Probability bounds analysis

<h2 id="history-of-bounding-probability">History of bounding probability</h2>
The idea of bounding probability has a very long tradition throughout the history of probability theory. Indeed, in 1854 <a href="/facts/George_Boole/fWJT2rek">George Boole</a> used the notion of interval bounds on probability in his <a href="/facts/The_Laws_of_Thought/kdkVw7Vj">The Laws of Thought</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Also dating from the latter half of the 19th century, the <a href="/facts/Chebyshev_inequality/jf5ReDJv">inequality</a> attributed to <a href="/facts/Chebyshev/LzRqLqRm">Chebyshev</a> described bounds on a distribution when only the mean and variance of the variable are known, and the related <a href="/facts/Markov_inequality/Re85kMQJ">inequality</a> attributed to <a href="/facts/Andrey_Markov/F409PDqD">Markov</a> found bounds on a positive variable when only the mean is known. <a href="/facts/Henry_E._Kyburg,_Jr./Go4P2K13">Kyburg</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> 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 <a href="/facts/John_Maynard_Keynes/4otHjBRu">Keynes</a>.
Of particular note is <a href="/facts/Maurice_Ren%C3%A9_Fr%C3%A9chet/QXrzxA1s">Fréchet</a>'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 <a href="/facts/Imprecise_probability/unnG18zY">imprecise probability</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>)
The methods of probability bounds analysis that could be routinely used in
risk assessments were developed in the 1980s. Hailperin<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> described a computational scheme for bounding logical calculations extending the ideas of Boole. Yager<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> described the elementary procedures by which bounds on <a href="/facts/Convolution_of_probability_distributions/l467HgVK">convolutions</a> can be computed under an assumption of independence. At about the same time, Makarov,<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> and independently, Rüschendorf<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> solved the problem, originally posed by <a href="/facts/Kolmogorov/bEKwtrs6">Kolmogorov</a>, 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.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> generalized the result of Makarov and expressed it in terms of <a href="/facts/Copula_(probability_theory)/9tgrLvLF">copulas</a>. 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.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h2 id="arithmetic-expressions">Arithmetic expressions</h2>
Arithmetic expressions involving operations such as additions, subtractions, multiplications, divisions, minima, maxima, powers, exponentials, logarithms, square roots, absolute values, etc., are commonly used in <a href="/facts/Probabilistic_risk_assessment/VIkIGuRp">risk analyses</a> 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.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h3>Mathematical details</h3>
Let 
 
 
 
 
 D
 
 
 
 {\displaystyle \mathbb {D} }
 
 denote the space of distribution functions on the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> 
 
 
 
 
 R
 
 ,
 
 
 {\displaystyle \mathbb {R} ,}
 
 i.e.,

D
 
 =
 {
 D
 
 |
 
 D
 :
 
 R
 
 →
 [
 0
 ,
 1
 ]
 ,
 D
 (
 x
 )
 ≤
 D
 (
 y
 )
 
  for all 
 
 x
 <
 y
 }
 .
 
 
 {\displaystyle \mathbb {D} =\{D|D:\mathbb {R} \to [0,1],D(x)\leq D(y){\text{ for all }}x<y\}.}

A p-box is a quintuple

{
          
            
              
                F
                ¯
              
            
            ,
            
              
                F
                _
              
            
            ,
            m
            ,
            v
            ,
            
              F
            
          
          }
        
        ,
      
    
    {\displaystyle \left\{{\overline {F}},{\underline {F}},m,v,\mathbf {F} \right\},}

where 
 
 
 
 
 
 F
 ¯
 
 
 
 
 {\displaystyle {\overline {F}}}
 
 and 
 
 
 
 
 
 F
 _
 
 
 ∈
 
 D
 
 
 
 {\displaystyle {\underline {F}}\in \mathbb {D} }
 
, 
 
 
 
 m
 
 
 {\displaystyle m}
 
 and 
 
 
 
 v
 
 
 {\displaystyle v}
 
 are real intervals, and 
 
 
 
 
 F
 
 ⊂
 
 D
 
 
 
