Probability bounds analysis: Difference between revisions

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{{Short description|Mathematical method of risk analysis}}

{{Use dmy dates|date=AprilOctober 20132022}}

'''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 box]]es, and constrain [[cumulative distribution function|cumulative probability distributions]] (rather than [[probability density function|densities]] or [[probability mass function|mass functions]]).

This [[upper and lower bounds|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 [[probability box|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.

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 [[Probabilistic risk assessment|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]]''.<ref name="BOOLE1854">{{cite book|url= https://www.gutenberg.org/ebooks/15114 |last=Boole |first=George |title=An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities |publisher=Walton and Maberly |year=1854 |___location=London}}</ref><ref name=Hailperin86>{{cite book |last=Hailperin |first=Theodore |title=Boole's Logic and Probability |publisher=North-Holland |year=1986 |___location=Amsterdam |isbn=978-0-444-11037-4 }}</ref> Also dating from the latter half of the 19th century, the [[Chebyshev inequality|inequality]] attributed to [[Chebyshev]] described bounds on a distribution when only the mean and variance of the variable are known, and the related [[Markov inequality|inequality]] attributed to [[Andrey Markov|Markov]] found bounds on a positive variable when only the mean is known. [[Henry E. Kyburg, Jr.|Kyburg]]<ref name="kyburg99">Kyburg, H.E., Jr. (1999). [https://sipta.org/documentation/interval_prob/kyburgnew.pdf Interval valued probabilities]. SIPTA Documentation on Imprecise Probability.</ref> 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 [[John Maynard Keynes|Keynes]].

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]].<ref name="BOOLE1854">{{cite book

Of particular note is [[Maurice René Fréchet|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]].<ref name="WALLEY1991">{{cite book|url= https://archive.org/details/statisticalreaso0000wall |last=Walley |first=Peter |title=Statistical Reasoning with Imprecise Probabilities |url-access=registration |publisher=Chapman and Hall |year=1991 |___location=London |isbn=978-0-412-28660-5 }}</ref>)

| last = Boole

| first = George

| title = An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities

| publisher = Walton and Maberly

| year = 1854

| ___location = London

| url = http://www.gutenberg.org/etext/15114

}}</ref><ref name=Hailperin86>{{cite book

| last = Hailperin

| first = Theodore

| title = Boole's Logic and Probability

| publisher = North-Holland

| year = 1986

| ___location = Amsterdam

| isbn = 0-444-11037-2 }}

</ref> Also dating from the latter half of the 19th century, the [[Chebyshev inequality|inequality]] attributed to [[Chebyshev]] described bounds on a distribution when only the mean and

variance of the variable are known, and the related [[Markov inequality|inequality]] attributed to [[Andrey Markov|Markov]] found bounds on a

positive variable when only the mean is known.

[[Henry E. Kyburg, Jr.|Kyburg]]<ref name="kyburg99">Kyburg, H.E., Jr. (1999). [http://www.sipta.org/documentation/interval_prob/kyburg.pdf Interval valued probabilities]. SIPTA Documention on Imprecise Probability.</ref> 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 [[John Maynard Keynes|Keynes]].

Of particular note is [[Maurice René Fréchet|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]].<ref name="WALLEY1991">{{cite book

| last = Walley

| first = Peter

| title = Statistical Reasoning with Imprecise Probabilities

| publisher = Chapman and Hall

| year = 1991

| ___location = London

| isbn = 0-412-28660-2 }}</ref>)

The methods of probability bounds analysis that could be routinely used in

risk assessments were developed in the 1980s. Hailperin<ref name=Hailperin86 /> described a computational scheme for bounding logical calculations extending the ideas of Boole. Yager<ref name=Yager>Yager, R.R. (1986). Arithmetic and other operations on Dempster–Shafer structures. ''International Journal of Man-machine Studies'' '''25''': 357–366.</ref> described the elementary procedures by which bounds on [[convolution of probability distributions|convolutions]] can be computed under an assumption of independence. At about the same time, Makarov,<ref name=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.</ref> and independently, Rüschendorf<ref>Rüschendorf, L. (1982). Random variables with maximum sums. ''Advances in Applied Probability'' '''14''': 623–632.</ref> 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.<ref name=Franketal87>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.</ref> generalized the result of Makarov and expressed it in terms of [[Copula (probability theory)|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.<ref name=WilliamsonDowns>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.</ref><ref name=Fersonetal03>Ferson, S., V. Kreinovich, L. Ginzburg, D.S. Myers, and K. Sentz. (2003). [http://www.ramas.com/unabridged.zip ''Constructing Probability Boxes and Dempster–Shafer Structures''] {{webarchive|url=https://web.archive.org/web/20110722073459/http://www.ramas.com/unabridged.zip |date=22 July 2011 }}. SAND2002-4015. Sandia National Laboratories, Albuquerque, NM.</ref><ref>Berleant, D. (1993). Automatically verified reasoning with both intervals and probability density functions. ''Interval Computations'' '''1993 (2) ''': 48–70.</ref><ref>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}}).</ref><ref name=BerleantGoodmanStrauss>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.</ref><ref name=Fersonetal04>Ferson, S., R. Nelsen, J. Hajagos, D. Berleant, J. Zhang, W.T. Tucker, L. Ginzburg and W.L. Oberkampf (2004). [http://www.ramas.com/depend.pdf ''Dependence in Probabilistic Modeling, Dempster–Shafer Theory, and Probability Bounds Analysis'']. Sandia National Laboratories, SAND2004-3072, Albuquerque, NM.</ref> <!--

It is possible to mix very different kinds of knowledge together in a bounding analysis. For instance,

In some cases, we may not know whether a quantity varies or is a fixed constant. Even if we know a quantity to be a constant, we may not know its value precisely. And, even if we know a quantity to be randomly varying, we may not know the statistical distribution that governs that variation, or the stochastic dependence it may have with other quantities.

