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Bayes and robust Bayes

9 October 2007, Stony Brook University, MAR 550, Challenger 165

• Introduction • Bayes’ rule • Distributions – Conjugate pairs • Model uncertainty • Updating • Multinomial sampling • Robust Bayes • Exercises

Outline

– Getting priors and specifying likelihoods • Advantages and disadvantages – Bounds on cumulatives, density-ratio bounds • Imprecise Dirichlet model (IDM) – Walley’s marbles, Laplace’s sunrises, (fault trees) • Conclusions and synopsis

Bayesians are like snowflakes

• Huge diversity among practitioners • Most, but not all, are subjectivists • Most, but not all, regard all statistical problems as part of decision theory • Most, but not all, believe that precise probabilities should and can be elicited • Most but, amazingly, not all employ Bayes’ rule to obtain results

Bayesian domains and concerns

• Statistics • Decision analysis • Risk analysis • Science itself • Updating and aggregation, rarely convolutions • Disparate lines of evidence, data, judgments • Little or no data

Derivation of Bayes’ rule P(

&

) = P(

|

) P(

) = P(

|

) P(

)

=

) P(

)

/ P(

)

A A

B B

Disease example

Prevalence in the general population is 0.01% Tests of diseased people are positive 99.9% of the time Test of healthy people are negative 99.99% of the time

Given that you’ve tested positive, what’s the chance that you actually have the disease?

Almost all doctors say 99% or greater, but the true answer is only about 50%.

Apply Bayes’ rule

P(disease)  P(positive | disease) P(disease | positive) = —————————————— P(positive) where the normalization factor P(positive) is computed as the sum P(disease)  P(positive | disease) + (1  P(disease) )  (1  P(negative | healthy)) prevalence = 0.01% = P(disease) sensitivity = 99.9% = P(positive | disease) specificity = 99.99% = P(negative | healthy) prevalence  sensitivity /( prevalence  sensitivity +(1  prevalence )  (1  specificity)) = 0.01%  99.9% / ( 0.01%  99.9% + (1  0.01% )  (1  99.99%)) = 0.4998

To see why this is so…

• Imagine 10,000 people without any risk factors • The prevalence of 0.01% suggests about one person from this group has the disease • The sensitivity of the test (99.9%) says that this person will test positive almost surely • The specificity of 99.99% suggests that, of the 9,999 people who do not have the disease, another will also test positive • Thus, we’d expect two people to test positive for the disease,

but only one of them actually has it

Paternity probability from blood tests

• • • • • Mother has type O blood; girl has type B blood (she got the B allele from her father) An alleged father has type AB so

could

have been the donor of the B allele P(B | F) = probability the girl would inherit the B allele if the man were her father P(B | not F) = probability she could inherit the B allele from another man P(F) = prior probability the man is the father of the girl P(F | B) = probability the man is her father given that they share the B allele Genetics says P(B | F) = ½ Background frequency of the B allele is 0.09, so P(B | not F) = 0.09

Prior unknown, so set P(F) = ½ P(F)  P(F) P(B|F)+(1   P(B | F) P(F) )  P(B|not F) ½  ½ ½  + (1  ½ ½ )  0.09

So is he the babydaddy?

Waiting for a bus

• • • • You got to the bus stop early, where the bus is now 10 minutes late. It might not come at all, and the next one is in two hours. You might need to walk. 90% of buses do their rounds, and 80% of those that do are less than 10 min late 10% chance that bus won’t show up at all Given that it’s already 10 minutes late, what’s the probability the bus will come?

A decision (which is where Bayesians really want to be) about whether to continue to wait or start walking would consider the costs of missing your appointment and how tired you are.

P(

|

) versus P(

|

)

• Not interchangeable – No matter how convenient that’d be • Error makers – Stupid people – Smart people (Laplace, de Morgan) invented Bayesian inference wrote the first treatise on probability in English • Bayes’ rule converts one to the other

Bayes’ rule on distributions posterior



prior



likelihood

posterior (normalized) likelihood prior

Density of normal(  ,

Example

)  2 1 πσ 2 exp    

 2 σ .