 {\displaystyle \mathbf {F} \subset \mathbb {D} }
 
. This quintuple denotes the set of distribution functions 
 
 
 
 F
 ∈
 
 F
 
 ⊂
 
 D
 
 
 
 {\displaystyle F\in \mathbf {F} \subset \mathbb {D} }
 
 such that:

∀
                x
                ∈
                
                  R
                
                :
                
              
              
                
                  
                    F
                    ¯
                  
                
                (
                x
                )
                ≤
                F
                (
                x
                )
                ≤
                
                  
                    F
                    _
                  
                
                (
                x
                )
              
            
            
              
              
                
                
                  ∫
                  
                    
                      R
                    
                  
                
                x
                d
                F
                (
                x
                )
                ∈
                m
              
              
              
                
                  expectation condition
                
              
            
            
              
              
                
                
                  ∫
                  
                    
                      R
                    
                  
                
                
                  x
                  
                    2
                  
                
                d
                F
                (
                x
                )
                −
                
                  
                    (
                    
                      
                        ∫
                        
                          
                            R
                          
                        
                      
                      x
                      d
                      F
                      (
                      x
                      )
                    
                    )
                  
                  
                    2
                  
                
                ∈
                v
              
              
              
                
                  variance condition
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\forall x\in \mathbb {R} :\qquad &{\overline {F}}(x)\leq F(x)\leq {\underline {F}}(x)\\[6pt]&\int _{\mathbb {R} }xdF(x)\in m&&{\text{expectation condition}}\\&\int _{\mathbb {R} }x^{2}dF(x)-\left(\int _{\mathbb {R} }xdF(x)\right)^{2}\in v&&{\text{variance condition}}\end{aligned}}}

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 
 
 
 
 {
 
 B
 
 1
 
 
 ,
 
 B
 
 2
 
 
 ,
 [
 −
 ∞
 ,
 ∞
 ]
 ,
 [
 0
 ,
 ∞
 ]
 ,
 
 D
 
 }
 
 
 {\displaystyle \{B_{1},B_{2},[-\infty ,\infty ],[0,\infty ],\mathbb {D} \}}
 
 can be denoted more compactly as [B1, B2]. This notation harkens to that of intervals on the real line, except that the endpoints are distributions rather than points.
The notation 
 
 
 
 X
 ∼
 F
 
 
 {\displaystyle X\sim F}
 
 denotes the fact that 
 
 
 
 X
 ∈
 
 R
 
 
 
 {\displaystyle X\in \mathbb {R} }
 
 is a random variable governed by the distribution function F, that is,

{
            
              
                
                  F
                  :
                  
                    R
                  
                  →
                  [
                  0
                  ,
                  1
                  ]
                
              
              
                
                  x
                  ↦
                  Pr
                  (
                  X
                  ≤
                  x
                  )
                
              
            
            
          
        
      
    
    {\displaystyle {\begin{cases}F:\mathbb {R} \to [0,1]\\x\mapsto \Pr(X\leq x)\end{cases}}}

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

H
        (
        z
        )
        =
        
          ∫
          
            z
            =
            x
            +
            y
          
        
        F
        (
        x
        )
        G
        (
        y
        )
        d
        z
        =
        
          ∫
          
            
              R
            
          
        
        F
        (
        x
        )
        G
        (
        z
        −
        x
        )
        d
        x
        =
        F
        ∗
        G
        .
      
    
    {\displaystyle H(z)=\int _{z=x+y}F(x)G(y)dz=\int _{\mathbb {R} }F(x)G(z-x)dx=F*G.}

This operation is called a <a href="/facts/Convolution/PVPrdz9J">convolution</a> on F and G. The analogous operation on p-boxes is straightforward for sums. Suppose

X
        ∼
        A
        =
        [
        
          A
          
            1
          
        
        ,
        
          A
          
            2
          
        
        ]
        ,
        
        
          and
        
        
        Y
        ∼
        B
        =
        [
        
          B
          
            1
          
        
        ,
        
          B
          
            2
          
        
        ]
        .
      
