bayesian inference and decision theoryseor.vse.gmu.edu/~klaskey/syst664/bayes_unit1.pdf · unit 1...

George Mason University

Unit 1 - 1 -

Department of Systems Engineering and Operations Research

©Kathryn Blackmond Laskey Spring 2019

Bayesian Inference andDecision Theory

Unit 1: A Brief Tour of Bayesian Inference and Decision Theory

Instructor: Kathryn Blackmond LaskeyRoom 2214 ENGR

(703) 993-1644Office Hours: Wednesday 3:30-5:30 PM, or by

appointment

Spring 2019


Unit 1 - 2 -



What this Course is About

• You will learn a way of thinking about problems of inference and decision-making under uncertainty

• You will learn to construct mathematical models for inference and decision problems

• You will learn how to apply these models to draw inferences from data and to make decisions

• These methods are based on Bayesian Decision Theory, a formal theory for rational inference and decision making


Unit 1 - 3 -



Logistics• Web site

– http://seor.vse.gmu.edu/~klaskey/SYST664/SYST664.html– Blackboard site: http://mymason.gmu.edu

• Textbook and Software– Hoff, A First Course in Bayesian Statistical Methods, Springer, 2009

» Free softcopy from Mason library– Other recommended texts on course web site– We will use R, a free open-source statistical computing environment: http://www.r-

project.org/. R code for many textbook examples is on author�s web site– Late in the semester we will use JAGS, an open-source package for Markov Chain

Monte Carlo simulation (interfaces with R): http://mcmc-jags.sourceforge.net/• Requirements

– Regular assignments (30%): can be handed in on paper or through Blackboard– Take-home midterm (35%) and final (35%)

• Office hours– Official office hours are 3:30-5:30 PM Wednesdays– I respond to questions by email and am available by appointment

• Course delivery– 4:30-7:10 Mondays, IN 209 or online via Blackboard Collaborate; all classes recorded

• Policies and Resources– Academic integrity policy– Read the policies and resources section of the syllabus

http://seor.vse.gmu.edu/~klaskey/SYST664/SYST664.html

http://www.r-project.org/


Unit 1 - 4 -



Course Outline

Unit 1: A Brief Tour of Bayesian Inference and Decision TheoryUnit 2: Random Variables, Parametric Models, and Inference

from ObservationUnit 3: Statistical Models with a Single ParameterUnit 4: Monte Carlo Approximation Unit 5: The Normal ModelUnit 6: Gibbs SamplingUnit 7: Hierarchical Bayesian ModelsUnit 8: Bayesian Regression and Analysis of VarianceUnit 9: Additional Monte Carlo MethodsUnit 10: Hypothesis Tests, Bayes Factors, and Bayesian Model

Averaging

(Later units are subject to change)


Unit 1 - 5 -



Learning Objectives for Unit 1

• Describe the elements of a decision model• Refresh knowledge of probability• Apply Bayes rule for simple inference problems and interpret

the results • Use a graph to express conditional independence among

uncertain quantities• Explain why Bayesians believe inference cannot be separated

from decision making• Compare Bayesian and frequentist philosophies of statistical

inference• Compute and interpret the expected value of information (VOI)

for a decision problem with an option to collect information• Download, install and use R statistical software


Unit 1 - 6 -



Bayesian Inference• Bayesians use probability to quantify rational degrees of belief• Bayesians view inference as belief dynamics

– Begin with prior beliefs– Use evidence to update prior beliefs to posterior beliefs– Posterior beliefs become prior beliefs for future evidence

• Inference problems are usually embedded in decision problems

• We will learn to buildmodels of inferenceand decision problems

• “All models are wrongbut some are useful”– George Box


Unit 1 - 7 -



Decision Theory• Decision theory is a formal theory of decision making under

uncertainty• A decision problem consists of:

– Possible actions: {a}aÎA

– States of the world (usually uncertain): {s}sÎS

– Possible consequences: {c}cÎC (of action in a state)

• Question: What is the best action?• Answer (according to decision theory):

– Measure �goodness� of consequences with a utility function u(c)– Measure likelihood of states with probability distribution p(s) – Best action with respect to model maximizes expected utility:

• Caveat emptor:– How good it is for you depends on fidelity of model to your beliefs

and preferences

€

a *= argmaxa

E[u(c) |a]{ } For brevity, we may write E[u(a)] for E[u(c) | a]