 ) is μ  2 2   • One observation

= 13 drawn from normal( q ,  2) • Prior for q • Multiply: is normal(6,1): • Likelihood of normal( q ,  2): P (

P ( θ |

)  P ( θ ) P (

| θ ) P ( θ )  1 exp    θ  6  2 | θ )   1 2 π exp 4 π exp    3 θ 2    2  4 

( 12   4 2 θ  2

) θ      • Posterior  exp((  3 q 2 +50 q )/4) = normal(8⅓,  ⅔ )

Answer

0.5

1 0 0 0 5 10 q 15 20

Interpretation of the posterior

• Estimates value of θ (a distribution mean) – Shrinkage (pull towards the prior’s mean) • Uncertainty expressed as distributions • Peaky density (steeper CDF) means

surer

• Posterior is more certain than the prior – Even though the datum is surprising to the prior

0.5

Credible intervals

1 0 0 0 5 10 q 90% credible interval (Bayesian confidence interval) 15 20

Bayesian confidence intervals

• Express probability the value is in an interval – No complex Neyman-Pearson interpretation • No guarantee about performance – Performance “90% of the time…” doesn’t apply – Don’t call them “confidence intervals” Prosecutor’s fallacy • Asymmetric intervals – Narrowest (mode) – Symmetric (median) – Mean-centered

Where does the likelihood come from?

• Mechanism that generated the data • ‘Inverse’ probability “Libby is Billy spelled sideways, sorta” – Density: probability of

as a function of – Likelihood: probability of

x x

, given θ as a function of θ, given

• Not a distribution (doesn’t integrate to one) – Equivalence class of functions, can scale however we want • Appropriateness of a probability model that justifies a particular likelihood function is always at best a matter of opinion (Edwards 1972, page 51)

Where do priors come from?

• Whatever

You

say • Maximum entropy • EDFs or empirical Bayes (Newman 2000) – Data used to estimate prior and also likelihood • Uniformative or reference priors – No prior can represent ignorance (Bernardo 1979) – Just a way to minimize the prior’s influence to discover what the data themselves are saying about the posterior • Conjugate pairs

Conjugate pairings

Likelihood

bernoulli bernoulli ( q ) poisson ( q ( ) q ) Likelihood parameters other than θ are assumed to be known normal ( q ,

) exponential ( q ) binomial (

, q ) uniform (0, q )

Prior

uniform (0, 1) beta (  , gamma (  ,  ) normal gamma beta (  ,  (  ( )  )  , ,   ) ) negbinomial normal ( q ,

) exponential ( q ) pareto (

) beta (  ,  ) gamma (  ,  ) invgamma ( multinomial (

, q

) dirichlet (

t j

)  ,  )

Posterior

beta (1+ 

x i

,1+



x i

) beta (  + 

x i

,  +



x i

) gamma (  + 

x i

,  +

) normal ((  /  2 + 

x i /s

2 )/

, 1/ 

) normal

=1/  2 +

2 gamma (  +

,  + 

x i

) beta pareto beta gamma inversegamma dirichlet (

n+s

x j

st j

)/(

n+s

))

= 

x j

 {1,...,

}

Model uncertainty

Bayesian model averaging

(Draper 1995) • Similar to the probabilistic mixture • Updates prior probabilities to get weights • Takes account of available data

Bayesian model averaging

• Assume it’s actually the first model • Compute probability distribution under that model • Read off probability density of observed data – Product if multiple data; it’s the likelihood for that model • Repeat above steps for each model • Compute posterior  prior  likelihood – This gives the Bayes’ factors • Use the posteriors as weights for the mixture

Example

Either

(

) =

Plus (

) = A + B or

(

) =

Times (

) = A  B where

~ normal(0,1),

~ normal(5, 1) Single datum: Priors:

(

) = 7.59

Plus ) = 0.6, P(

Times ) = 0.4

0.3

0.2

0.1

Likelihoods

Times

Plus

7.59

0 -10 0

f ( A , B )

10 20

Plus (

) ~

A+B f

Times (

) ~

 ~ normal(5,  2)

~ normal(0,  26) L L Plus (7.59) = 0.05273

Times (7.59) = 0.02584

R: dnorm(7.59,5,sqrt(2)) Excel: =NORMDIST(7.59, 5, SQRT(2), FALSE)

Bayes’ factors

Posterior probabilities for the two models posterior  prior  likelihood Plus 0.6

 0.05273

/( 0.6

 0.05273

+ 0.4

 0.02584

)=0.7538 Times normalization factor 0.4

 0.02584

/( 0.6

 0.05273

+ 0.4

 0.02584

)=0.2462

These are the weights for the mixture distribution

0.3

0.2

0.1

0 -20 -15 -10

Times

-5 1 0.75

0.5

0.25

0 -20 -15 -10 -5

Times

f (A ,B )