    
    {\displaystyle X\sim A=[A_{1},A_{2}],\quad {\text{and}}\quad Y\sim B=[B_{1},B_{2}].}

If X and Y are stochastically independent, then the distribution of Z = X + Y is inside the p-box

[
          
            
              A
              
                1
              
            
            ∗
            
              B
              
                1
              
            
            ,
            
              A
              
                2
              
            
            ∗
            
              B
              
                2
              
            
          
          ]
        
        .
      
    
    {\displaystyle \left[A_{1}*B_{1},A_{2}*B_{2}\right].}

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<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a><a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> showed that

Z
        ∼
        
          [
          
            
              sup
              
                z
                =
                x
                +
                y
              
            
            max
            (
            F
            (
            x
            )
            +
            G
            (
            y
            )
            −
            1
            ,
            0
            )
            ,
            
              inf
              
                z
                =
                x
                +
                y
              
            
            min
            (
            F
            (
            x
            )
            +
            G
            (
            y
            )
            ,
            1
            )
          
          ]
        
      
    
    {\displaystyle Z\sim \left[\sup _{z=x+y}\max(F(x)+G(y)-1,0),\inf _{z=x+y}\min(F(x)+G(y),1)\right]}

These bounds are implied by the <a href="/facts/Copula_(probability_theory)/9tgrLvLF">Fréchet–Hoeffding</a> <a href="/facts/Copula_(probability_theory)/9tgrLvLF">copula</a> bounds. The problem can also be solved using the methods of <a href="/facts/Mathematical_programming/oRn8Iv5I">mathematical programming</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>
The convolution under the intermediate assumption that X and Y have positive dependence is likewise easy to compute, as is the convolution under the extreme assumptions of <a href="/facts/Comonotonicity/3fgaQ2NR">perfect positive</a> or <a href="/facts/Countermonotonicity/3fgaQ2NR">perfect negative</a> dependency between X and Y.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>
Generalized convolutions for other operations such as subtraction, multiplication, division, etc., can be derived using transformations. For instance, p-box subtraction A − B can be defined as A + (−B), where the negative of a p-box B = [B1, B2] is [B2(−x), B1(−x)].

<h2 id="logical-expressions">Logical expressions</h2>
Logical or <a href="/facts/Boolean_function/uT7E8pqC">Boolean expressions</a> involving <a href="/facts/Logical_conjunction/62uyyn1d">conjunctions</a> (<a href="/facts/AND_gate/tVKQD0w1">AND</a> operations), <a href="/facts/Logical_disjunction/jbI2EGlM">disjunctions</a> (<a href="/facts/OR_gate/ox2lHgIM">OR</a> operations), exclusive disjunctions, equivalences, conditionals, etc. arise in the analysis of fault trees and event trees common in risk assessments. If the probabilities of events are characterized by intervals, as suggested by <a href="/facts/George_Boole/fWJT2rek">Boole</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> and <a href="/facts/John_Maynard_Keynes/4otHjBRu">Keynes</a><a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> among others, these binary operations are straightforward to evaluate. For example, if the probability of an event A is in the interval P(A) = a = [0.2, 0.25], and the probability of the event B is in P(B) = b = [0.1, 0.3], then the probability of the <a href="/facts/Logical_conjunction/62uyyn1d">conjunction</a> is surely in the interval

  P(A & B) = a × b
= [0.2, 0.25] × [0.1, 0.3]
= [0.2 × 0.1, 0.25 × 0.3]
= [0.02, 0.075]
so long as A and B can be assumed to be independent events. If they are not independent, we can still bound the conjunction using the classical <a href="/facts/Frechet_inequalities/TWvlM7kS">Fréchet inequality</a>. In this case, we can infer at least that the probability of the joint event A & B is surely within the interval

  P(A & B) = env(max(0, a+b−1), min(a, b))
= env(max(0, [0.2, 0.25]+[0.1, 0.3]−1), min([0.2, 0.25], [0.1, 0.3]))
= env([max(0, 0.2+0.1–1), max(0, 0.25+0.3–1)], [min(0.2,0.1), min(0.25, 0.3)])
= env([0,0], [0.1, 0.25])
= [0, 0.25]
where env([x1,x2], [y1,y2]) is [min(x1,y1), max(x2,y2)]. Likewise, the probability of the <a href="/facts/Logical_disjunction/jbI2EGlM">disjunction</a> is surely in the interval

