Conditional Probability

Understanding independent and conditional probability is basic for a right utilization of numerous probabilistic and measurable ideas and strategies.This article is also focusing on Baye’s theorem which means the direct application of conditional probability. Hence, the model explains the individuals which evaluatesconditional probability i.e, P(A|B) (the probability of A given that B has happened) by a procedure that takes after standardfrequentistprobability hypothesis yet is liable to irregular commotion.

But the epistemic perspective has two further features. First, it considers many agents together, with their mutual information. This would be like having my probability about your probabilities, etc. But even more importantly, product update does not just select subzones of the current information space, but it transforms the latter much more drastically as required by relevant information-carrying actions. Probabilistic theory speaks about events A on which we conditionalize, which seems a similar ambition. One wants to combine conditional probability with an account of how actions change the current probability model. Let's see how this works out by continuing with the earlier example of probability tree diagrams. 5 Computing updates on probabilities with public actions

Analysis of multiple noninvasive tests offers the promise of more accurate diagnosis of coronary artery disease, but discordant test responses can occur frequently and, when observed, result in diagnostic uncertainty. Accordingly, 43 patients undergoing diagnostic coronary angiography were evaluated by noninvasive testing and the results subjected to analysis using Bayes' theorem of conditional probability. The procedures used included electrocardiographic stress testing for detection of exercise-induced ST segment

The quantum mechanical model of a composite system consisting of two copies is the Hilbert space tensor prod- uct H ⊗ H. The self-adjoint projection operators e on H are mapped to two copies on H ⊗H by π 1 (e) := e ⊗ I and π 2 (e) := I ⊗ e. Note that (I) and (J) are then satisfied. The time evolution of the composite system is described by unitary transformations of H ⊗ H and therefore the cloning operation should be a unitary transformation. It defines an automorphism of the quantum logic of H ⊗H . Theorem 1 thus includes the quantum mechanical no- cloning theorem for pure states as a special case. Instead of the Hilbert space and tensor product formalism, The- orem 1 requires only a few very basic principles; these are the existence and the uniqueness of the conditional probabilities and the existence of two compatible copies of the system in a larger system. Nevertheless, the proof of the no-cloning theorem in the quantum mechanical Hilbert space formalism can be mimicked, replacing the Hilbert space inner product h | i by the specific state- independent conditional probability P ( | ).

roblem be- mes rather obvious using basic conditional probability, this was not the approach by those arguing their case in the controversy over the Monty Hall problem. Both sides in the controversy used much less preciser, and some- times casual language, to state their thinking. Some used the word “probability”, some used the word “odds”, some used the word “chance”, and yet others used a mix of these simultaneously as if all these terms refer to the same thing. But did such an agreement, as Wittgenstein would have probably wondered, exist between the parties to the controversy that all these terms mean the same thing? The possibility of that being the case seems not only remote but also highly unlikely. Hence, faulty rea- soning goes undiscovered and communication itself be- gins to become ineffective—as Wittgenstein had sug- gested. Consider the following examples of what was said:

In conclusion, we have studied the relation between a hidden variables theory and the existence of the Bloch sphere and with a new type of conditional probability and Fisher information as metric for information space we show that the hidden variable are not possible. Now, we have derived some proposition concerning a quantum expected val- ue under an assumption about the existence of the Bloch sphere in N spin-1/2 systems. However, the hidden variables theory has violated the proposition with a magnitude that grows exponential with the number of particles. Therefore, we have had to give up either the existence of the Bloch sphere or the hidden variables theory. The hidden va- riables theory does not have depictured physical phenomena using the existence of the Bloch sphere with a violation factor that grows exponentially with the number of par- ticles. Now, we point out the problem that when we cannot measure an observable we cannot say nothing on this measure as in the non-commutative case. So we have con- tradictions. In classical interpretation of quantum mechanics does not exist conditional probability and we cannot measure the probability but with the introduction of the in- formation space and Fisher metric we show that conditional probability is possible but limited to statistical parameters as average value or other parameters. So contradiction is eliminated. Now, entanglement and Bell theorem can be understood in a new type of set theory that includes copula [27] and information [28]. Maybe we are right that pro- jection operator is not sufficient to understand quantum mechanics so we cannot give Hilbert space axiomatic structure. Now, axiomatic Hilbert space is useful but cannot completely explain the meaning of the quantum mechanics. With information space, we can give a meaning with the axiomatic Hilbert that is always a useful mathematical instrument to use information and probability together.

Statistics and probability has an abstract, uncertain nature. Where conclusions in mathematical problems are mostly certain, in probability it is only possible to have ‘probabilistic’ conclusions that approximate the theoretical probability in order to make decisions. Because of its uncertain nature there have been serious cognitive issues in teaching and learning of probability. Conditional probability is specifically difficult for students to learn and it is also difficult for teachers to teach. With the help of simulation tools where students can participate in the process of building probability models and collecting data generated from those models, there are great opportunities to improve teaching and learning of conditional probability. In this thesis, research-based misconceptions and suggestions for overcoming these misconceptions are discussed. Using research-based suggestions and previous frameworks, a new framework about creating conditional probability tasks (i.e. simulation tools) has been constructed for others to use. Two TinkerPlots tasks are presented as examples in which task content and problems are aligned with the framework.