Plus Plus

10 10 15 20 15 20 25 25

Bayesian model averaging

• Produces single distribution as answer • Can account for differential prior credibility • Takes account of available data • Must enumerate all possible models • May be computationally challenging • Confounds variability and incertitude • Averages together incompatible theories • Underestimates tail risks

Updating from relational constraints

Updating with only constraints

Suppose

 Any triples

W H





A W H A

   [23, 33] [112, 150] [2000, 3200] get excluded with a likelihood of zero Mass concentrated onto manifold of feasible combinations Likelihood

(





)   (





) Prior Pr(

) 

(

 [ 23 , 33 ]) 33  23 

(

 [ 112 , 150 ]) 150  112 

(

 [ 2000 , 3200 ]) 3200  2000 Posterior

(





)   (





)  Pr(

)  (  ) Dirac delta function

(  ) indicator function

1 0 1 0 20 1

120

0 1500 30 2500 140

Original interval ranges Interval constraint analysis

40 160 3500

Multinomial sampling

• Bag of marbles colored red , blue , or green • Bag is shaken and marbles are randomly drawn • Its color is recorded and it’s put back in the bag • What’s the chance the next marble is red ?

– If we knew how many marbles of each color are in the bag, we could compute the probability – If we haven’t seen inside the bag, what can we say given the colors of marbles drawn so far?

Multinomial sampling

• Consider

random observations (sampled with replacement) • As each marble is drawn, we record its color 

  = { red , blue , green } • This is just the standard multinomial with P( 

) = q

, where

= 1 , 2 , 3

0  q

, and q

= 1

Bayesian analysis

• Given

n j

= number of marbles colored 

, where 

n j

= N, the likelihood function is

j k

  1 θ

n j j

• A Dirichlet prior is conjugate

j k

  1 θ

st j j

 1 Dirichlet(

t j

) where 0 <

, 0 <

t j

< 1, 

t j

= 1 (

t j

is mean for q

)

Solution

j k

  1 θ

n j j



st j

 1

j k

  1 θ

n j j



st j

 1

, (

n j

st j

)/(

)) • Bayes-Laplace prior is uniform Dirichlet(

=3,

t j

=1/3) • Posterior for this prior is Dirichlet(

+3, (

n j

+1)/(

+3)) • The vector of means for a Dirichlet(

t j

) is

t j

, so the posterior expected value of q

is (

n j

+1)/(

+3) • Suppose we had 7 draws: blue , green , blue , blue , red , green , red • Predictive probability the next marble is red is (2+1)/(7+3) = 0.3

• Compare to the observed frequency of red marbles 2/7  0.2857

• Before any data,

=0, the probability is (0+1)/(0+3) = 0.333

shrinkage

Beyond colored marbles

• Multinomial sampling occurs in many risk analytic and statistical problems with random iid trials with finitely many possible outcomes • Analysis, and solutions, for all such problems are just the same as for the marbles

Advantages of Bayesian statistics

• Ascendant – Fashionable and iconoclastic – Answers easier since computer revolution • Decision making – Integrated into Bayesian framework (Berry 1996; 1997) – Frequentists don’t even balance costs of Type II errors • Rationality – Maximizes expected utility, avoids sure loss (Dutch book) – Rational agents seeing same info eventually agree (Dawid) • Subjective information – Legitimizes and takes account of prior beliefs • Working without any empirical data • Naturalness • Data mining

Naturalness

Bayesian Frequentist • Any question (OJ’s guilt, God) • Intuitive credibility intervals • Distributions represent uncertainty • Can use all information, even if it’s from outside the experiment • Likelihood calculated only from the data actually seen • Posterior depends on data only through the likelihood • Free to stop whenever so can enjoy serendipity, windfalls • Can use scientific stopping rule • Can only study repeatable events • Torturous confidence intervals • No parameter distributions (fixed) • Can only take account of what’s observed during the experiment • Consider probabilities of hypothetical data

never observed

• Probability measures depend on the experimental design • Must follow experimental design to compute

• Can’t use scientific stopping rule

Data mining

Bayesian Frequentist • Perfectly okay • Doesn’t affect results • Scientifically improper (can’t compute

) • Scientists free to peek at data before and during • Data contaminates the scientist’s mind • Can have many more parameters than samples • Can estimate lots of parameters at once • Bad to create a model with many parameters • Bad to estimate more that a few parameters at a time

