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### Verified computation with probabilities

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Scott Ferson, [email protected]

Applied Biomathematics

Scientific Computing, Computer Arithmetic and Verified Numerical Computations

El Paso, Texas, 3 October 2008

Perspective

- Very elementary methods of interval analysis
- Low-dimensional, static
- Verified computing (but not roundoff error)
- Huge uncertainties
- Intervals combined with probability theory
- Total probabilities (events)
- Probability distributions (random variables)

Bounding probability is an old idea

- Boole and de Morgan
- Chebyshev and Markov
- Borel and Fréchet
- Kolmogorov and Keynes
- Berger and Walley
- Williamson and Downs

Terminology

- Dependence = stochastic dependence
- More general than repeated variables
- Independence = stochastic independence
- Best possible = tight (almost)
- some elements in the set may not be possible

Incertitude

- Arises from incomplete knowledge
- Incertitude arises from
- limited sample size
- measurement uncertainty or surrogate data
- doubt about the model
- Reducible with empirical effort

Variability

- Arises from natural stochasticity
- Variability arises from
- scatter and variation
- spatial or temporal fluctuations
- manufacturing or genetic differences
- Not reducible by empirical effort

They must be treated differently

- Variability is randomness

Needs probability theory

- Incertitude is ignorance

Needs interval analysis

- Imprecise probabilities can do both at once

Risk assessment applications

- Environmental pollution

heavy metals, pesticides, PMx, ozone, PCBs, EMF, RF, etc.

- Engineered systems

traffic safety, bridge design, airplanes, spacecraft, nuclear plants

- Financial investments

portfolio planning, consultation, instrument evaluation

- Occupational hazards

manufacturing and factory workers, farm workers, hospital staff

- Food safety and consumer product safety

benzene in Perrier, E. coli in beef, salmonella in tomato, children’s toys

- Ecosystems and biological resources

endangered species, fisheries and reserve management

$

Total probabilities (events)

- Fault or event trees
- Logical expressions (Hailperin 1986)
- Reliability analyses
- Nuclear power plants
- Aircraft safety system design
- Gene technology release assessments
- etc.

Interval arithmetic for probabilities

x + y = [x1 + y1, x2 + y2]

xy = [x1 y2, x2 y1]

xy = [x1 y1, x2 y2]

xy = [x1 y2, x2 y1]

min(x, y) = [min(x1, y1), min(x2, y2)]

max(x, y) = [max(x1, y1), max(x2, y2)]

Rules are simpler because intervals confined to [0,1]

Probabilistic logic

- Conjunctions (and)
- Disjunctions (or)
- Negations (not)
- Exclusive disjunctions (xor)
- Modus ponens (if-then)
- etc.

Conjunction (and)

P(A |&| B) = P(A) P(B)

Example: P(A) = [0.3, 0.5]

P(B) = [0.1, 0.2]

A and B are independent

P(A |&| B) = [0.03, 0.1]

Fréchet case

P(A & B)=[max(0, P(A) + P(B)–1), min(P(A), P(B))]

- Makes no assumption about the dependence
- Rigorous (guaranteed to enclose true value)
- Best possible (cannot be any tighter)

uncertain

Fréchet examplesExamples: P(A) = [0.3, 0.5]

P(B) = [0.1, 0.2]

P(A & B) = [0, 0.2]

P(C) = 0.29

P(D) = 0.22

P(C & D) = [0, 0.22]

S1

Relay

K2

Relay

K1

Timer

relay R

Motor

Pressure

switch S

From

Pump

reservoir

Pressure tank T

Outlet

valve

Example: pump systemWhat’s the risk the tank ruptures under pumping?