Abstract—This paper proposes a direct model for conditional probability density forecasting of residential loads, based on a deep mixture network. Probabilistic residential load forecasting can provide comprehensive information about future uncertain- ties in demand. An end-to-end composite model comprising convolution neural networks (CNNs) and gated recurrent unit (GRU) is designed for probabilistic residential load forecasting. Then, the designed deep model is merged into a mixture density network (MDN) to directly predict probability density functions (PDFs). In addition, several techniques, including adversarial training, are presented to formulate a new loss function in the direct probabilistic residential load forecasting (PRLF) model. Several state-of-the-art deep and shallow forecasting models are also presented in order to compare the results. Furthermore, the effectiveness of the proposed deep mixture model in char- acterizing predicted PDFs is demonstrated through comparison with kernel density estimation, Monte Carlo dropout, a combined probabilistic load forecasting method and the proposed MDN without adversarial training.

This brief proposes an accuracy-adjustment fixed-width Booth multiplier that compensates the truncation error using a multilevel conditional probability (MLCP) estimator and derives a closed form for various bit widths L and column information w. Compared with the exhaustive simulations strategy, the proposed MLCP estimator substantially reduces simulation time and easily adjusts accuracy based on mathematical derivations. Unlike previous conditional-probability methods, the proposed MLCP uses entire nonzero code, namely MLCP, to estimate the truncation error and achieve higher accuracy levels. Furthermore, the simple and small MLCP compensated circuit is proposed in this brief. The results of this brief show that the proposed MLCP Booth multipliers achieve low-cost high-accuracy performance. Hough transform is widely used for detecting

Abstract—with the wide application of probabilistic systems, the performance analysis for probabilistic system with model checking have attracted wide attention. For conditional probability formulae of complex parametric system, this paper proposes a counterexample generation method of conditional probability properties based on continuous time probabilistic model. We use continuous time Markov reward model with comprehensive feature representation ability as the system model need to be verified, give satisfiability probability solution algorithm of probabilistic computation tree logic multiple constraints until formulae path properties after model pretreatment, put forward the counterexample generation method of conditional probability on multiple constraints until formulae and give the example analysis. The theoretical analysis and example result show that the feasibility and validity of the method.

The notion of conditional probability is indispensable within many walks of life, and in diverse realms of scientific disciplines. In particular, this notion is of critical importance in many medical contexts, and constitutes an invaluable useful piece of knowledge for the physician and the patient alike. However, there are many contemporary complaints of collective probabilistic illiteracy, and even a widespread inability to merely understand the meaning of numbers [1]. Currently, there are many research efforts aiming at making the concepts of conditional probability more transparent and accessible to ordinary and professional people [1-37]. This paper is a serious attempt to review the aforementioned research topic via four avenues, to be depicted as representations or tools. Some of these avenues are novel methods and some have already appeared in the literature but are still subject to ongoing vigorous research. This research topic is of an obviously interdisciplinary nature as it combines elements from widely diverse areas such as mathematics, clinical medicine, epidemiological diagnostic testing, probability and statistics, problem solving, and educational psychology. We hope that this paper is of general interest to researchers in the aforementioned areas. However, we mainly target a clinical audience, in general, and medical researchers and educators, in particular. In addition, we hope that the paper might be of some benefit, at least partially, to practicing physicians as well as to medical students (perhaps with some graceful help from their competent educators).

the incidence and mortality of each cancer effects different populations (e.g., different races, SEER population at dif- ferent times), while controling for the effect of differing age distributions between populations being compared. A disadvantage of the DSR is that it is hard to relate to an individual's risk. For example, Table I-4 of the SEER Can- cer Statistics Review, 1975–2000 [2] states that the DSR for breast cancer for females for the years 1996–2000 is 135 per 100,000 person-years. The average American woman may wonder, how does that relate to my risk? Will I be likely to get breast cancer in my lifetime? If I am 40 years old now, what is my risk of getting breast cancer in the next 10 years given that I have survived to this old without getting it? These questions are the motivation for using the age conditional probability of developing dis- ease (ACPDvD), and in order to estimate the ACPDvD for female breast cancer, we require information not only about the rate of female breast cancer but also about the rates of dying from female breast cancer and dying from other causes.

In their papers [12], [13] Battigalli and Siniscalchi develop their notions of conditional and strong belief using conditional probability systems. Their theory follows the prob- ability 1 principle of quantitative representation of belief, equating “believing H given E” with “assigning probability 1 to H when conditioning on E”. Throughout the thesis, we will be comparing (whenever possible) our setting and theories of conditional and r-stable beliefs with B-S’s work. In particular, we will see that there is a direct analogy between B-S’s work and ours, since we will show that our notion of r-stability can express B-S’s notion of strong belief. Furthermore, this comparison shows that there is a tight connection between the theory presented in this thesis and B-S’s work, a connection that opens directions for future applications of stable beliefs in epistemic game theory, as we will discuss in chapter 9.

l Segoviano (2006) using the conditional probability of default methodology investigates the probability of default for small business enterprises as function of macroeconomics variables. The dataset used in this study is represented by non-performing loans ratio registered in Mexico and Norway.

