Bayess Theorem

The Bayes rule simply states that a sample or object should be assigned to that group having the highest conditional probability and application of this rule to parametric classification schemes provides optimum discriminating capability. An explanation of the term conditional probability is perhaps in order here. 

Similarly, the probability of tossing two coins resulting in both showing heads is given by 

however, one coin is already showing heads, then the conditional probability of spinning the other coin and both showing heads is now 

Which is to be read as the probability of both coins being heads given that one coin is heads is 0.5 , i.e. the probability of an event is modified, for better or worse, by prior knowledge. 

Returning to our analytical problem, of the 21 samples analysed and listed in Table 1, over 50% (11 of the 21) are known to belong to Group A. Thus, in the absence of any analytical data it would seem reasonable to assign any unknown 

Bayes theorem provides a method to calculate the probability of a disease after the addition of new information to previously obtained information. These formulas can be incorporated into computer spreadsheets and other computer aids to help in the estimation of updated probabilities. One method, which can be performed without a computer, involves using the odds ratio version of Bayes theorem. The odds ratio of the occurrence of a disease is calculated before the test result is known this information is then combined with the results of the test in the form of a likelihood ratio. The final result is again in the form of an odds ratio this can be converted into a probability, if desired. The advantages of this method are that it is relatively easily memorized and that it requires little mathematical calculation. The odds ratio is useful when it is combined with the likelihood ratio in a more memorable form of Bayes theorem  

Odds ratio after = Odds ratio before x Likelihood ratio 

Step 1. Calculate the odds ratio that carcinoma is present before performing the ultrasound. Given a PSA of 4.0 to lO.Ojig/L, there is an estimated probability of 12% in a BPH population with biopsy-verifiable disease thus the probability of no disease is (1 0.12) = 0.88. The 

Step 2. Calculate the likelihood ratio of the new information (findings of the transrectal ultrasound). Screening studies on urology patients report sensitivities of approximately 92% and specificities that range from 30% to 70% (average 50%) for transrectal ultrasound. The likelihood ratio positive is the sensitivity divided by (1 - specificity), or 0.92/0.50 - 1.8. 

Step 3. Calculate the odds ratio after incorporation of the new information. The revised odds ratio estimate (the product of Steps 1 and 2) is 0.25. 

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  

Here H is a. hypothesis and D is some data, and so P(H I D) is the probability of the hypothesis given the data. Bayes theorem relates this to the probability of the hypothesis before the data became available, P(H), the probability of the data given the hypothesis, P(D H), and finally the probability of the data P D). The term P H) is usually called the prior probability, or simply the prior, as it is obviously the probability that is assigned before the data is available. As the probability P D) is independent of the hypothesis, knowledge of this is not needed when comparing the relative probability of different hypotheses. In this sense, it is a normalization term that is often not important to an argument. 

Expressed mathematically, therefore, and applying Bayes rule, a sample is assigned to Group A, G(A), on the condition that 

Unfortunately, to determine these conditional probability values, 

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J Cornfield. In DL Meyer, RO Collier, eds. The Frequency Theory of Probability, Bayes Theorem, and Sequential Clinical Trials. Bloomington, In Phi Delta Kappa, 1970, pp 1-28. 

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... 

Consider n mutually exclusive events A, A2,. .., A whose union is tlie sample space S. Let B be any given event. Then Bayes theorem states... 

As a example of the application of Bayes theorem, suppose tliat 50% of a company s manufactured output comes from a New York plant, 30% from a Permsylvania plant, and 20% from a Delaware plant. On die basis of plant records it is estimated diat defective items constitute 1% of the output of the New York plant, 3% of the Pennsylvania plant, and 4% of die Delaware plant. If an item selected at random from die company s manufactured output is found to be defective, what are die revised probabilities diat die item was produced, by each of die diree plants ... 

For anotlier example of the use of Bayes theorem, suppose that the probability is 0.80 tliat an airplane crash due to structural failure is diagnosed correctly. Suppose, in addition, tliat tlie probability is 0.30 tliat an airplane crash not due to structural failure is incorrectly attributed to structural failure. If 35% of all airplane crashes are due to structural failure, wliat is tlie probability that an airplane crash was due to structural failure, given tliat it lias been so diagnosed Let Ai be tlie event tliat structural failure is tlie cause of tlie airplane crash. Let A2 be tlie event tliat tlie cause is otlier tlian structural failure. Let B be tlie event tliat tlie airplane crash is diagnosed as being due to structural failure. Tlien... 