  P(A v B) = a + b − a × b = 1 − (1 − a) × (1 − b)
= 1 − (1 − [0.2, 0.25]) × (1 − [0.1, 0.3])
= 1 − [0.75, 0.8] × [0.7, 0.9]
= 1 − [0.525, 0.72]
= [0.28, 0.475]
if A and B are independent events. If they are not independent, the Fréchet inequality bounds the disjunction

  P(A v B) = env(max(a, b), min(1, a + b))
= env(max([0.2, 0.25], [0.1, 0.3]), min(1, [0.2, 0.25] + [0.1, 0.3]))
= env([0.2, 0.3], [0.3, 0.55])
= [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<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> to get best-possible bounds on the expression. A similar problem one presents in the case of <a href="/facts/Probabilistic_logic/4MIPxPIJ">probabilistic logic</a> (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.

<h2 id="magnitude-comparisons">Magnitude comparisons</h2>
The probability that an uncertain number represented by a p-box D is less than zero is the interval Pr(D < 0) = [F(0), F̅(0)], where F̅(0) is the left bound of the probability box D and F(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),
A ≤ B = Pr(A − B ≤ 0), and
A ≥ B = Pr(B − A ≤ 0).
Thus the probability that A is less than B is the same as the probability that their difference is less than zero, and this probability can be said to be the value of the expression A < B.
Like arithmetic and logical operations, these magnitude comparisons generally depend on the stochastic dependence between A and B, and the subtraction in the encoding should reflect that dependence. If their dependence is unknown, the difference can be computed without making any assumption using the Fréchet operation.

<h2 id="sampling-based-computation">Sampling-based computation</h2>
Some analysts<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a><a class="footnote-ref" id="fnref:29" href="#fn:29">29</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a><a class="footnote-ref" id="fnref:31" href="#fn:31">31</a><a class="footnote-ref" id="fnref:32" href="#fn:32">32</a><a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> use sampling-based approaches to computing probability bounds, including <a href="/facts/Monte_Carlo_simulation/AkHjY7jc">Monte Carlo simulation</a>, <a href="/facts/Latin_hypercube/4A1BRTGA">Latin hypercube</a> methods or <a href="/facts/Importance_sampling/jrbUnMXa">importance sampling</a>. These approaches cannot assure mathematical rigor in the result because such simulation methods are approximations, although their performance can generally be improved simply by increasing the number of replications in the simulation. Thus, unlike the analytical theorems or methods based on mathematical programming, sampling-based calculations usually cannot produce <a href="/facts/Verified_computing/Mtj01FQb">verified computations</a>. However, sampling-based methods can be very useful in addressing a variety of problems which are computationally <a href="/facts/NP-hard/mQeNPv4R">difficult</a> to solve analytically or even to rigorously bound. One important example is the use of Cauchy-deviate sampling to avoid the <a href="/facts/Curse_of_dimensionality/dk7SnY6L">curse of dimensionality</a> in propagating <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> uncertainty through high-dimensional problems.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>

<h2 id="relationship-to-other-uncertainty-propagation-approaches">Relationship to other uncertainty propagation approaches</h2>
PBA belongs to a class of methods that use <a href="/facts/Imprecise_probability/unnG18zY">imprecise probabilities</a> to simultaneously represent <a href="/facts/Uncertainty_quantification/YzIJjhEE">aleatoric and epistemic uncertainties</a>. PBA is a generalization of both <a href="/facts/Interval_analysis/1ylnKraf">interval analysis</a> and probabilistic <a href="/facts/Convolution_of_probability_distributions/l467HgVK">convolution</a> such as is commonly implemented with <a href="/facts/Monte_Carlo_simulation/AkHjY7jc">Monte Carlo simulation</a>. PBA is also closely related to <a href="/facts/Robust_Bayes_analysis/uwlavFQy">robust Bayes analysis</a>, which is sometimes called <a href="/facts/Bayesian_sensitivity_analysis/uwlavFQy">Bayesian sensitivity analysis</a>. PBA is an alternative to second-order Monte Carlo simulation.