Problems and technical issues

• Controversial • Needs priors, sampling model/likelihood • Computational difficulty: It can be very hard to derive answers analytically • Zero preservation: If either the likelihood or the prior is zero for a given q , then the posterior is zero • Subjectivity required: Beliefs needed for priors may be inconsistent with public policy/decision making • Inadequate model of ignorance: Doesn’t distinguish between incertitude and equiprobability

Fear of the prior

• Many Bayesians are reluctant to employ subjective priors in scientific work (Yang and Berger 1998) • Hamming (1991, page 298) wrote bluntly “

If the prior distribution, at which I am frankly guessing, has little or no effect on the result, then why bother; and if it has a large effect, then since I do not know what I am doing how would I dare act on the conclusions drawn?

” [emphasis in the original]

Everything’s in the prior

1 0.8

0.6

0.4

0.2

0 0 The strength of the evidence is reflected in the function’s curvature 0.2

0.4

0.6

0.8

Prior probability of paternity

Groups

• Bayesian theory does not work for groups – Rationality inconsistent with democratic process • Some even say “groups can’t make decisions” – Despite the fact they come to verdicts, conclusions, findings, choices, … Geo. Hazelrigg, NSF • Must imagine a personalist “You” – Teams, agencies, collaborators, companies, clients – Reviewers, peers

Is coherence enough?

• Bayesians say their reasoning is rational and their conclusions sound if their beliefs and inferences are

coherent

(consistent to avoid sure loss) • Standard Bayesian inferences are coherent, but coherence is only a

minimal

requirement of rationality (Walley 1991, 396f) • Beliefs ought to also

conform to evidence

, which Bayesian inferences rarely do (ibid.) • Abandoning objectivity forsakes the essential connection to evidence and accountability in the real world • Evidence comes from measurement and objectivity

Robust Bayes

Robustness

• An answer is robust if it does not depend sensitively on the assumptions and calculation inputs on which it’s based • Related to important ideas in other areas of statistics such as robust and resistance estimators • Robust Bayes analysis, also called Bayesian sensitivity analysis, investigates the robustness of answers

Bayes’ rule made safe for uncertainty

sensitivity  prevalence prevalence + (1   sensitivity specificity)  (1  prevalence) 1 1    1 prevalence   1   1 sensitivit  y specificit y  1 / (1 + ((1/prevalence  1)  (1  specificity)) / sensitivity)

sensitivity = 99.9% specificity = 99.99% prevalence = [0.005, 0.015]% ( [0.005,0.015]% [0.005, 0.015]% 99.9% + (1   99.9% [0.005,0.015]% )  (1  99.99%)) = [19.99, 99.95]% 1 / (1 + ((1/ [0.005, 0.015]%  1)  (1  99.99%)) / 99.9% ) = [33.31, 59.98]%

Uncertainty about the prior

class of prior distribution  class of posteriors posteriors priors likelihood

Uncertainty about the likelihood

class of likelihood functions  class of posteriors posteriors likelihoods prior

Uncertainty about both

posteriors priors likelihoods -5 0 5 q 10 15 20

Moments of the normals

Prior: normal(μ,σ) μ = [1, 6] σ = [2.5, 6] Likelihood: normal(

)

= [5,13]

= [4, 5] Posterior: normal(θ 1 ,

1 ) mean = ( μ / σ 2 +

2 )/(1/ σ 2 + 1/

2 ) = [2.28, 10.35] stdev = 1/  (1/ σ 2 + 1/

2 ) = [1.24, 1.66] Conjugacy facilitates the calculation, but note that σ 2 and s 2 are repeated so be careful evaluating the mean

Robust Bayes can make a p-box

Priors Likelihoods -5 0 5 10 class of priors, class of likelihoods  15 20 class of posteriors

Uncertainty about decisions

class of probability models  class of utility functions  class of decisions class of decisions If you end up with a single decision, you’re in like flint.

If the class of decisions is large and diverse, then you should be somewhat tentative about making any decision.

Bayesian dogma of ideal precision

• Robust Bayes is plainly inconsistent with the Bayesian idea that uncertainty should be measured by a

single

additive probability measure and values should always be measured by a

precise

utility function.