Bayes theorem provides tlie mechanism for revising prior probabilities, i.e., for converting tliem into posterior probabilities on tlie basis of the observed occurrence of some given event. ... 

Bayes theorem provides the mechaiiism for converting tlie prior pdf of Z to the posterior pdf of Z on tlie basis of tlie occurrence of event B (i.e., no failures in 10 years). Applying Bayes tlieoreiii and denoting tlie posterior pdf of Z by f(z B) yields... 

If a well proves productive, the ensuing completion operation may require an area in excess of the drilling area. This may mean allocations for frac tank placement, blenders, pump trucks, bulk trucks and nitrogen trucks. In today s economic climate, the operator should weigh the probability of success, Bayes theorem (Equation 4-373), with the cost of constructing and reclaiming an additional area needed for stimulation (Equation 4-374). Plans such as these... 

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... 

The general problem is then to estimate 0 and u knowing the values of the measurements, y, and the probability distribution function of e (measurement error). If P(e) is the error distribution, then y will be distributed according to P y - x 9, u). Thus, according to Bayes theorem, (Alburquerque and Biegler, 1996), the posterior... 

According to Bayes theorem, the probability of the states, given the data, will be... 

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. 

Bayes theorem, which was first described centuries ago by the English clergyman after whom it is named, is one of the most imposing statistical formulas in the biomedical sciences (Lindley, 1971). Put in symbols more meaningftd for researchers such as pathologists, the formula is... 

Most researchers, even those who can deal with sensitivity, specificity, and predictive values, throw in the towel when it comes to Bayes theorem. This is odd, because a close look at the equation reveals that Bayes theorem is merely the formula for the positive predictive value (Box and Tiao, 1973). 

The numerator of Bayes theorem merely describes cell a (the tme-positive results). The probability of being in cell a is equal to the prevalence times the sensitivity, where XD+) is the prevalence (the probability of being in the effected column) and where XT + D+) is the sensitivity (the probability of being in the top row, given the fact of being in the effected column). The denominator of Bayes theorem consists of two terms, the first of which once again describes cell a (the true-positive results) and the second of which describes cell b (the false-positive error rate, or X I + D—), is multiplied by the prevalence of noneffected animals, or... 

In genetics, an even simpler-appearing formula for Bayes theorem is sometimes used. The numerator is the same, but the denominator is merely p 1+). This makes sense because the denominator in a/(a + b) is equal to all of those who have positive test results, whether they are true-positive or false-positive results. 

Bayes Theorem and Evaluation of Safety Assessment Studies... 

In a population with a low prevalence of a particular toxicity, most of the positive results in a screening program for that lesion or effect would be falsely positive. Although this does not automatically invalidate a study or assessment program, it raises some concerns about cost-effectiveness, and these can be explored using Bayes theorem (Racine et al., 1986)... 

A program employing a immunochemical stain based test to screen tissues for a specific effect will be discussed as an example. This test uses small amounts of antibody tissues for a specific effect, and the presence of an immunologically bound stain is considered a positive result. If the sensitivity and specificity of the test and the prevalence of biochemical effect are known, Bayes theorem can be used to predict what proportion of the tissues with positive test results will have true-positive results (actually be effected). 

EXAMPLE 22.3. Use of Bayes Theorem or a 2x2 Table to Determine the Positive Predictive Value of a Hypothetical Tuberculin Screening Program... 

Under the circumstances, Bayes theorem could be used to make a second estimate of probability, which is called the posterior probability, reflecting the fact... 

In light of the 27% posterior probability, the pathologist decides to order a parathyroid hormone radioimmunoassay, even though this test is expensive. If the radioimmunoassay had a sensitivity of 95% and a specificity of 98% and the results turned out to be positive, the Bayes theorem could again be used to calculate the... 

Maximum entropy (ME) is a tool of Bayesian Statistics, and thus is built around Bayes Theorem. Since, as diffractionists, we are interested in maps and particularly in obtaining an optimum map from measured data, we can state this theorem in the following way... 

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. ... 

While the use of Bayes Theorem in this context is not generally controversial its use more generally in medical and clinical research has not always been positively received." It is not the scope of the present chapter to illustrate the use of Bayesian statistics in a more general context and interested readers should read the excellent introduction to the use of Bayesian methods in health-care evaluation provided by Spiegelhalter et alP... 

In recent years, a method of causality assessment based on Bayes theorem has been developed by a number of workers in the United States. " Its application to the evaluation... 

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