<h2 id="applications">Applications</h2>
Main article: <a href="/facts/Applications_of_p-boxes_and_probability_bounds_analysis/GtUmmG7B">Applications of p-boxes and probability bounds analysis</a>
<a href="/facts/Probability_box/MoQRSlGW">P-boxes</a> and probability bounds analysis have been used in many applications spanning many disciplines in engineering and environmental science, including:

<ul><li>Engineering design<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a></li>
<li>Expert elicitation<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a></li>
<li>Analysis of species sensitivity distributions<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a></li>
<li><a href="/facts/Sensitivity_analysis/X9x3WAnK">Sensitivity analysis</a> in <a href="/facts/Aerospace_engineering/2Ib02w7f">aerospace engineering</a> of the buckling load of the frontskirt of the <a href="/facts/Ariane_5/H78wlLG4">Ariane 5</a> launcher<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a></li>
<li><a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ODE</a> models of <a href="/facts/Chemical_reactor/asqApwa4">chemical reactor</a> dynamics<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a><a class="footnote-ref" id="fnref:40" href="#fn:40">40</a></li>
<li><a href="/facts/Pharmacokinetics/x4CUUnle">Pharmacokinetic</a> variability of inhaled <a href="/facts/Volatile_organic_compound/BFMlKwgt">VOCs</a><a class="footnote-ref" id="fnref:41" href="#fn:41">41</a></li>
<li><a href="/facts/Groundwater_model/1LXF0pD4">Groundwater modeling</a><a class="footnote-ref" id="fnref:42" href="#fn:42">42</a></li>
<li>Bounding <a href="/facts/Failure_analysis/C5MVRgn6">failure</a> probability for series systems<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a></li>
<li><a href="/facts/Heavy_metal_(chemistry)/Cblmy8qI">Heavy metal</a> <a href="/facts/Soil_contamination/BzPwKXhM">contamination in soil</a> at an <a href="/facts/Ironworks/bdzImAzP">ironworks</a> <a href="/facts/Brownfield_land/E1su5meG">brownfield</a><a class="footnote-ref" id="fnref:44" href="#fn:44">44</a><a class="footnote-ref" id="fnref:45" href="#fn:45">45</a></li>
<li>Uncertainty propagation for <a href="/facts/Salinity_in_Australia/2JfYxq4Y">salinity</a> risk models<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a></li>
<li>Power supply system safety assessment<a class="footnote-ref" id="fnref:47" href="#fn:47">47</a></li>
<li><a href="/facts/Contaminated_land/howElJxn">Contaminated land</a> risk assessment<a class="footnote-ref" id="fnref:48" href="#fn:48">48</a></li>
<li>Engineered systems for drinking <a href="/facts/Water_treatment/nIstBCdG">water treatment</a><a class="footnote-ref" id="fnref:49" href="#fn:49">49</a></li>
<li>Computing soil screening levels<a class="footnote-ref" id="fnref:50" href="#fn:50">50</a></li>
<li>Human health and ecological risk analysis by the <a href="/facts/United_States_Environmental_Protection_Agency/ghI985SI">U.S. EPA</a> of <a href="/facts/Polychlorinated_biphenyl/R3sC5PEf">PCB</a> contamination at the <a href="/facts/Housatonic_River/Vh5zugeL">Housatonic River</a> <a href="/facts/Superfund/FFJJqmmu">Superfund</a> site<a class="footnote-ref" id="fnref:51" href="#fn:51">51</a><a class="footnote-ref" id="fnref:52" href="#fn:52">52</a></li>
<li><a href="/facts/Environmental_impact_assessment/LWNWaKJC">Environmental assessment</a> for the <a href="/facts/Calcasieu_River/sBX0RJbz">Calcasieu Estuary</a> Superfund site<a class="footnote-ref" id="fnref:53" href="#fn:53">53</a></li>