• Some Bayesians justify it as a convenience • Others suggest it accounts for uncertainty beyond probability theory

Desirable properties of the class

• Easy to understand and elicit • Easy to compute with • Sufficiently big to reflect one’s uncertainty • Generalization to multiple dimensions (Berger 1994) • Near ignorance (vacuous, but not in

all

ways)

Defining classes of priors

• Parametric conjugate families • Parametric but non-conjugate families • • Density-ratio (bounded density distributions)  -contamination, mixture, quantile classes, etc.

• Bounds on cumulatives (trivializes results)

0 1

Parametric class

Parameters can be chosen to make the class wide , but it will often still be too sparse in the sense that the distributions it admits are too similar to reflect what we think might be possible.

Bounded cumulative distributions

• Shouldering: Bounds on CDFs (p-boxes) admit density distributions that may be zero anywhere.

• Renormalization: Posteriors have unit area.

• Zero preservation: Posterior’s range is inside the intersection of the ranges of the prior and likelihood.

• Triviality: Together, these imply that all you can ever conclude about the posterior is its range

Shouldering and zero preservation

A shoulder on a CDF corresponds to zero density.

Because the posterior can be zero almost everywhere, all the mass could be pushed to any point.

Likelihood Prior  Thus posterior is zero everywhere 

Result is trivial

posterior likelihood 0 prior

Bounded densities

0.6

0.5

0.4

0.3

0.2

0.1

0 0 5 10 15 q 20 25 30 35 q Area below prior’s lower bound is smaller than 1; area below upper bound is larger than 1. There are no restrictions on the likelihood bounds.

Un-normalized posterior bounds

How can we normalize?

Normalized posterior bounds

Normalization trick: divide the lower bound by the area of the upper and vice versa

Bounds on posterior densities

• Imply bounds on cumulative tail risks • Imply bounds on moments • Imply bounds on the mode bounds on mode

Take-home lessons

• Many ways depending on uncertainty • Parametric uncertainty can be too sparse • Yet too much generality trivializes the result • Often not so easy to do the calculations

Probability of an event

• Imagine a gamble that pays one dollar if an event occurs (but nothing otherwise) – How much would you pay to buy this gamble?

– How much would you be willing to sell it for?

• Probability theory requires the same price for both – By asserting the probability of the event, you agree to buy any such gamble offered for this amount or less,

and

to sell the same gamble for any amount less than or equal to this ‘fair’ price…and to do so for

every

event!

• IP just says, sometimes, your highest buying price might be smaller than your lowest selling price

Imprecise Dirichlet model

Walley’s (1996) bag of marbles

• What’s the probability of drawing a red marble if you’ve never seen inside the bag?

• First, consider a model with

random (iid) observations (sampled with replacement). As each of the

marbles is drawn, we can see its color 

  = {  1 , …, 

} • This is just the standard multinomial with P( 

) = q

, where

=1,…,

0  q

, and q

= 1

Bayesian analysis

• Given

n j



n j

= number of marbles that were colored  = N, the likelihood function is proportional to

n j j j

  1 • A Dirichlet prior is convenient (it’s a multivariate generalization of a beta), with density proportional to

j k

  1 θ

st j j

 1 Dirichlet(

t j

) where 0 <

, 0 <

t j

< 1, 

t j

= 1 (

t j

is mean for q

)

The answer seems easy

• Posterior is also Dirichlet, in density proportional to

j k

  1 θ

n j j



st j

 1

j k

  1 θ

n j j



st j

 1

, (

n j

st j

)/(

)) • But what if you don’t even know how many colors

there are in the bag? – Letting  letting  = {red, not red} gives a different answer from = {blue, green, yellow, red} –

Lots

of ways to form  , lots of different answers • Peeking (using the data to form  ) is incoherent

Walley’s “Imprecise Dirichlet model”

• Likelihood is multinomial, as before • Prior is the class of

all

Dirichlet distributions M 0

= {

= Dirichlet(

t j

)} for a fixed

> 0 that does not depend on  • The corresponding posterior class is M

N s

= {

= Dirichlet(

, (

n j

st j

)/(

N+s

))}

Predictive probabilities

• Under this posterior, the predictive probability that the next observation will be color 

is the posterior mean of q

which is (

n j

st j

)/(

) • By extremizing this quantity on

t j

, we get posterior upper and lower probabilities P = _

n j

P = (

n j

/ ( +

N s

) / (

)

) ( (

t t j j

  0) 1) • Easily generalizes to other events

What if there’s no or little data?

• Before any observations, P = [P,  P] = [0,1] .

