Conditional probability theory

Waller (NUREG/CR-4314) provides a concise review of USC with 143 references and relevant papers. He quotes Evans (1975) that no new theory is needed to analyze. system dependencies. The problem persists because the conditional probabilities are not known. Except for the bounding method used in WASH-1400, the other two methods presented below can be shown to be derivable from the theory of Marshall and Olkin (1967). Waller reviews methods other than the three presented here. They are not presented for absence of physical insight to the problem. 

Bayes rule, Eq. (3-164), finds many applications in problems of statistical inference86 and signal detection theory,36 where the conditional probability on the right can be calculated directly in terms of the physical parameters of the problem, but where the quantity of real interest is the conditional probability on the left. 

In this section we consider how to express the response of a system to noise employing a method of cumulant expansions [38], The averaging of the dynamical equation (2.19) performed by this technique is a rigorous continuation of the iteration procedure (2.20)-(2.22). It enables one to get the higher order corrections to what was found with the simplest perturbation theory. Following Zatsepin [108], let us expound the above technique for a density of the conditional probability which is the average... 

The reliability of screening methods is usually expressed in terms of probability theory. In this regard, the conditional probability, P(B A), characterizing the probability of an event B given that another event A occurs, plays an important role. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

One of the most important concepts of any probability theory is the conditional probability. In the density-based approach we can introduce the conditional density. If densities p D) and p(H) (24) exist, p(H) 0 and the following limit p D IH) exists, then we call it conditional density ... 

For polyhedra the situation is similar to usual probability theory densities p(D) and p(H) always exist and if p H) / 0 then conditional density exists too. For general measurable sets the situation is not so simple, and existence of p(D) and p(H)= 0 does not guarantee existence of p(D IH). 

Having read the theory, subjects responded to six questions. The first four questions assessed the conditional probabilities hypothesized in part to motivate the vote (1) if you vote, how likely is it that the other supporters of Party A will vote in larger numbers than the supporters of Party B (2) If you abstain, how likely is it that the other supporters of Party A will vote in larger numbers than the supporters of Party B (3) If you vote, how likely is it that Party A will defeat Party B And (4), if you abstain, how likely is it that Party A will defeat Party B Responses were made on 9-point scales labelled in the middle and at the endpoints. On a similar scale, subjects were asked, How likely are you to vote if the theory were true and voting in Delta were costly and, finally, subjects checked yes or no to the question, Would you vote if the theory were true and voting in Delta were costly ... 

The Ufson-Roig matrix theory of the helix-coil transition In polyglycine is extended to situations where side-chain interactions (hydrophobic bonds) are present both In the helix and in the random coil. It is shown that the conditional probabilities of the occurrence of any number and size of hydrophobic pockets In the random coil can be adequately described by a 2x2 matrix. This is combined with the Ufson-Roig 3x3 matrix to produce a 4 x 4 matrix which represents all possible combinations of any amount and size sequence of a-helix with random coil containing all possible types of hydrophobic pockets In molecules of any given chain length. The total set of rules is 11) a state h preceded and followed by states h contributes a factor wo to the partition function 12) a state h preceded and followed by states c contributes a factor v to the partition function (3) a state h preceded or followed by one state c contributes a factor v to the partition function 14) a state c contributes a factor u to the partition function IS) a state d preceded by a state other than d contributes a factor s to the partition function 16) a state d preceded by a state d contributes a factor r to the partition function. 

R. A. Marcus Even though solvents and solvent-solute interactions or interactions with a protein can be very complicated and the resulting motion can be highly anharmonic, under a particular condition there can be a great simplification because of the many coordinates (perhaps analogous to the central-limit theorem in probability theory). 

A solid solution of silver and silver halide. He considers the most probable theory is one assuming a condition of solid solution. 

For the penultimate model, we just have the triad conditional probabilities we started with, of course. What we re actually doing is using the theory of Markov chains, named after a Russian mathematician who studied the probability of mutually dependent events. In this general approach we would write (Equation 6-30) ... 

The Cochran formula, Equation (9), estimates the triplet phase only exploiting the information contained in the three moduli hl, kU h+kl- The representation theory proposed by Giacovazzo " indicates how the information contained in all reciprocal space could be used to improve the Cochran s estimate of The conclusive conditional probability distribution has again a von Mises expression ... 