<li><a href="/facts/Aerospace_engineering/2Ib02w7f">Aerospace engineering</a> for <a href="/facts/Supersonic_nozzle/5LcDcs69">supersonic nozzle</a> thrust<a class="footnote-ref" id="fnref:54" href="#fn:54">54</a></li>
<li><a href="/facts/Verification_and_validation/hgqysrzb">Verification and validation</a> in scientific computation for engineering problems<a class="footnote-ref" id="fnref:55" href="#fn:55">55</a></li>
<li>Toxicity to small mammals of environmental <a href="/facts/Mercury_(element)/WlxfoHva">mercury</a> contamination<a class="footnote-ref" id="fnref:56" href="#fn:56">56</a></li>
<li>Modeling travel time of pollution in <a href="/facts/Groundwater/JYM2IDS2">groundwater</a><a class="footnote-ref" id="fnref:57" href="#fn:57">57</a></li>
<li><a href="/facts/Reliability_engineering/cd6AS5w3">Reliability analysis</a><a class="footnote-ref" id="fnref:58" href="#fn:58">58</a></li>
<li><a href="/facts/Endangered_species/L7cIweSW">Endangered species</a> assessment for reintroduction of <a href="/facts/Leadbeater%27s_possum/z2PsEKxb">Leadbeater's possum</a><a class="footnote-ref" id="fnref:59" href="#fn:59">59</a></li>
<li>Exposure of <a href="/facts/Insectivore/ukA7UnLr">insectivorous</a> birds to an agricultural <a href="/facts/Pesticide/Uhf4KnwJ">pesticide</a><a class="footnote-ref" id="fnref:60" href="#fn:60">60</a></li>
<li><a href="/facts/Climate_change/rEN4BDE7">Climate change</a> projections<a class="footnote-ref" id="fnref:61" href="#fn:61">61</a><a class="footnote-ref" id="fnref:62" href="#fn:62">62</a><a class="footnote-ref" id="fnref:63" href="#fn:63">63</a></li>
<li>Waiting time in <a href="/facts/Queueing_theory/8NDl56EM">queuing systems</a><a class="footnote-ref" id="fnref:64" href="#fn:64">64</a></li>
<li><a href="/facts/Extinction/24p5NP2q">Extinction</a> risk analysis for <a href="/facts/Northern_spotted_owl/6IyaFQzQ">spotted owl</a> on the <a href="/facts/Olympic_Peninsula/nW8FJa47">Olympic Peninsula</a><a class="footnote-ref" id="fnref:65" href="#fn:65">65</a></li>
<li><a href="/facts/Biosecurity/Q1EDEjsD">Biosecurity</a> against introduction of <a href="/facts/Invasive_species/Ya1v6RGx">invasive species</a> or agricultural <a href="/facts/Pest_(organism)/mXGHJ8s7">pests</a><a class="footnote-ref" id="fnref:66" href="#fn:66">66</a></li>
<li><a href="/facts/Finite_element_method/eUcfP1vn">Finite-element</a> <a href="/facts/Structural_analysis/phRJdfup">structural analysis</a><a class="footnote-ref" id="fnref:67" href="#fn:67">67</a><a class="footnote-ref" id="fnref:68" href="#fn:68">68</a><a class="footnote-ref" id="fnref:69" href="#fn:69">69</a></li>
<li><a href="/facts/Cost_estimation_models/tMRvhkNg">Cost estimates</a><a class="footnote-ref" id="fnref:70" href="#fn:70">70</a></li>
<li><a href="/facts/Nuclear_stockpile/urhNKh4Q">Nuclear stockpile</a> certification<a class="footnote-ref" id="fnref:71" href="#fn:71">71</a></li>
<li><a href="/facts/Fracking/cCTzfwkg">Fracking</a> risks to <a href="/facts/Water_pollution/SCUhUYqB">water pollution</a><a class="footnote-ref" id="fnref:72" href="#fn:72">72</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Probability_box/MoQRSlGW">Probability box</a></li>
<li><a href="/facts/Robust_Bayes_analysis/uwlavFQy">Robust Bayes analysis</a></li>
<li><a href="/facts/Imprecise_probability/unnG18zY">Imprecise probability</a></li>
<li>Second-order Monte Carlo simulation</li>
<li><a href="/facts/Monte_Carlo_simulation/AkHjY7jc">Monte Carlo simulation</a></li>
<li><a href="/facts/Interval_arithmetic/1ylnKraf">Interval analysis</a></li>
<li><a href="/facts/Probability_theory/mFBn51rE">Probability theory</a></li>
<li><a href="/facts/Risk_analysis/xGhxtxdp">Risk analysis</a></li></ul>