E5

or

K2

E4

K1

or

E2

or

S1

E1

E3

or

and

T

S

Fault treeBoolean expression of “tank rupturing under pressure” E1

Unsure about the probabilities and their dependencies

Results

Points, independence

Mixed dependencies

Fréchet

Intervals and Fréchet

105

104

103

Probability of tank rupturing under pumping

Interval probabilities

- Allow verified computing of reliabilities
- Distinguish two forms of uncertainty
- Rigorous bounds always easy to get
- Best possible bounds may need mathematical programming because of repeated variables

Typical problems

- Sometimes little or even no actual data
- Updating is rarely used
- Very simple arithmetic calculations
- Occasionally, finite element meshes or differential equations
- Usually small in number of inputs
- Nuclear power plants are the exception
- Results are important and often high profile
- But the approach is being used ever more widely

Example: pesticides & farmworkers

- Total dose is decomposed by pathway
- Dermal exposure on hands
- Exposures to rest of body
- Inhalation

(concentration in air, exposure duration, breathing rate, penetration factor, absorption efficiency)

- Takes account of related factors
- Acres, gallons per acre, mixing time
- Body mass, frequency of hand washing, etc.

Worst-case analysis

- Traditional method
- Mix of deterministic and extreme values
- Actually a kind of interval analysis
- Says how bad it could it be, but not how unlikely that outcome is

Probabilistic analysis

- State-of-the-art method
- Usually via Monte Carlo simulation
- Requires the full joint distribution
- All the distributions for every input variable
- All their intervariable dependencies
- Often we need to guess about a lot of it

What’s needed

- Reliable, conservative assessments of tail risks
- Using available information but without forcing analysts to make unjustified assumptions
- Neither computationally expensive nor intellectually taxing

Probability bounds analysis

- Marries intervals with probability theory
- Distinguishes variability and incertitude
- Solves many problems in uncertainty analysis
- Input distributions unknown
- Imperfectly known correlation and dependency
- Large measurement error, censoring, small sample sizes
- Model uncertainty

Calculations

- All standard mathematical operations
- Arithmetic operations(+, , ×, ÷, ^, min, max)
- Logical operations(and, or, not, if, etc.)
- Transformations(exp, ln, sin, tan, abs, sqrt, etc.)
- Backcalculation (tolerance solutions)(deconvolutions, updating)
- Magnitude comparisons(<, ≤, >, ≥, )
- Other operations(envelope, mixture, etc.)
- Faster than Monte Carlo
- Guaranteed to bounds answer
- Good solutions often easy to compute

Probability box (p-box)

Interval bounds on an cumulative distribution function (CDF)

1

Cumulative probability

0

0.0

1.0

2.0

3.0

X

Duality

- Bounds on the probability at a value

Chance the value will be 15 or less is between 0 and 25%

- Bounds on the value at a probability

95th percentile is between 40 and 70

1

Cumulative

Probability

0

0

20

40

60

80

X

1

0

0

10

20

30

40

10

20

30

Uncertain numbersProbability

distribution

Probability

box

Interval

1

Cumulative probability

0

0

10

20

30

40

Probability bounds arithmetic

1

1

A

B

Cumulative Probability

Cumulative Probability

0

0

0

1

2

3

4

5

6

0

2

4

6

8

10

12

14

What’s the sum of A+B?

prob=1/9

A+B[5,13]

prob=1/9

A+B[7,13]

prob=1/9

A+B[8,14]

prob=1/9

A+B[9,15]

prob=1/9

A+B[9,15]

prob=1/9

A+B[10,16]

prob=1/9

A+B[11,17]

prob=1/9

Cartesian productA = mix(1,[1,3], 1,[2,4], 1, [3,5])

B = mix(1, [2,8], 1, [6,10], 1,[8,12])

A |+| B

~(range=[3,17], mean=[7.3,14], var=[0,22])

A + B

~(range=[3,17], mean=[7.3,14], var=[0,38])

A+B

independence

A[1,3]

p1 = 1/3

A[2,4]

p2 = 1/3

A[3,5]

p3 = 1/3

B[2,8]

q1 = 1/3

A+B[3,11]

prob=1/9

B[6,10]

q2 = 1/3

B[8,12]

q3 = 1/3

Generalization of methods

- When inputs are distributions, the answers conform with probability theory
- When inputs are intervals, it agrees with interval analysis

Where do we get p-boxes?