Unit 1 - 8 -



Illustrative Example:Highly Oversimplified Decision Problem

• Decision problem: Should patient be treated for disease?– We suspect she may have disease but do not know– Without treatment the disease will lead to long illness– Treatment has unpleasant side effects

• Decision model:– Actions: aT (treat) and aN (don�t treat)– States of world: sD (disease now) and sW (well now)– Consequences: cWN (well shortly, no side effects), cWS (well shortly, side

effects), cDN (disease for long time, no side effects)– Probabilities and Utilities:

» P(sD) = 0.3» u(cWN) = 100, u(cWS) = 90; u(cDN) = 0

• Expected utility:– Treat: .3´90 + .7´90 = 90– Don't treat: .3´0 + .7´100 = 70

• Best action is aT (treat patient)

u(cWN) 100

u(cDN) 0

u(cWS) 90

P(sD) = 0.3 P(sW) = 0.7

EU(aN) 70


Unit 1 - 9 -



Sensitivity Analysis: Optimal Decision as Function of Sickness Probability• Expected utility of not treating depends on the probability p = P(sD)

of having the disease– E[U|aT] = 90– E[U|aN] = 0p + 100(1-p) = 100(1 – p)– We should treat if p > 0.1, don�t treat if p < 0.1

• If we are unsure about the value of p we may want to explore how the optimal decision changes as we vary p

• We may want to gatherinformation to refine ourestimate of p, especially if our estimate is near the crossover point

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exp

ect

ed

Uti

lity

P(Sick)

Expected Utility for Disease Problem

E[U(Treat)]

E[U(NoTreat)]

Expected gain from treatment at p = 0.3

No dependence on p

Decreases as p increases


Unit 1 - 10 -



Why be a Bayesian?• Arguments from theory

– A coherent decision maker uses probability to represent uncertainty, uses utility to represent value, and maximizes expected utility

– If you are not coherent then someone can make "Dutch book" on you (turn you into a "money pump")

• Pragmatic arguments– Decision theory provides a useful and principled methodology for modeling

problems of inference, decision and learning from experience– Engineering tradeoffs between accuracy, complexity and cost can be analyzed

and evaluated– Both empirical data and informed engineering judgment can be explicitly

represented and incorporated into a model– Bayesian methods can handle small, moderate and large sample sizes; small,

moderate and large numbers of parameters– With other approaches it may be more difficult to understand why you got the

results you did and how to improve your model• Arguments from experience

– Successful applications– Success attributed to decision theoretic technology

• Caution:– Uncritical application of �cookbook� methods can lead to disaster!– Good modelers iteratively assess, check and revise assumptions


Unit 1 - 11 -



• A probability distribution is a function P(×) applied to sets such that:

– P(A) ≥ 0 for all subsets A of the universal set W– P(W) = 1

– If AiÇAj = Æ then P(A1ÈA2È…) = P(A1) + P(A2)+ …

• From this basic definition we can deduce other identities of probability theory, e.g.:

– P(Æ) = 0

– P(A) ≤ 1 for all subsets A of W– P(AÈB) = P(A) + P(B) - P(AÇB) for any A and B

– If AiÇAj = Æ and A=A1ÈA2È…, then P(AÇB) = Σi P(B|Ai)P(Ai)

• The conditional probability of A given B for any two sets A and B is defined as a number P(A|B) satisfying:

– P(A|B)P(B) = P(AÇB)

– If P(B)≠0 this is equivalent to the traditional formula:

– A is independent of B if P(A|B) = P(A)

Review: Probability Basics

P(A | B) = P(A∩ B)P(B)

Law of total

probability


Unit 1 - 12 -



Extending the Disease Example: Gathering Information

• We may be able to perform a test before deciding whether to

treat the patient

– Test has two outcomes: tP

(positive) and tN

(negative)

– Quality of test is characterized by two numbers:

» Sensitivity: Probability that test is positive if patient has disease

» Specificity: Probability that test is negative if patient does not have

disease

• Test characteristics:

– Sensitivity: P(tP

| sD

) = 0.95

– Specificity: P(tN

| sW

) = 0.85

• How does the model change if test results are available?

