Transcript Where do the inputs come from?

Untangling equations
involving uncertainty
Scott Ferson, Applied Biomathematics
Vladik Kreinovich, University of Texas at El Paso
W. Troy Tucker, Applied Biomathematics
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
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
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This is an interval, not
a uniform distribution
Probability bounds analysis (PBA)
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independence
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CDF
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independence
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assuming
nothing
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PBA handles common problems
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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
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
Example
• Suppose
W = [23, 33]
H = [112, 150]
A = [2000, 3200]
• Does knowing WH=A let us to say any more?
Answer
• Yes, we can infer that
W = [23, 28.57]
H = [112, 139.13]
A = [2576, 3200]
• 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.
Bayesian strategy
Prior
I (W  [23,33]) I ( H  [112 ,150 ]) I ( A  [2000 ,3200 ])
Pr(W , H , A) 
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
33  23
150  112
3200  2000
Likelihood
L( A  W  H | W , H , A)   ( A  W  H )
Posterior
f (W , H , A | A  W  H )   ( A  W  H )  Pr(W , H , A)
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
Updating with p-boxes
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W
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Answers
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intersect(W, A/H)
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intersect(H, A/W)
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intersect(A, WH)
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
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
Can’t just invert the equation
conc  intake
dose =
body mass
dose  body mass
conc =
intake
When conc is put back into the forward equation,
the dose is wider than planned
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!
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
Backcalculation with p-boxes
Suppose A + B = C, where
A = normal(5, 1)
C = {0  C, median  15, 90th %ile  35, max  50}
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Getting the answer
• The backcalculation algorithm basically
reverses the forward convolution
• Not hard at all…but a little messy to show
• Any distribution
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totally inside B is
sure to satisfy the
constraint … it’s
“kernel”
B
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Check by plugging back in
A + B = C*  C
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C*
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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, B
A, C
B ,C
A, B
A, C
B ,C
A, B
A, C
B ,C
A, B
A, C
B ,C
A
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)
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)
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
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
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])
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)
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
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
Acknowledgments
• Janos Hajagos, Stony Brook University
• Lev Ginzburg, Stony Brook University
• David Myers, Applied Biomathematics
• National Institutes of Health SBIR program
End
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