- Assumption
- Modeling
- Robust Bayes analysis
- Constraint propagation
- Data with incertitude
- Measurement error
- Sampling error
- Censoring

Empirical distributions

1

1

1

Skinny

Puffy

Cumulative probability

Cumulative probability

0

0

0

0

0

2

2

4

4

6

6

8

8

10

10

0

0

2

2

4

4

6

6

8

8

10

10

x

x

x

x

Fitted distributions

1

1

Skinny

Puffy

Cumulative probability

0

0

0

10

20

0

10

20

x

x

Statisticians often ignore the incertitude, or treat it as though it were (uniform) variability.

Constraint propagation

(lognormal with interval parameters)

Maximum entropy erases uncertainty

1

1

1

mean, std

positive, quantile

min,max

0

0

0

2

4

6

8

10

-10

0

10

20

0

10

20

30

40

50

1

1

1

min,

max, mean

sample data, range

integral, min,max

0

0

0

2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

1

1

1

(lognormal with interval parameters)

min, mean

min,max, mean,std

p-box

0

0

0

0

10

20

30

40

50

2

4

6

8

10

2

4

6

8

10

Example: PCBs and duck hunters

Location: Massachusetts and Connecticut

Receptor: Adult human hunters of waterfowl

Contaminant: PCBs (polychorinated biphenyls)

Exposure route: dietary consumption of contaminated waterfowl

Based on the assessment for non-cancer risks from PCB to adult hunters who consume contaminated waterfowl described in Human Health Risk Assessment: GE/Housatonic River Site: Rest of River, Volume IV, DCN: GE-031203-ABMP, April 2003, Weston Solutions (West Chester, Pennsylvania), Avatar Environmental (Exton, Pennsylvania), and Applied Biomathematics (Setauket, New York).

Hazard quotient

EF = mmms(1, 52, 5.4, 10) meals per year // exposure frequency, censored data, n = 23

IR = mmms(1.5, 675, 188, 113) grams per meal //poultry ingestion rate from EPA’s EFH

C = [7.1, 9.73] mg per kg //exposure point (mean) concentration

LOSS = 0 // loss due to cooking

AT = 365.25 days per year //averaging time (not just units conversion)

BW = mixture(BWfemale, BWmale) // Brainard and Burmaster (1992)

BWmale = lognormal(171, 30) pounds // adult male n = 9,983

BWfemale = lognormal(145, 30) pounds // adult female n = 10,339

RfD = 0.00002 mg per kg per day // reference dose considered tolerable

HQ = (EF |*| IR |*| C |*| (1|-| LOSS)) |/| (AT |*| BW |*| RfD)

Automatically verified results

EF = mmms(1, 52, 5.4, 10) *units(\'meals per year\') // exposure frequency, censored data, n = 23

IR = mmms(1.5, 675, 188, 113) *units(\'grams per meal\') // poultry ingestion rate from EPA’s EFH

C = [7.1, 9.73] *units(\'mg per kg\') // exposure point (mean) concentration

LOSS = 0 // loss due to cooking

AT = 365.25 *units(\'days per year\') // averaging time (not just units conversion)

BWmale = lognormal(171, 30) *units(\'pounds\') // adult male n = 9,983

BWfemale = lognormal(145, 30) *units(\'pounds\') // adult female n = 10,339

BW = mixture(BWfemale, BWmale) // Brainard and Burmaster (1992)

RfD = 0.00002 *units(\'mg per kg per day\') // reference dose considered tolerable

HQ = (EF |*| IR |*| C |*| (1|-| LOSS)) / (AT |*| BW |*| RfD)

HQ

~(range=[5.52769,557906], mean=[1726,13844], var=[0,7e+09]) meals grams meal-1 days-1 pounds-1 day

