Prior probability

In this expression, p(H) is referred to as the prior probability of the hypothesis H. It is used to express any information we may have about the probability that the hypothesis H is true before we consider the new data D. p(D H) is the likelihood of the data given that the hypothesis H is true. It describes our view of how the data arise from whatever H says about the state of nature, including uncertainties in measurement and any physical theory we might have that relates the data to the hypothesis. p(D) is the marginal distribution of the data D, and because it is a constant with respect to the parameters it is frequently considered only as a normalization factor in Eq. (2), so that p(H D) x p(D H)p(H) up to a proportionality constant. If we have a set of hypotheses that are exclusive and exliaus-tive, i.e., one and only one must be true, then... 

However, equation 2.(i-2 IS valid because A, B are commuting variables that lead to equation 2.6-3. Rearranging, results in one of the usual forms of the Bayes equation (equation 2.6-4). PiA E) is the prior probability of A given E. P(B A E is probability that is... 

Suppose that an explosion at a chemical plant could have occurred as a result of one of tliree mutually exclusive causes equipment nialfimction, carelessness, or sabotage. It is estimated tliat such an explosion could occur witli probability 0.20 as a result of equipment malfunction, 0.40 as a result of carelessness, and 0.75 as a result of sabotage. It is also estimated tliat tlie prior probabilities of the tliree possible causes of the explosion are, respectively, 0.50,0.35, and 0.15. Using Bayes tlieorem, deteniiine tlie most likely cause of the explosion. 

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. ... 

The key point is that the underdetermined system of linear equations is rendered soluble by an assumption of the prior probabilities of the unknown coefficients. It is important to realize that truncating the number of modes creates... 

Friedman [12] introduced a Bayesian approach the Bayes equation is given in Chapter 16. In the present context, a Bayesian approach can be described as finding a classification rule that minimizes the risk of misclassification, given the prior probabilities of belonging to a given class. These prior probabilities are estimated from the fraction of each class in the pooled sample ... 

Jaynes, E. (1968) Prior probabilities, In IEEE Transactions on Systems Science and Cybernetics, Vol. SSC-4, pp. 227-241. 

How to judge the relevance of these asphericities Are they really compatible with the data or are they simply the biased result of an ill-adapted model The best way to answer this question is to use this result as a prior probability for a new MaxEnt reconstruction. The map thus obtained, which is given in Figure 4, is striking the... 

Why did the posterior probability increase so much the second time One reason was that the prior probability was considerably higher in the second calculation than in the first (27% versus 2%), based on the fact that the first test yielded positive results. Another reason was that the specificity of the second test was quite high (98%), which markedly reduced the false-positive error rate and therefore increased the positive predictive value. 

The criterion D is a measure of divergence among the models, obtained from information theory. The quantity nt is the prior probability associated with model / after the nth observation is obtained o2 is the common variance of the n observations y(l), y( 2), , y(n — 1), y(n) a2 is the variance for the predicted value of y(n + 1) by model i. When we have two models, D simplifies to... 

An initial nine data points were taken at 35°C in an adiabatic flow reactor. The initial prior probabilities were taken to be equal (equal probability of... 

Two concepts of probabilities are important in classification. The groups to be distinguished have so-called prior probabilities (priors), the theoretical or natural probabilities (before classification) of objects belonging to one of the groups. After classification, we have posterior probabilities of the objects to belong to a group, which are in general different from the prior probabilities, and hopefully allow a clear... 

FIGURE 5.2 Visualization of the Bayesian decision mle in the univariate case, where the prior probabilities of the three groups are equal. The dashed lines are at the decision boundaries between groups 1 and 2 (x12) and between groups 2 and 3 ( 23). 

The right plot in Figure 5.3 shows a linear discrimination of three groups. Here all three groups have the same prior probability, but their covariance matrices are not equal (different shape and orientation). The resulting mle is no longer optimal in the sense defined above. An optimal rule, however, could be obtained by quadratic discriminant analysis which does not require equality of the group covariances. 

FIGURE 5.3 An optimal discriminant mle is obtained in the left picture, because the group covariances are equal and an adjustment is made for different prior probabilities. The linear mle shown in the right picture is not optimal—in terms of a minimum probability of misclassification—because of the different covariance matrices. 

FIGURE 5.4 Linear discriminant scores dj for group j by the Bayesian classification rule based on (Equation 5.2). mj, mean vector of all objects in group j Sp1, inverse of the pooled covariance matrix (Equation 5.3) x, object vector (to be classified) defined by m variables Pj, prior probability of group j. 

By rearranging Equation 5.2 for the Bayesian mle, it can be seen that in case of two groups the solution is exactly the same as for the Fisher mle if the prior probabilities are equal. However, since the prior probabilities p and p2 are not considered in the Fisher mle, the results will be different for the Bayesian mle if p / p2. 

One has to be careful with the use of the misclassification error as a performance measure. For example, assume a classification problem with two groups with prior probabilities pi = 0.9 and p2 = 0.1, where the available data also reflect the prior probabilities, i.e., nx k, npi and n2 np2. A stupid classification rule that assigns all the objects to the first (more frequent) group would have a misclassification error of about only 10%. Thus it can be more advisable to additionally report the misclassification rates per group, which in this case are 0% for the first group but 100% for the second group which clearly indicates that such a classifier is useless. 

This measure depends on the relative group sizes (estimated prior probabilities), and thus it is in general not suited to characterize the prediction performance of a classifier. 

Assuming uniform prior probabilities, we maximise S subject to these constraints. This is a standard variation problem solved by the use of Lagrangian multipliers. A numerical solution using standard variation methods gives i.p6j=. 05435, 0.07877, 0.11416, 0.16545, 0.23977, 0.34749 with an entropy of 1.61358 natural units. 

---