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
A probability box (or p-box) is a characterization of uncertain numbers consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be pe ...
es, and constrain
cumulative probability distributions (rather than
densities
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
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 theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. 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
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...
used the notion of interval bounds on probability in his ''
The Laws of Thought
''An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities'' by George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole was a professor of mathem ...
''.
Also dating from the latter half of the 19th century, the
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
attributed to
Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
described bounds on a distribution when only the mean and variance of the variable are known, and the related
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
attributed to
Markov Markov ( Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include:
Academics
*Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology at ...
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
John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946), was an English economist whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originally trained in m ...
.
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 Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. There ...
.
)
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
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
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 solved the problem, originally posed by
Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, 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., 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.
Arithmetic expressions
Arithmetic expressions involving operations such as additions, subtractions, multiplications, divisions, minima, maxima, powers, exponentials, logarithms, square roots, absolute values, etc., are commonly used in
risk analyses and uncertainty modeling. Convolution is the operation of finding the probability distribution of a sum of independent random variables specified by probability distributions. We can extend the term to finding distributions of other mathematical functions (products, differences, quotients, and more complex functions) and other assumptions about the intervariable dependencies. There are convenient algorithms for computing these generalized convolutions under a variety of assumptions about the dependencies among the inputs.
Mathematical details
Let
denote the space of distribution functions on the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s
i.e.,
:
A p-box is a quintuple
:
where
are real intervals, and
This quintuple denotes the set of distribution functions
such that:
:
If a function satisfies all the conditions above it is said to be ''inside'' the p-box. In some cases, there may be no information about the moments or distribution family other than what is encoded in the two distribution functions that constitute the edges of the p-box. Then the quintuple representing the p-box
can be denoted more compactly as
1, ''B''2">'B''1, ''B''2 This notation harkens to that of intervals on the real line, except that the endpoints are distributions rather than points.
The notation
denotes the fact that
is a random variable governed by the distribution function ''F'', that is,
:
Let us generalize the tilde notation for use with p-boxes. We will write ''X'' ~ ''B'' to mean that ''X'' is a random variable whose distribution function is unknown except that it is inside ''B''. Thus, ''X'' ~ ''F'' ∈ ''B'' can be contracted to X ~ B without mentioning the distribution function explicitly.
If ''X'' and ''Y'' are independent random variables with distributions ''F'' and ''G'' respectively, then ''X'' + ''Y'' = ''Z'' ~ ''H'' given by
:
This operation is called a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
on ''F'' and ''G''. The analogous operation on p-boxes is straightforward for sums. Suppose
:
If ''X'' and ''Y'' are stochastically independent, then the distribution of ''Z'' = ''X'' + ''Y'' is inside the p-box
:
Finding bounds on the distribution of sums ''Z'' = ''X'' + ''Y'' ''without making any assumption about the dependence'' between ''X'' and ''Y'' is actually easier than the problem assuming independence. Makarov
[ showed that
:]