification has been proposed of why this should be the case. This paper attempts to fill this apparent gap by studying asymmetries among probabilities of various states of uncertain models. By rewriting the joint probability distribution over a model's variables into a product of individual variables' prior and conditional probability distributions and applying central limit theorem to this product, we can demonstrate that the probabilities of individual states of the model can be expected to be drawn from highly skewed lognormal distributions. With sufficient asymmetry in individual prior and conditional probability distributions, a small fraction of states can be expected to cover a large portion of the total probability space with the remaining states having practically negligible probability. Theoretical discussion is supplemented by simulation results and an illustrative real-world example.

It is well known that conditional independence can be used to factorize a joint probability into a multiplication of conditional probabilities. This paper proposes a constructive definition of intercausal independence, which can be used to further factorize a conditional probability. An inference algorithm is developed, which makes use of both conditional independence and intercausal independence to reduce inference complexity in Bayesian networks.

The criticism on the usage of Bayesian Networks in expert systems was centered around the claim that the use of probability requires a massive amount of data in the form of conditional probabilities. This paper shows that given information easily obtained from experts, the dependence model and some observations, the conditional probabilities can be estimated using backpropagation, such that during training the Bayesian characteristic of the network is preserved. Applying the Occam's razor principal results in defining a partial order among neural network structures. Experiments show that for the Multiplexer problem, the network compiled from the more succinct causal model generalized better than the one compiled from the less succinct model.

Publisher Summary This chapter focuses on conditional probability and conditional expectation, which are important concepts in probability theory. In calculating probabilities and expectations, when some partial information is available, the desired probabilities and expectations are conditional ones. On the other hand, in calculating a desired probability or expectation, it is often extremely useful to first condition on some appropriate random variable. If X and Y have a joint probability density function f(x, y), then the conditional probability density function of X, given that Y = y, is defined for all values of y such that fY(y) > 0 by fX |Y (x | y) =.

An approach to reasoning with default rules where the proportion of exceptions, or more generally the probability of encountering an exception, can be at least roughly assessed is presented. It is based on local uncertainty propagation rules which provide the best bracketing of a conditional probability of interest from the knowledge of the bracketing of some other conditional probabilities. A procedure that uses two such propagation rules repeatedly is proposed in order to estimate any simple conditional probability of interest from the available knowledge. The iterative procedure, that does not require independence assumptions, looks promising with respect to the linear programming method. Improved bounds for conditional probabilities are given when independence assumptions hold.

The belief network is a well-known graphical structure for representing independences in a joint probability distribution. The methods, which perform probabilistic inference in belief networks, often treat the conditional probabilities which are stored in the network as certain values. However, if one takes either a subjectivistic or a limiting frequency approach to probability, one can never be certain of probability values. An algorithm should not only be capable of reporting the probabilities of the alternatives of remaining nodes when other nodes are instantiated; it should also be capable of reporting the uncertainty in these probabilities relative to the uncertainty in the probabilities which are stored in the network. In this paper a method for determining the variances in inferred probabilities is obtained under the assumption that a posterior distribution on the uncertainty variables can be approximated by the prior distribution. It is shown that this assumption is plausible if their is a reasonable amount of confidence in the probabilities which are stored in the network. Furthermore in this paper, a surprising upper bound for the prior variances in the probabilities of the alternatives of all nodes is obtained in the case where the probability distributions of the probabilities of the alternatives are beta distributions. It is shown that the prior variance in the probability at an alternative of a node is bounded above by the largest variance in an element of the conditional probability distribution for that node.

Publisher Summary This chapter discusses the conditional Markov processes. Relationships are given among the probabilities of conditional Markov chains for neighbouring tests. The conditional probabilities at the end of the observation interval—the final probabilities—are satisfied with first-type equations corresponding to an increase in the observation interval. The second-type equations for the conditional probabilities within the observation interval are written in terms of these final probabilities. The following special cases are considered: (1) Gaussian noise with independent values that becomes a delta-correlational process when the moments of time are compacted, and (2) a continuous Markov process. The related problem of the time sign reversal of ordinary, a priori, Markov processes is also treated.

Probabilities are numbers assigned to events. This chapter considers the nature of the assignment of probabilities to events, which, in effect, defines a function whose domain is the class of events, which are subsets of Ω. To be satisfactory, the assignment must preserve the essential properties of the “classical” probability function and must be consistent with the various interpretations, such as the relative-frequency interpretation, associated with the classical theory. the extensive theory of measures is available to aid in the development of the probability model and its properties. The physical analogy of probability with mass provides an important aid in visualizing various properties of the probability function. The very simplicity and generality of the probability model has an important consequence. Although the formal system determines how probabilities of various compound events formed from a given class are related, it does not determine how the probabilities of the various members of the class are to be assigned—except in very special cases.