The principle of probability calculation under the conditional probability approach was similar to the serial screening test in disease screening. As Fig. 1 illustrates, when the first screen test was negative, there would be no second and third tests, and these patients were counted as “missing.” By the additional two screening processes, the data of patients who completed at least two screening tests were used to estimate the probability of each possible outcome of the second screening, given the first screening result, and the data of patients who completed all the three screening tests were used to estimate the probability of each possible outcome of the third screening, given the first and second screening results. Then, the final outcome probability could be calculated by multiplying the third screening proba- bility with the given condition probability. Finally, we added up the probabilities of all possible final outcomes that indicated the diagnosis of MAU as the so-called “the adjusted preva- lence,” as Table 1 demonstrates. The gray cells in Table 1 show that the probability, unreasonably affected by the sparse count of the third screening, had been corrected by contingency factor or the estimation from the second screening.

After all, we had the concept of conditional probability long before we donned our philosopher’s or mathematician’s caps. In ordinary language, conditional probability statements can be made using locutions of the form “it’s likely that p, given q”, “it’s improbable that r, if s ”, and variations on these. In The Piano Man, Billy Joel sings, “I’m sure that I could be a movie star, if I could get out of this place”: his probability for becoming a movie star is high, not unconditionally, but conditionally on his getting out of this place. And so forth. Conditional probability is not just a technical notion such as ‘zero-sum game’ or ‘categorical imperative’ or ‘Turing comput- ability’. Rather, it is a familiar concept, like ‘game’, or ‘moral duty’, and I daresay more familiar than ‘computability’. It is there to be analyzed, if possible. (If it is not possible, then it follows immediately that the ratio does not provide an adequate analysis of it.) Our choice of words in reading P (A | B) as ’the probability of A, given B’ is not so innocent after all. And far from being a stipulative definition of that concept, I will argue that the ratio formula is not even an adequate analysis of it. 3

This paper suggests the use of the conditional probability integral transformation (CPIT) method as a goodness of fit (GOF) technique in the field of accelerated life testing (ALT), specifically for validating the underlying distributional assumption in accelerated failure time (AFT) models. The CPIT method is based on transforming the data into independent and identically distributed (i.i.d) Uniform (0, 1) random variables and then applying a certain GOF technique to test the uniformity of the transformed random variables. In this paper, the CPIT method is used to validate each of the exponential and lognormal distributions' assumptions in an AFT model under constant stress and complete sampling. The performance of this method is investigated via a simulation study. Moreover, a real life example is presented to illustrate the application of it. Concluding comments about the good performance of the CPIT method are made.

We elaborate on an alternative representation of conditional probability to the usual tree diagram. We term the representation “turtleback diagram” for its resemblance to the pattern on turtle shells. Adopting the set theoretic view of events and the sample space, the turtleback diagram uses elements from Venn diagrams—set intersection, complement and partition—for condition- ing, with the additional notion that the area of a set indicates probability whereas the ratio of areas for conditional probability. Once parts of the dia- gram are drawn and properly labeled, the calculation of conditional probabil- ity involves only simple arithmetic on the area of relevant sets. We discuss turtleback diagrams in relation to other visual representations of conditional probability, and detail several scenarios in which turtleback diagrams prove useful. By the equivalence of recursive space partition and the tree, the tur- tleback diagram is seen to be equally expressive as the tree diagram for ab- stract concepts. We also provide empirical data on the use of turtleback dia- grams with undergraduate students in elementary statistics or probability courses.

Efficient haplotyping in pedigrees is important for the fine mapping of quantitative trait locus (QTL) or complex disease genes. To reconstruct haplotypes efficiently for a large pedigree with a large number of linked loci, two algorithms based on conditional probabilities and likelihood computations are presented. The first algorithm (the conditional probability method) produces a single, approximately optimal haplo- type configuration, with computing time increasing linearly in the number of linked loci and the pedigree size. The other algorithm (the conditional enumeration method) identifies a set of haplotype configura- tions with high probabilities conditional on the observed genotype data for a pedigree. Its computing time increases less than exponentially with the size of a subset of the set of person-loci with unordered genotypes and linearly with its complement. The size of the subset is controlled by a threshold parameter. The set of identified haplotype configurations can be used to estimate the identity-by-descent (IBD) matrix at a map position for a pedigree. The algorithms have been tested on published and simulated data sets. The new haplotyping methods are much faster and provide more information than several existing stochastic and rule-based methods. The accuracies of the new methods are equivalent to or better than those of these existing methods.

_______________________________________________________ This study actually draws from and builds on an earlier paper (Kumar and Bhattacharya, 2002). Here we have basically added a neutrosophic dimension to the problem of determining the conditional probability that a financial fraud has been actually committed, given that no Type I error occurred while rejecting the null hypothesis H 0 : The observed first-digit frequencies approximate a Benford