1. Untangling equations involving uncertainty
Scott Ferson, Applied Biomathematics
Vladik Kreinovich, University of Texas at El Paso
W. Troy Tucker, Applied Biomathematics

2. Overview Three kinds of operations Deconvolutions Backcalculations
Updates (oh, my!)
Very elementary methods of interval analysis
Low-dimensional
Simple arithmetic operations
But combined with probability theory

3. Probability box (p-box)
Bounds on a cumulative distribution function (CDF)
Envelope of a Dempster-Shafer structure
Used in risk analysis and uncertainty arithmetic
Generalizes probability distributions and intervals
Cumulative probability 10 20 30 40
0.5 1
This is an interval, not
a uniform distribution

4. Probability bounds analysis (PBA)
a =T( 0 , 10 , 20) + [0, 5]
b =N([20,23],[1,12])
Disagreement between theoretical and observed variance
c = a |+| b
c = a + b
Probability bounds analysis (PBA)
assuming
independence 80 1 40 1 20
CDF
assuming
independence 80 1
assuming
nothing 1 80

5. PBA handles common problems
Imprecisely specified distributions
Poorly known or unknown dependencies
Non-negligible measurement error
Inconsistency in the quality of input data
Model uncertainty and non-stationarity
Plus, it’s much faster than Monte Carlo

6. Updating
Using knowledge of how variables are related to tighten their estimates
Removes internal inconsistency and explicates unrecognized knowledge
Also called constraint updating or editing
Also called natural extension

7. Example Suppose W = [23, 33] H = [112, 150] A = [2000, 3200]
Does knowing W  H = A let us to say any more?

8. Answer Yes, we can infer that W = [23, 28.57] H = [112, 139.13]
The formulas are just W = intersect(W, A/H), etc.
To get the largest possible W, for instance, let A be as large
as possible and H as small as possible, and solve for W =A/H.

9. Bayesian strategy
Prior
Likelihood
Posterior

10. Bayes’ rule
Concentrates mass onto the manifold of feasible combinations of W, H, and A
Answers have the same supports as intervals
Computationally complex
Needs specification of priors
Yields distributions that are not justified (come from the choice of priors)
Expresses less uncertainty than is present

11. Updating with p-boxes 1 1 1 H A W 20 30 40
120
140
160
2000
3000
4000

12. A H W Answers 1 intersect(W, A/H) intersect(H, A/W) intersect(A, WH)
2000
3000
4000 1 A 20 30 40 W
120
140
160 H
intersect(W, A/H)
intersect(H, A/W)
intersect(A, WH)

13. Calculation with p-boxes
Agrees with interval analysis whenever inputs are intervals
Relaxes Bayesian strategy when precise priors are not warranted
Produces more reasonable answers when priors not well known
Much easier to compute than Bayes’ rule

14. Backcalculation
Find constraints on B that ensure C = A + B satisfies specified constraints
Or, more generally, C = f(A1, A2,…, Ak, B)
If A and C are intervals, the answer is called the tolerance solution

15. Can’t just invert the equation
When conc is put back into the forward equation, the dose is wider than planned
conc  intake
body mass
dose =
dose  body mass
intake
conc =

16. Example dose = [0, 2] milligram per kilogram intake = [1, 2.5] liter
mass = [60, 96] kilogram
conc = dose * mass / intake
[ 0, 192] milligram liter-1
dose = conc * intake / mass
[ 0, 8] milligram kilogram-1
Doses 4 times larger
than tolerable levels!

17. Backcalculating probability distributions
Needed for engineering design problems, e.g., cleanup and remediation planning for environmental contamination
Available analytical algorithms are unstable for almost all problems
Except in a few special cases, Monte Carlo simulation cannot compute backcalculations; trial and error methods are required

18. Backcalculation with p-boxes
Suppose A + B = C, where
A = normal(5, 1)
C = {0  C, median  15, 90th %ile  35, max  50}
-10 10 20 30 40 50 60 1 C 2 3 4 5 6 7 8 1 A

19. Getting the answer
The backcalculation algorithm basically reverses the forward convolution
Not hard at all…but a little messy to show
Any distribution totally inside B is sure to satisfy the constraint … it’s “kernel”
-10 10 20 30 40 50 1 B

20. Check by plugging back in
A + B = C*  C
-10 10 20 30 40 50 60 1 C* C

21. When you
Know that
A + B = C
A – B = C
A  B = C
A / B = C
A ^ B = C
2A = C
A² = C
And you have
estimates for
A, B
A, C
B ,C A C
Use this formula
to find the unknown
C = A + B
B = backcalc(A,C)
A = backcalc (B,C)
C = A – B
B = –backcalc(A,C)
A = backcalc (–B,C)
C = A * B
B = factor(A,C)
A = factor(B,C)
C = A / B
B = 1/factor(A,C)
A = factor(1/B,C)
C = A ^ B
B = factor(log A, log C)
A = exp(factor(B, log C))
C = 2 * A
A = C / 2
C = A ^ 2
A = sqrt(C)

22. Kernels Existence more likely if p-boxes are fat
Wider if we can also assume independence
Answers are not unique, even though tolerance solutions always are
Different kernels can emphasize different properties
Envelope of all possible kernels is the shell (i.e., the united solution)

23. Precise distributions
Precise distributions can’t express the nature of the target
Finding a conc distribution that results in a prescribed distribution of doses says we want some doses to be high (any distribution to the left would be even better)
We need to express the dose target as a p-box

24. Deconvolution Uses information about dependence to tighten estimates
Useful, for instance, in correcting an estimated distribution for measurement uncertainty
For instance, suppose Y = X + 
If X and  are independent, Y² = X² + ²
Then we do an uncertainty correction

25. Example Y = X +  Y,  ~ normal
X ~ N(decon(Y, X), sqrt(decon(², Y²))
Y ~ N([5,9], [2,3]);  ~ N([1,+1], [½,1])
X ~ N(dcn([1,1],[5,6]), sqrt(dcn([¼,1],[4,9])))
X ~ N([6,8], sqrt([3, 63])

26. Deconvolutions with p-boxes
As for backcalculations, computation of deconvolutions is troublesome in probability theory, but often much simpler with p-boxes
Deconvolution didn’t have an analog in interval analysis (until now via p-boxes)

27. Relaxing over-determination
Most constraint problems almost never have solutions with probability distributions
The constraints are too numerous and strict
P-boxes relax these constraints so that many problems can have solutions

28. P-boxes in interval analysis
P-boxes bring probability distributions into the realm of intervals
Express and solve backcalculation problems better than is possible in probability theory by itself
Generalize the notion of tolerance solutions (kernels)
Relax unwarranted assumptions about priors in updating problems needed in a Bayesian approach
Introduce deconvolution into interval analysis

29. Acknowledgments Janos Hajagos, Stony Brook University
Lev Ginzburg, Stony Brook University
David Myers, Applied Biomathematics
National Institutes of Health SBIR program

30. End

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