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Verified computation with probabilities

Verified computation with probabilities. This presentation is animated. (Press F5 to display.). Scott Ferson, scott@ramas.com Applied Biomathematics Scientific Computing, Computer Arithmetic and Verified Numerical Computations El Paso, Texas, 3 October 2008. Perspective.

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Verified computation with probabilities

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  1. Verified computation with probabilities This presentation is animated. (Press F5 to display.) Scott Ferson, scott@ramas.com Applied Biomathematics Scientific Computing, Computer Arithmetic and Verified Numerical Computations El Paso, Texas, 3 October 2008

  2. 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)

  3. 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

  4. 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

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

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

  7. They must be treated differently • Variability is randomness Needs probability theory • Incertitude is ignorance Needs interval analysis • Imprecise probabilities can do both at once

  8. 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 $

  9. Probabilistic logic

  10. 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.

  11. Interval arithmetic for probabilities x + y = [x1 + y1, x2 + y2] xy = [x1 y2, x2 y1] xy = [x1 y1, x2 y2] xy = [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]

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

  13. 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]

  14. Stochastic dependence Independence Probabilities are areas in the Venn diagrams

  15. 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)

  16. }certain uncertain Fréchet examples Examples: 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]

  17. Switch S1 Relay K2 Relay K1 Timer relay R Motor Pressure switch S From Pump reservoir Pressure tank T Outlet valve Example: pump system What’s the risk the tank ruptures under pumping?

  18. R E5 or K2 E4 K1 or E2 or S1 E1 E3 or and T S Fault tree Boolean expression of “tank rupturing under pressure” E1 Unsure about the probabilities and their dependencies

  19. Results Points, independence Mixed dependencies Fréchet Intervals and Fréchet 105 104 103 Probability of tank rupturing under pumping

  20. 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

  21. Probabilistic arithmetic

  22. 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

  23. 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.

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 1 1 0 0 10 20 30 40 10 20 30 Uncertain numbers Probability distribution Probability box Interval 1 Cumulative probability 0 0 10 20 30 40

  32. 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?

  33. A+B[4,12] 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 product A = 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

  34. A+B under independence 1.00 0.75 0.50 Cumulative probability 0.25 0.00 15 0 3 6 9 12 18 A+B

  35. Generalization of methods • When inputs are distributions, the answers conform with probability theory • When inputs are intervals, it agrees with interval analysis

  36. Where do we get p-boxes? • Assumption • Modeling • Robust Bayes analysis • Constraint propagation • Data with incertitude • Measurement error • Sampling error • Censoring

  37. A tale of two data sets Skinny (n = 6) Puffy (n = 9) 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

  38. 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

  39. 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.

  40. Constraint propagation (lognormal with interval parameters)

  41. 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

  42. 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).

  43. 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)

  44. = 1 - CDF Inputs 1 1 Exceedance risk EF IR 0 0 0 10 20 30 40 50 60 0 200 400 600 meals per year grams per meal 1 1 males females C BW 0 0 0 100 200 300 0 10 20 pounds mg per kg

  45. 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

  46. 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

  47. Stochastic dependence

  48. 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?

  49. 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

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