Statistical decision theory

Berger JO. 1985. Statistical decision theory and Bayesian analysis. New York Springer. 

J.O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd edn. (Springer-Verlag, New York, 1985)... 

Raiffa, H. A. and Schlaifer, R. S. (1961). Applied Statistical Decision Theory. MIT Press, Cambridge. 

Fritsch, K. and Hsu, J. C. (1997). On analysis of means. In Advances in Statistical Decision Theory and Methodology. Editors N. Balakrishnan and S. Panchapakesan. Birkhauser, Boston, 114-119. 

Stefansson, G., Kim, W., and Hsu, J. C. (1988). On confidence sets in multiple comparisons. In Statistical Decision Theory and Related Topics IV, volume 2. Editors S. S. Gupta and J. O. Berger, pages 89-104. Springer-Verlag, New York. 

The recognition accuracy estimation described above faces one very important problem what is the best choice for the threshold value 0 To solve this problem, statistical decision theory is used. ° The basis for this is an analysis of the so-called the Received Operating Characteristic (ROC) curve. By tradition, ROC is plotted as a function of true positive rate TPj TP + FN) (or sensitivity) versus false positive rate FPj TN+FP) (or 1-Specificity) for all possible threshold values 0. Figure 6.5 presents an example of such a ROC curve for the results obtained with our computer program PASS in predicting antineoplastic activity. 

To obtain the qualitative ( Yes/No ) results of prediction, it is necessary to define the threshold Bk values for each kind of activity Ak- On the basis of statistical decision theory (Section 6.3.4) it is possible using the risk functions minimization, but nobody can a priori determine such functions for all kinds of activity and for all possible real-world problems. Therefore the predicted activity spectrum is presented in PASS by the list of activities with probabilities to be active Pa and to be inactive Pi calculated for each activity. The list is arranged in descending order of Pa—Pi, thus, the more probable activities are at the top of the list. The list can be shortened at any desirable cutoff value, but Pa>Pi is used by default. If the user chooses a rather high value of Pa as a cutoff for selection of probable activities, the chance to confirm the predicted activities by the experiment is high too, but many activities will be lost. For instance, if Pq>80% is used as a threshold, about 80% of real activities will be lost for Pq>70%, the portion of lost activities is 70%, etc. 

Condensation of information is obtained through formal calculations of central values and dispersions without prejudice as to the type of distribution. Interpretation of the meaning of the information obtained is another matter. Here knowledge of statistical decision theory is helpful.Statistical tools should be employed as aids to common sense. 

In conclusion, there is much room for improvement - for instance, in the application of statistical decision theory, improved risk assessment methods and generally better data. 

Frank, I. E. Pungor, E. Veress, G. E. "Statistical Decision Theory Applied to Analytical Chemistry" Anal. Chim. Acta 133 (1981) 433. 

Engineering-Statistical methods. 2. Structural engineering-Mathematics. 3. Bayesian statistical decision theory. I. Tide. 

Berger, J. O. 1980. Statistical decision theory foundations, concepts and methods. NY Springer-Verlag. 

The theory supporting Bayesian Belief networks rests on a rich tradition of probability theory, and statistical decision theory and it is supported by excellent axiomatic and behavioural arguments [Pearl, 1998]. A Bayesian belief network for a set of variables X = Xj, X2,. .., X consists of a) a directed network structure that encodes a set of conditional independence assertions about variables in X and b) a set P of local probability distributions associated with each variable, describing the distribution of the variable conditioned on its parent variables. The nodes in the network structure are in one-to-one correspondence with the variables in the probabilistic model. 

The loss-function is the cost for estimating with estimator 0 when the true parameter value is 9. The posterior mean is the Bayesian estimator that minimizes the squared-error loss-function, while the posterior median is the Bayesian estimator that minimizes the absolute value loss-function. One of the strengths of Bayesian statistics, is that we could decide on any particular loss function, and find the estimator that minimizes it. This is covered in the field of statistical decision theory. We will not pursue this topic further in this book. Readers are referred to Berger (1980) and DeGroot (1970). 

Bayesian statistical decision theory—Data processing. I. Title. 

Berger, J.O. Statistical Decision Theory and Bayesian Analysis. Springer, New York (1985) Bischof, C., Carle, A., Khademi, P. Mauer the ADIFOR 2.0 system for the automatic differentiation of FORTRAN 77 programes. IEEE J. Comput. Sci. Eng. 3, 18-32 (1996)... 

The development of tools to ensure the use of such objective behef systems, and computational methods such as Monte Carlo simulation have brought the Bayesian approach to the fore in recent years in areas such as parameter estimation, statistical learning, and statistical decision theory. Here, we focus our attention primarily upon parameter estimation, first restricting our discussion to single-response data. 

---