• This is appropriately vacuous (duh!) • The width of the interval decreases as sample size

increases, but

doesn’t depend

on space  at all

or the sample • In the limit

  , the Walley, Bayesian and frequentist answers all converge to

n j

What should

be?

• Determines the prior’s influence on posterior If

 0, interval  point   , interval width  1 (overconfident) (ineducable) Using

= 1 or 2 might be reasonable • Letting it depend on

would be incoherent • Letting it vary would be incoherent • Different values of

give

consistent

inferences that are nested (unlike Bayesian inconsistency)

Bayesian inconsistency

• Different priors • Bayes (

n j

• Jeffreys (1946, 1961) (

n j

• Perks (1947) (

n j

• Haldane (1948)

n j

+ ½ + m /

) / (

m /k N

) / ( ) / ( +

N N k

) + ½ + 1)

) • Different models for  • Bayes [0, 1] • Jeffreys (1946, 1961) [0, 1] • Perks (1947) • Haldane (1948) [

n j n j

/ / (

N N

+ 1), (

n j

+ 1) / (

+ 1)] IDM

= 

= 1

= 0

 2

What if the event has never happened?

• If

n j

= 0, then P = 0 and  P =

) – No reason to bet on such an event at any odds – But should be willing to bet

against

and shorter odds as

increases it at shorter • Contrast this to Laplace’s famous calculation of the probability the sun wouldn’t rise the next day via his “rule of succession” (

n j

+1)/(

+2) = 1/1826215

Conclusions and synopsis

Bayesian analysis

How?

decide on your “prior” belief, P(model) compute likelihood function, P(data | model), as a function of the model multiply to get posterior belief, P(model | data) today’s posterior is tomorrow’s prior

Why?

“rational” acknowledges inescapability of subjectivity updating rule supports the coherent accumulation of knowledge

Why not?

tolerates (requires) subjectivity Bayesian rationality does not extend to group decisions inadequate model of ignorance (confounds with equiprobability)

Robust Bayes (sensitivity analysis)

How?

use a class of distributions to represent uncertainty about the prior use a class of functions to represent uncertainty about the likelihood get class of posteriors by applying Bayes’ rule to every possible combination

Why?

accounts for the analyst’s doubts about required inputs can be cheaper and easier than insisting on precise inputs consistent with robust statistics (doesn’t require analyst’s omniscience) expresses the reasonableness of the posterior

Why not?

can be computationally difficult does not obey Bayesian dogma of ideal precision still has zero preservation problem

• • • • • • • • • • • • • •

References

Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances.

Philosophical Transactions of the Royal Society

53: 370-418. Reprinted in 1958 in

Biometrika

45: 293-315.

Berger, J.O. (1985).

Statistical Decision Theory and Bayesian Analysis

. Springer-Verlag, New York.

Bernardo, J.M. (1979). Reference posterior distributions for Bayesian inference.

Journal of the Royal Statistical Society, Series B

41: 113-147 (with discussion).

Berry, D.A. (1996).

Statistics: A Bayesian Perspective.

Duxbury Press, Belmont, California.

Berry, D.A. (1997). Bayesian Statistics. Institute of Statistics and Decisions Sciences, Duke University, http://www.pnl.gov/Bayesian/Berry/ .

Dawid, A.P. (1982). Intersubjective statistical models, In

Exchangeability in Probability and Statistics

, edited by G. Koch and F. Spizzichino, North-Holland, Amsterdam.

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Likelihood

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Exercises

1. Check the math in the example on slide 13.

2. Sketch [P,  P] for { -,B,G,B,B,G,R,G,…,(35/100)R,…(341/1000)R } .

3. What can be said about the posterior if the prior is uniform(

) and the likelihood function is an a non zero constant for values of q between

and

and zero elsewhere? Sketch the answer for

a d c

= 21. What is the posterior if  [4,6],

 [20,22]?

 = 2,

[1,3],

= 14,

 = 5, [11,17], 4. How could you do Bayesian backcalculation?

5. What is the statistical ensemble corresponding to a prior probability distribution?

End

Missing failure data

Failure data is often incomplete

• We know how many times C failed, but don’t whether it was A or B or both that made it fail • We know from component testing how often A failed, but not whether B would have too A C or B

• Traditional Bayesian approach (Vesely, et al.) – Makes assumptions about missing data – Requires independence • IDM can relax these assumptions – Computes chance of C, A or B failing at next test – Uses only failure data

C ,

A ,

B ,

B – And hardly any assumptions