In his treatise "The local structure of turbulence in an incompressible viscous liquid at very high Reynolds numbers , Kolmogorov [289] considered the elements of free turbulence as random variables, which are in general terms accessible to probability theory. This assumes local isotropic turbulence. Thus the probability distribution law is independent of time, since a temporally steady-state condition is present. For these conditions Kolmogorov postulated two similarity hypotheses ... 

We can expand this theory to the more general cases as follows. Consider the case that the stress state of the body is not homogeneous and, therefore, each of cells is placed at different stress states according to its position in the body and more over, each of those stress states are varied also with time. For this case, divide the time into n of the short intervals. Then P(tg) can be obtained as follows applying the concept of conditional probability. 

Problem 7.1 The above derivation of the copolymer composition equation [Eq. (7.11)] involves the steady-state assumption for each type of propagating species. Show that the same equation can also be derived from elementary probability theory without invoking steady-state conditions [6-8]. 

This chapter introduces three recent advances to the state-of-the-art, extending the abilities of attribute correspondences. Contextual attribute correspondences associate selection conditions with attribute correspondences. Semantic matching extends attribute correspondences to be specified in terms of ontological relationship. Finally, probabilistic attribute correspondences extend attribute correspondences by generating multiple possible models, modeling uncertainty about which one is correct by using probability theory. 

This is as far as the theory of probability can carry us. We must now introduce a model from which the conditional probabilities P(A B C) can be calculated. 

In probability theory, we are concerned with propositions, or assertions about the occurrence of events. Generally speaking, these assertions will be conditional that is, we assert that such and such is true if this and that are true. Some formal scheme of writing propositions is required, and we adopt the following (A B) means that A is true if B is true. (Another way of putting this is to say that A is true given B, or A is true on data B. )... 

The formation of polymer networks by step-growth polymerization has been modeled using statistical theories, such as the Flory-Stockmayer classical theory [61-64], the Macosko-Miller conditional probability model [65-70], and Gordon s cascade theory [71-74]. However, statistical methods have not been successful for modeling of polymer network formation in chain-growth polymerization systems. 

According to one point of view, expressed by Laplace, dynamical systems like the Solar System are completely deterministic, so probability theory can have no relevance. But this point of view requL a God-like omniscience in being able to determine initial conditions exactly. This requires an infinite number of digits and is beyond the capacity of anybody or anything ctf finite size, including the observable Universe (Ford 1983) [Ref. 59]. In reality measurement is only able to determine the state of a classical system to a finite number of digits, and even this determination is subject to errors, without quantum mechanics, and whether this determination is made by human or machine. Such measurements limit the known or recorded motion to a range of possible orbits. 

The Reverend Thomas Bayes [1702-1761] was a British mathematician and Presbyterian minister. He is well known for his paper An essay towards solving a problem in the doctrine of chances [14], which was submitted by Richard Price two years after Bayes death. In this work, he interpreted probability of any event as the chance of the event expected upon its happening. There were ten propositions in his essay and Proposition 3,5 and 9 are particularly important. Proposition 3 stated that the probability of an event X conditional on another event Y is simply the ratio of the probability of both events to the probability of the event Y. This is the definition of conditional probability. In Proposition 5, he introduced the concept of conditional probability and showed that it can be expressed regardless of the order in which the events occur. Therefore, the concern in conditional probability and Bayes theorem is on correlation but not causality. The consequence of Proposition 3 and 5 is the Bayes theorem even though this was not what Bayes emphasized in his article. In Proposition 9, he used a billiard example to demonstrate his theory. The work was republished in modern notation by G. A. Barnard [13]. In 1774, Pierre-Simon Laplace extended the results by Bayes in his article Memoire sur la probabilite des causes par les evenements (in French). He treated probability as a tool for filling up the gap of knowledge. The Bayes theorem is one of the most frequently encountered eponyms in the literature of statistics. 

Markovian pair conditional probability density with the basic concepts of the catastrophe theory, he succeeded in introducing new Markovian ELF classes which generalize the previous Becke-Edgecombe definition. Going beyond the actual interpretation of ELF as the error in electron locahzation, this new approach provides a quantum step-function indicating where the electrons are trapped rather than where they have peaks of spatial density. 

For practical applications, we will not consider T(r) itself but rather the definite positive kinetic energy density of independent particles r (r) which appears in the exact density functional theory[31j. Within this framework, the non-von Weizsacker term accounts only for the Fermi correlation and is usually referred to as Pauli kinetic energy density[32]. Another propery of r ff(r) is its relationship to the conditional probability rO for... 

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