<h2 id="further-references">Further references</h2>
<ul><li>Bernardini, Alberto; Tonon, Fulvio (2010). Bounding Uncertainty in Civil Engineering: Theoretical Background. Berlin: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-11189-1.</li>
<li>Ferson, Scott (2002). RAMAS Risk Calc 4.0 Software : Risk Assessment with Uncertain Numbers. Boca Raton, Florida: Lewis Publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56670-576-9.</li>
<li>Gerla, G. (1994). "Inferences in Probability Logic". Artificial Intelligence. 70 (1–2): 33–52. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0004-3702%2894%2990102-3">10.1016/0004-3702(94)90102-3</a>.</li>
<li>Oberkampf, William L.; Roy, Christopher J. (2010). Verification and Validation in Scientific Computing. New York: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-11360-1.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.ramas.com/pbawhite.pdf">Probability bounds analysis in environmental risk assessments</a></li>
<li><a href="http://ualr.edu/jdberleant/intprob/">Intervals and probability distributions</a></li>
<li><a href="https://web.archive.org/web/20120210155925/http://www.sandia.gov/epistemic/">Epistemic uncertainty project</a></li>
<li><a href="http://www.sipta.org/">The Society for Imprecise Probability: Theories and Applications</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Boole, George (1854). An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities. London: Walton and Maberly. <a href="https://www.gutenberg.org/ebooks/15114" target="_blank">https://www.gutenberg.org/ebooks/15114</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hailperin, Theodore (1986). Boole's Logic and Probability. Amsterdam: North-Holland. ISBN 978-0-444-11037-4. <a href="978-0-444-11037-4" target="_blank">978-0-444-11037-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kyburg, H.E., Jr. (1999). Interval valued probabilities. SIPTA Documentation on Imprecise Probability. <a href="https://sipta.org/documentation/interval_prob/kyburgnew.pdf" target="_blank">https://sipta.org/documentation/interval_prob/kyburgnew.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Walley, Peter (1991). Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall. ISBN 978-0-412-28660-5. <a href="978-0-412-28660-5" target="_blank">978-0-412-28660-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Hailperin, Theodore (1986). Boole's Logic and Probability. Amsterdam: North-Holland. ISBN 978-0-444-11037-4. <a href="978-0-444-11037-4" target="_blank">978-0-444-11037-4</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
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<li id="fn:14">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. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
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<li id="fn:16">Yager, R.R. (1986). Arithmetic and other operations on Dempster–Shafer structures. International Journal of Man-machine Studies 25: 357–366. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
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<li id="fn:19">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. <a href="http://www.ramas.com/depend.pdf" target="_blank">http://www.ramas.com/depend.pdf</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">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. <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">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. <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">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. <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">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. <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">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. <a href="http://www.ramas.com/depend.pdf" target="_blank">http://www.ramas.com/depend.pdf</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Boole, George (1854). An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities. London: Walton and Maberly. <a href="https://www.gutenberg.org/ebooks/15114" target="_blank">https://www.gutenberg.org/ebooks/15114</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Kyburg, H.E., Jr. (1999). Interval valued probabilities. SIPTA Documentation on Imprecise Probability. <a