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Probability bounds analysis (PBA) is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used to project partial information about random variables and other quantities through mathematical expressions. For instance, it computes sure bounds on the distribution of a sum, product, or more complex function, given only sure bounds on the distributions of the inputs. Such bounds are called probability boxes, and constrain cumulative probability distributions (rather than densities or mass functions). This bounding approach permits analysts to make calculations without requiring overly precise assumptions about parameter values, dependence among variables, or even distribution shape. Probability bounds analysis is essentially a combination of the methods of standard interval analysis and classical probability theory. Probability bounds analysis gives the same answer as interval analysis does when only range information is available. It also gives the same answers as Monte Carlo simulation does when information is abundant enough to precisely specify input distributions and their dependencies. Thus, it is a generalization of both interval analysis and probability theory. The diverse methods comprising probability bounds analysis provide algorithms to evaluate mathematical expressions when there is uncertainty about the input values, their dependencies, or even the form of mathematical expression itself. The calculations yield results that are guaranteed to enclose all possible distributions of the output variable if the input p-boxes were also sure to enclose their respective distributions. In some cases, a calculated p-box will also be best-possible in the sense that the bounds could be no tighter without excluding some of the possible distributions. P-boxes are usually merely bounds on possible distributions. The bounds often also enclose distributions that are not themselves possible. For instance, the set of probability distributions that could result from adding random values without the independence assumption from two (precise) distributions is generally a proper subset of all the distributions enclosed by the p-box computed for the sum. That is, there are distributions within the output p-box that could not arise under any dependence between the two input distributions. The output p-box will, however, always contain all distributions that are possible, so long as the input p-boxes were sure to enclose their respective underlying distributions. This property often suffices for use in risk analysis and other fields requiring calculations under uncertainty. ==History of bounding probability== The idea of bounding probability has a very long tradition throughout the history of probability theory. Indeed, in 1854 George Boole used the notion of interval bounds on probability in his The Laws of Thought.〔 〕 Also dating from the latter half of the 19th century, the inequality attributed to Chebyshev described bounds on a distribution when only the mean and variance of the variable are known, and the related inequality attributed to Markov found bounds on a positive variable when only the mean is known. Kyburg〔Kyburg, H.E., Jr. (1999). (Interval valued probabilities ). SIPTA Documention on Imprecise Probability.〕 reviewed the history of interval probabilities and traced the development of the critical ideas through the 20th century, including the important notion of incomparable probabilities favored by Keynes. Of particular note is Fréchet's derivation in the 1930s of bounds on calculations involving total probabilities without dependence assumptions. Bounding probabilities has continued to the present day (e.g., Walley's theory of imprecise probability.) The methods of probability bounds analysis that could be routinely used in risk assessments were developed in the 1980s. Hailperin〔 described a computational scheme for bounding logical calculations extending the ideas of Boole. Yager〔Yager, R.R. (1986). Arithmetic and other operations on Dempster–Shafer structures. ''International Journal of Man-machine Studies'' 25: 357–366.〕 described the elementary procedures by which bounds on convolutions can be computed under an assumption of independence. At about the same time, Makarov,〔Makarov, G.D. (1981). Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed. ''Theory of Probability and Its Applications'' 26: 803–806.〕 and independently, Rüschendorf〔Rüschendorf, L. (1982). Random variables with maximum sums. ''Advances in Applied Probability'' 14: 623–632.〕 solved the problem, originally posed by Kolmogorov, of how to find the upper and lower bounds for the probability distribution of a sum of random variables whose marginal distributions, but not their joint distribution, are known. Frank et al.〔Frank, M.J., R.B. Nelsen and B. Schweizer (1987). Best-possible bounds for the distribution of a sum—a problem of Kolmogorov. ''Probability Theory and Related Fields'' 74: 199–211.〕 generalized the result of Makarov and expressed it in terms of copulas. Since that time, formulas and algorithms for sums have been generalized and extended to differences, products, quotients and other binary and unary functions under various dependence assumptions.〔Williamson, R.C., and T. Downs (1990). Probabilistic arithmetic I: Numerical methods for calculating convolutions and dependency bounds. ''International Journal of Approximate Reasoning'' 4: 89–158.〕〔Ferson, S., V. Kreinovich, L. Ginzburg, D.S. Myers, and K. Sentz. (2003). (''Constructing Probability Boxes and Dempster–Shafer Structures'' ). SAND2002-4015. Sandia National Laboratories, Albuquerque, NM.〕〔Berleant, D. (1993). Automatically verified reasoning with both intervals and probability density functions. ''Interval Computations'' 1993 (2) : 48–70.〕〔Berleant, D., G. Anderson, and C. Goodman-Strauss (2008). Arithmetic on bounded families of distributions: a DEnv algorithm tutorial. Pages 183–210 in ''Knowledge Processing with Interval and Soft Computing'', edited by C. Hu, R.B. Kearfott, A. de Korvin and V. Kreinovich, Springer (ISBN 978-1-84800-325-5).〕〔Berleant, D., and C. Goodman-Strauss (1998). Bounding the results of arithmetic operations on random variables of unknown dependency using intervals. ''Reliable Computing'' 4: 147–165.〕〔Ferson, S., R. Nelsen, J. Hajagos, D. Berleant, J. Zhang, W.T. Tucker, L. Ginzburg and W.L. Oberkampf (2004). (''Dependence in Probabilistic Modeling, Dempster–Shafer Theory, and Probability Bounds Analysis'' ). Sandia National Laboratories, SAND2004-3072, Albuquerque, NM.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Probability bounds analysis」の詳細全文を読む スポンサード リンク
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