Conditional probability/Bayes’ theorem

In Bayesian probability, Bayes theorem is an important tool used to incorporate information to refine the probability assigned to a hypothesis. Bayes theorem can be stated as a relationship between conditional probabilities ... 

After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

Three algorithms have been implemented in both single and multiperspective environments. In this way any bias introduced by a single algorithm should be removed. The first is the statistical Naive Bayesian Classifier, ft reduces the decision-making problem to simple calculations of feature probabilities, ft is based on Bayes theorem and calculates the posterior probability of classes conditioned on the given unknown feature... 

It is usefid to know the sensitivity and specificity of a test. Once a researcher decides to use a certain test, two important questions require answers If the test results are positive, what is the probability that the researcher has the condition of interest If the test results are negative, what is the probability that the patient does not have the disease Bayes theorem provides a way to answer these questions. 

Suppose that a woman has a positive mammogram. What is the probability that she in fact has breast cancer To solve this problem statisticians use Bayes Theorem, a theorem in conditional probability introduced by the non-conformist minister the Reverend Thomas Bayes in 1763. Gigerenzer explains how Bayes theorem works by converting the problem into natural frequencies. ... 

Equation (5.10) is a statement of Bayes theorem. Since the theorem is proved using results or axioms valid for both frequentist and Bayesian views, its use is not limited to Bayesian applications. Note that it relates 2 conditional probabilities where the events A and B are interchanged. 

Probabilistic methods. Other QSAR-like probabilistic approaches have also been developed for compound database mining. Binary QSAR (BQ) is discussed here as an example (Labute 1999). BQ is based on Bayes theorem of conditional probabilities ... 

Bayes theorem. For the same settings, the Bayes theorem gives the conditional probability... 

For model Mj. let p(Y Mj. a) denote the predictive probability density of prospective data Y predicted in the manner of Eq. (6.1-10). with a known. Bayes theorem then gives the posterior probability of model Mj conditional on given Y and cr as... 

For a given experimental design and expectation model, let p Y Mj, S) denote the probability density of prospective data in Y-space, predicted via Eq. (7.1-2). According to Bayes theorem, the posterior probability of Model Mj conditional on given arrays Y and S is then... 

P(m/d) is a conditional density of probability for a model m, given the data d. According to the Bayes theorem, the following equation holds ... 

Population models describe the relationship between individuals and a population. Individual parameter sets are considered to arise from a joint population distribution described by a set of means and variances. The conditional dependencies among individual data sets, individual variables, and population variables can be represented by a graphical model, which can then be translated into the probability distributions in Bayes theorem. For most cases of practical interest, the posterior distribution is obtained via numerical simulation. It is also the case that the complexity of the posterior distribution for most PBPK models is such that standard MC sampling is inadequate leading instead to the use of Markov Chain Monte Carlo (MCMC) methods... 

Unfortunately, to determine these conditional probability values, i.e. confirm that a particular group is characterized by a specific set of variate values, involves the analysis of all potential samples in the parent population. This is obviously unrealistic in practice, and it is necessary to apply Bayes theorem which provides an indirect means of estimating the conditional probability,... 

One important use of conditional probabilities occurs in Bayes theorem. The conditional probability of an event A given an event is ... 

We introduced you to Bayes theorem in Chapter 6. According to this theorem, the conditional probability of A given B can be written as ... 

Bayes theorem which provides an indirect means of estimating the conditional probability, /"(G(A). v)-... 

Bayes theorem. A theorem due to Thomas Bayes (1701-1761) relating conditional probabilities of the sort P(X 7), where this is the probability X given Y, and which can be expressed in many forms. A simple form is... 

A sample space is generally defined and all probabilities are calculated with respect to that sample space. In many cases, however, we ate in a position to update the sample space based on new information. For example, like the fourth example of Example 2.3, if we just consider the case that two outcomes from roUing a die twice are the same, the size of the sample space is reduced from 36 to 6. General definitions of conditional probability and independence are introduced. The Bayes theorem is also introduced, which is the basis of a statistical methodology called Bayesian statistics. 

Inference on Bayesian Network Once we have constructed a Bayesian network, we can infer the information of each variable from those of other variables based on the network structure S and corresponding conditional probability distributions P. Then, the Bayes theorem is needed for our subsequent derivations in Example 11.5. 

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. 

Keywords Bayes theorem conditional probability information entropy Kalman Filter Markov Chain Monte Carlo simulation model identifiability particulate matter regression problem reliability structural health monitoring... 

An imperfect dice was drawn independently for N times whereas 1 appeared N times. The aim here is to update the probability of the occurrence of 1 in a single draw (denoted as Pi) and this can be achieved by the Bayes theorem. The conditional PDF of the uncertain parameter Pi given the value of N is ... 

This behavior can be explained as follows. Consider three random variables x, y and z and assume that one is interested in the conditional probability density p(x y, z) with p(z y) > 0 for the prediction of x by y and z. By using the Bayes theorem, the following relationship can be obtained ... 

Let T> denote the input-output or output-only data from a physical system or phenomenon. The goal is to use T> to select the most plausible/suitable class of models representing the system out of Nc given classes of models Ci, C2,..., Cjvc- Since probability may be interpreted as a measure of plausibility based on specified information [63], the probability of a class of models conditional on the set of dynamic data T> is required. This can be obtained by using the Bayes theorem as follows ... 

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