href="https://sipta.org/documentation/interval_prob/kyburgnew.pdf" target="_blank">https://sipta.org/documentation/interval_prob/kyburgnew.pdf</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Hailperin, Theodore (1986). Boole's Logic and Probability. Amsterdam: North-Holland. ISBN 978-0-444-11037-4. <a href="978-0-444-11037-4" target="_blank">978-0-444-11037-4</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Alvarez, D. A., 2006. On the calculation of the bounds of probability of events using infinite random sets. International Journal of Approximate Reasoning 43: 241–267. <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Baraldi, P., Popescu, I. C., Zio, E., 2008. Predicting the time to failure of a randomly degrading component by a hybrid Monte Carlo and possibilistic method. IEEE Proc. International Conference on Prognostics and Health Management. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
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<li id="fn:44">Augustsson, A., M. Filipsson, T. Öberg, B. Bergbäck (2011). Climate change—an uncertainty factor in risk analysis of contaminated land. Science of the Total Environment 409: 4693–4700. <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Baudrit, C., D. Guyonnet, H. Baroudi, S. Denys and P. Begassat (2005). Assessment of child exposure to lead on an ironworks brownfield: uncertainty analysis. 9th International FZK/TNO Conference on Contaminated Soil – ConSoil2005, Bordeaux, France, pages 1071–1080. <a href="http://www.univ-orleans.fr/mapmo/membres/baudrit/consoil05.pdf" target="_blank">http://www.univ-orleans.fr/mapmo/membres/baudrit/consoil05.pdf</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
<li id="fn:46">Dixon, W.J. (2007). Uncertainty Propagation in Population Level Salinity Risk Models. Technical Report Technical Report Series No. 164, Arthur Rylah Institute for Environmental Research. Heidelberg, Victoria, Australia <a href="http://www.depi.vic.gov.au/__data/assets/pdf_file/0017/333116/ARI-Technical-Report-164-Uncertainty-propagation-in-population-level-salinity-risk-models.pdf" target="_blank">http://www.depi.vic.gov.au/__data/assets/pdf_file/0017/333116/ARI-Technical-Report-164-Uncertainty-propagation-in-population-level-salinity-risk-models.pdf</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></li>
<li id="fn:47">Karanki, D.R., H.S. Kushwaha, A.K. Verma, and S. Ajit. (2009). Uncertainty analysis based on probability bounds (p-box) approach in probabilistic safety assessment. Risk Analysis 29: 662–75. <a href="#fnref:47" class="footnote-back-ref">↩</a></li>
<li id="fn:48">Sander, P., B. Bergbäck and T. Öberg (2006). Uncertain numbers and uncertainty in the selection of input distributions—Consequences for a probabilistic risk assessment of contaminated land. Risk Analysis 26: 1363–1375. <a href="/wiki/Probabilistic_risk_assessment" target="_blank">/wiki/Probabilistic_risk_assessment</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></li>
<li id="fn:49">Minnery, J.G., J.G. Jacangelo, L.I. Boden, D.J. Vorhees and W. Heiger-Bernays (2009). Sensitivity analysis of the pressure-based direct integrity test for membranes used in drinking water treatment. Environmental Science and Technology 43(24): 9419–9424. <a href="#fnref:49" class="footnote-back-ref">↩</a></li>
<li id="fn:50">Regan, H.M., B.E. Sample and S. Ferson (2002). Comparison of deterministic and probabilistic calculation of ecological soil screening levels. Environmental Toxicology and Chemistry 21: 882–890. <a href="#fnref:50" class="footnote-back-ref">↩</a></li>
<li id="fn:51">U.S. Environmental Protection Agency (Region I), GE/Housatonic River Site in New England <a href="http://www.epa.gov/ne/ge/" target="_blank">http://www.epa.gov/ne/ge/</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></li>
<li id="fn:52">Moore, Dwayne R.J.; Breton, Roger L.; Delong, Tod R.; Ferson, Scott; Lortie, John P.; MacDonald, Drew B.; McGrath, Richard; Pawlisz, Andrzej; Svirsky, Susan C.; Teed, R. Scott; Thompson, Ryan P.; Whitfield Aslund, Melissa (2016). "Ecological risk assessment for mink and short-tailed shrew exposed to PCBS, dioxins, and furans in the Housatonic River area". Integrated Environmental Assessment and Management. 12 (1): 174–184. Bibcode:2016IEAM...12..174M. doi:10.1002/ieam.1661. PMID 25976918. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></li>
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</ol>

Probability bounds analysis open-in-new

Probability bounds analysis