In some cases, the shape or family of the distribution of a quantity may be known from mechanistic or physics-based arguments, but its parameters may be in doubt. In others cases, some summary statistical characteristics of a quantity may have been recorded in the scientific literature, but other details and the original data are unavailable so that we do not know the family of the statistical distribution even though we know some of its parameters. In some cases, there may be sample data available but the small sample may be size, or the data values may have non-negligible measurement uncertainty.

Further suppose that sparse data were used to form the 95% confidence limits for the distribution of ''C''. And the variable ''D'' is known to be well described by a precise distribution.

Probability bounds analysis includes the important special case of [[dependency bounds analysis]]<<__Williamson and Downs>> to compute bounds on the cumulative distribution of a function of random variables when only the marginal distributions of the variables are known, which is a problem originally posed by [[Kolmogorov]].

-->

==Arithmetic expressions==

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:<math>\left \{ \overline{F}, \underline{F}, m, v, \mathbf{F} \right \},</math>

where <math>\overline{F}, </math> and <math>\underline{F} \in \mathbb{D}</math>, <math>m,</math> and <math>v</math> are real intervals, and <math>\mathbf{F} \subset \mathbb{D}.</math>. This quintuple denotes the set of distribution functions <math>F \in \mathbf{F} \subset \mathbb{D}</math> such that:

:<math>\begin{align}

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:::: = [0.2, 0.55].

It is also possible to compute interval bounds on the conjunction or disjunction under other assumptions about the dependence between A and B. For instance, one might assume they are positively dependent, in which case the resulting interval is not as tight as the answer assuming independence but tighter than the answer given by the Fréchet inequality. Comparable calculations are used for other logical functions such as negation, exclusive disjunction, etc. When the Boolean expression to be evaluated becomes complex, it may be necessary to evaluate it using the methods of mathematical programming<ref name=Hailperin86 /> to get best-possible bounds on the expression. A similar problem one presents in the case of [[probabilistic logic]] (see for example Gerla 1994). If the probabilities of the events are characterized by probability distributions or p-boxes rather than intervals, then analogous calculations can be done to obtain distributional or p-box results characterizing the probability of the top event. <!--

Prob(A and B) = Prob(A) * Prob(B).

Prob(A or B) = Prob(A) + Prob(B) – Prob(A) * Prob(B)

Operation Formula

conjunction [ max(0, a+b–1), min(a, b) ],

disjunction [ max(a, b), min(1, a+b) ],

a = [0.2, 0.25]

b = [0.1, 0.3]

a |&| b

[ 0.02, 0.075]

a & b

[ 0, 0.25]

a ||| b

[ 0.28, 0.475]

a | b

[ 0.2, 0.55]

-->

==Magnitude comparisons==

The probability that an uncertain number represented by a p-box ''D'' is less than zero is the interval Pr(''D'' < 0) = [<u>''F''</u>''(0), ''F̅''(0)], where ''F̅''(0) is the left bound of the probability box ''D'' and <u>''F''</u>(0) is its right bound, both evaluated at zero. Two uncertain numbers represented by probability boxes may then be compared for numerical magnitude with the following encodings:

:''A'' < ''B'' = Pr(''A'' − ''B'' < 0),

:''A'' > ''B'' = Pr(''B'' − ''A'' < 0),

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==Further references==

* {{cite book | last1 = Bernardini | first1 = Alberto | last2 = Tonon | first2 = Fulvio | title = Bounding Uncertainty in Civil Engineering: Theoretical Background | publisher = Springer | ___location = Berlin | year = 2010 | isbn = 978-3-642-11189-01 }}

* {{cite book | last = Ferson | first = Scott | title = RAMAS Risk Calc 4.0 Software : Risk Assessment with Uncertain Numbers | publisher = Lewis Publishers | ___location = Boca Raton, Florida | year = 2002 | isbn = 978-1-56670-576-29 }}

* {{cite journal |first=G. |last=Gerla |title=Inferences in Probability Logic |journal=Artificial Intelligence |volume=70 |issue=1–2 |pages=33–52 |year=1994 |doi=10.1016/0004-3702(94)90102-3 }}

* {{cite book | last1 = Oberkampf | first1 = William L. | last2 = Roy | first2 = Christopher J. | title = Verification and Validation in Scientific Computing | publisher = Cambridge University Press | ___location = New York | year = 2010 | isbn = 978-0-521-11360-1 }}<!-- In an email dated 28 March 2011, William Oberkampf stated "PBA is the only UQ method we discuss and apply in our examples in the book." -->

==External links==

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* [http://www.sipta.org/ The Society for Imprecise Probability: Theories and Applications]

[[Category:Probability bounds analysis| ]]

[[Category:Mathematical analysis]]