HQ = HQ + 0

HQ

~(range=[0.0121865,1229.97], mean=[3.8,30.6], var=[0,34557])

mean(HQ)

[ 3.8072899212, 30.520271393]

sd(HQ)

[ 0, 185.89511004]

median(HQ)

[ 0.66730580299, 54.338212132]

cut(HQ,95%)

[ 3.5745830321, 383.33656983]

range(HQ)

[ 0.012186461038, 1229.9710013]

1

mean [3.8, 31]

standard deviation [0, 186]

median [0.6, 55]

95th percentile [3.5 , 384]

range [0.01, 1230]

Exceedance risk

0

0

500

1000

HQ

Uncertainty about distribution shape

- PBA propagates uncertainty about distribution shape comprehensively
- Neither sensitivity studies nor second-order Monte Carlo can really do this
- Maximum entropy hides uncertainty

Dependence

- Not all variables are independent
- Body size and skin surface area
- Common-cause variables
- Default risks for mortgages
- Known dependencies should be modeled
- What can we do when we don’t know them?

Uncertainty about dependence

- Sensitivity analyses usually used
- Vary correlation coefficient between 1 and +1
- But this underestimates the true uncertainty
- Example: suppose X, Y ~ uniform(0,25) but we don’t know the dependence between X and Y

Varying correlation between 1 and +1

1

Pearson

Cumulative probability

X,Y ~ uniform(0,25)

0

0

10

20

30

40

50

X + Y

Uncertainty about dependence

- Can’t be studied with sensitivity analysis since it’s an infinite-dimensional problem
- Fréchet bounding lets you be sure
- Intermediate knowledge can be exploited
- Dependence can have large or small effect

Backcalculation

Generalization of tolerance solutions

E.g., from p-boxes for A and C, finds B such that A+B C

Needed for planning environmental cleanups, designing structures, etc.

Backcalculation with p-boxes

A = normal(5, 1)

C = {0 C, median 1.5, 90th%ile 35, max 50}

1

A

0

2

3

4

5

6

7

8

1

C

0

0

10

20

30

40

50

60

Getting the answer

Basically reverses the forward convolution

Any distribution totally inside B is sure to satisfy the constraint

Many possible B’s

1

B

0

-10

0

10

20

30

40

50

Backcalculation

Backcalculation wider under independence (narrower under Fréchet)

Monte Carlo methods don’t generally work except in a trial-and-error approach

Precise distributions can’t express the target

Probability vs. intervals

- Probability theory
- Handles likelihoods and dependence well
- Has an inadequate model of ignorance
- Lying: saying more than you really know
- Interval analysis
- Handles epistemic uncertainty (ignorance) well
- Inadequately models frequency and dependence
- Cowardice: saying less than you know

Probability bounds analysis

- Generalizes them to escape the limits of each
- Makes verified calculations about probabilities
- Using whatever knowledge is available
- Without requiring unjustified assumptions
- Well developed methodology
- But plenty of interesting and important questions remain for study

Diverse applications

- Human health risk analyses
- Conservation biology extinction/reintroduction
- Wildlife contaminant exposure analyses
- Chemostat dynamics
- Global warming forecasts
- Design of spacecraft
- Safety of engineered systems (e.g., bridges)

What p-boxes can’t do

- Give best-possible bounds on non-tail risks
- Conveniently get best-possible bounds when dependencies are subtle
- Show what’s most likely within the box

Lev Ginzburg

Rüdiger Kuhn

Vladik Kreinovich

David Myers

Bill Oberkampf

Janos Hajagos

Dan Berleant

Chris Paredis

Mark Stadtherr

NIH

Sandia National Labs

NASA

Applied Biomathematics

Electric Power Research Institute

AcknowledgmentsResearch topics

- Incorporation of ancillary information
- Propagation through black boxes
- Handling subtle dependencies
- Computing non-tail risks
- Combination with fuzzy numbers
- Decision theory
- Info-gap models
- Finite-element models
- ODEs and PDEs

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