– Take test, observe outcome t– Revise prior beliefs P(s

D

) to obtain posterior beliefs P(sD

|t)– Re-compute optimal decision using P(s

D

|t)


Unit 1 - 13 -



Bayes Rule: The Law of Belief Dynamics• Objective: use evidence to update beliefs about a hypothesis

– H1, … Hn: exclusive and exhaustive hypotheses– E: evidence

• Procedure: apply Bayes Rule:

• Bayes Rule (odds likelihood form):

Terminology:» P(H) - The prior probability of H P(E|H) - The likelihood for E given H» P(E) - The predictive probability of E P(H|E) - The posterior probability of H given E

» - The likelihood ratio for H1 versus H2 - The prior odds ratio for H1 versus H2

• The posterior probability of H1 increases relative to H2 if the evidence is more likely given H1 than given H2

€

P(E |H1)P(E |H2)

€

P(H1)P(H2)

P(E)>0, P(H2)>0 P(H1 | E)P(H2 | E)

= P(E |H1)P(H1)P(E |H2 )P(H2 )

HiÇHj = Æ, Ω=H1ÈH2È…

P(E)>0

! "# $ = %('(∩*)%(*) = % *|'( %('()%(*) = % *|'( %('()

∑. % *|' % '


Unit 1 - 14 -



Disease Example with Test

• Review of Problem Ingredients:– P(sD) = 0.3 (prior probability of disease)– P(tP | sD) = 0.95; P(tN | sW) = 0.85 (sensitivity & specificity of test)– u(cWN) = 100, u(cWS) = 90; u(cDN) = 0 (utilities)

• If test is negative:– P(sD | tN) = (0.3 x 0.05)/(0.3 x 0.05 + 0.7 x 0.85) = 0.025– EU(aN | tN) = 0.025 ´ 0 + 0.975 ´ 100 = 97.5– EU(aT | tN) = 0.025 ´ 90 + 0.975 ´ 90 = 90– Best action is not to treat

• If test is positive:

– P(sD | tP) = (0.3 x 0.95)/(0.3 x 0.95 + 0.7 x 0.15) = 0.731– EU(aN | tP) = 0.731 ´ 0 + 0.269 ´ 100 = 26.9– EU(aT | tP) = 0.731 ´ 90 + 0.269 ´ 90 = 90– Best action is to treat

• Optimal policy is to treat if positive; don’t treat if negative– We will call this strategy aF (FollowTest)

EU(aN | tN)

97.5

EU(aN | tP) 26.9

EU(aT) = EU(aT|tP) = EU(aT|tP)

90

EU(aN) 70

P(!"|#$) = P(!"&#$)P(!"&#$)+ P(!'&#$)

P(!"|#)) = P(!"&#))P(!"&#))+ P(!'&#))


Unit 1 - 15 -



Expected Value of Sample Information

• Reminder of problem ingredients:– P(sD) = 0.3 (prior probability of disease)– P(tP | sD) = 0.95; P(tN | sW) = 0.85 (sensitivity & specificity of test)– u(cWN) = 100, u(cWS) = 90; u(cDN) = 0 (utilities)

• Probability test will be positive (law of total probability):– P(tP) = P(tP | sD) P(sD) + P(tP | sW) P(sW) = 0.95×0.3 + 0.15×0.7 = 0.39

• If test is positive we should treat, with EU(aT) = 90• If test is negative we should not treat, with EU(aN | tN) = 97.5• Expected utility of FollowTest strategy (treat if test is positive, otherwise

don’t):– EU(aF) = P(tP) × EU(aT | tP) + P(tN) EU(aN | tN)

= 0.39× 90 + (1-0.39) 97.5 = 94.575

• If we do not perform the test, our best action is to treat everyone and EU(aT) = 90

• Gain in expected utility from doing the test is 94.575 – 90 = 4.575• This is called the Expected Value of Sample Information (EVSI)


Unit 1 - 16 -



Expected Value of Perfect Information

• Expected Value of Perfect Information (EVPI) is increase in utility from perfect knowledge of an uncertain variable

– Suppose an oracle will tell us whether patient is sick» An oracle has Sensitivity = Specificity = 1

– 30% chance we discover she is sick and treat - utility 90– 70% chance we discover she is well and don’t treat - utility 100– Expected utility if we ask the oracle 0.3 x 90 + 0.7 x 100 = 97– EVPI = 97 - 90 = 7

• EVPI ≥ EVSI ≥ 0– In our example, EVPI = 7 > EVSI = 4.6

• EVPI = EVSI = 0 if the test will not change your decision


Unit 1 - 17 -



Should We Collect Information?• General Principle: Free information can never hurt

• To analyze decision of whether to collect information D on outcome variable V:

– Find maximum expected utility option if we don't collect information

– Compute its expected utility U0

– Find EVPI» For each possible value V=v, assume it is known to be the true outcome, find optimal

decision, calculate its expected utility, and multiply by the probability that V=v» Add these values and subtract from no-information expected utility to get EVPI

– Compare EVPI with cost of information

– If EVPI is too small in relation to cost then stop; otherwise, compute EVSI» For each possible result D=d of the experiment, find the maximum expected utility

action a(d) and its utility u(a(d))» For each outcome V=v and result D=d of the experiment, find the joint probability P(v,d)» Calculate the expected utility with information USI = ∑v,d P(v,d)u(a(d))

» Compute EVSI = USI – U0 (difference in expected utility between no-information decision and decision with information)

– Compare EVSI with cost of information

– Collect information if expected utility gain is greater than cost of information


Unit 1 - 18 -



• FollowTest strategy treats if test is positive and otherwise not

E[U|aF] = 0.95π´90 + .05π´0 + 0.15 (1-π)´90 + 0.85 (1-π)´100= 98.5 – 13π

World State

Probability Action Utility

Sick, Positive

.95π Treat 90

Sick, Negative

.05π NoTreat 0

Well, Positive

.15(1-π) Treat 90

Well, Negative

.85(1-π) NoTreat 100

Expected Utility of FollowTest Policyas Function of Prior Probability π = P(sD)

Before doing test, we think:• There are four possibilities for

disease status and test results. Their probabilities P(s)P(t | s) are shown in the table

• We treat if test is positive and don’t treat if test is negative, with utilities as shown in the last column.

• We multiply probability times utility for each world state and sum to get the expected utility of FollowTest


Unit 1 - 19 -



• FollowTest strategy treats if test is positive and otherwise not

E[U|aF] = P(sD|tP)P(tP)´90 + . P(sD|tN)P(tN)´0 + P(sW|tP)P(tP)´90 + P(sW|tN)P(tN)´100

= P(tP) [P(sD|tP)´90 + P(sW|tP)´90] + P(tN) [P(sD|tN)´0 + P(sW|tN)´100]

= P(tP) EU[aT | tP] + P(tN) EU[aN | tN]

World State

Probability Action Utility

Sick, Positive

P(sD|tP)P(tP) Treat 90

Sick, Negative

P(sD|tN)P(tN) NoTreat 0

Well, Positive

P(sW|tP)P(tP) Treat 90

Well, Negative

P(sW|tN)P(tN) NoTreat 100

Another Way to FindExpected Utility of FollowTest Policy

(Compare with Slide 15)

Before doing test, we think:• We are uncertain about the test

result: it will be positive with probability P(tP) and negative with probability P(tN).

• If it is positive our expected utility will be EU[aT | tP].

• If it is negative our expected utility will be EU[aN | tN].

• So our expected utility of following the test is P(tP) EU[aT|tP] + P(tN) EU[aN|tN]


Unit 1 - 20 -



• FollowTest: EU(aF) = 98.5 – 13π• AlwaysTreat: EU(aT) = 90

– FollowTest is better when 98.5 – 13π > 90 or π < 8.5/13 = 0.654

• NeverTreat: EU(aN) = 100(1 – π)– FollowTest is better when 98.5 - 13π > 100(1-π)

or π > 1.5/87 = 0.017

Strategy Regions

75

80

85

90

95

100

0 0.2 0.4 0.6 0.8 1

Exp

ecte

d U

tility

P(Sick)

Sensitivity Analysis for EVSI with Costless Test

E[U(Treat)]

E[U(NoTreat)]

E[U[FollowTest)]

EVSI

EVSI is positive for 0.017 < ps < 0.654

π=

Region Optimal Strategyπ < 0.017 NeverTreat0.017 < π < 0.654 FollowTestπ > 0.654 AlwaysTreat


Unit 1 - 21 -



• FollowTest: EU(aF) = 98.5 – 13π – c (c is cost of test)• AlwaysTreat: EU(aT) = 90

– FollowTest is better when 98.5 – 13π – c > 90 or π < (8.5-c)/13• NeverTreat: EU(aN) = 100(1 – π)

– FollowTest is better when 98.5 - 13π – c > 100(1-π) or π > (1.5+c)/87

Strategy Regions for Costly Test

88"

90"

92"

94"

96"

98"

100"

0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" 0.8" 0.9" 1"

Expected(U*lity(of(Op*mal(Strategy(with(Costly(Test(

c=0"

c=1"

c=4"

c≥7.2"

Gain from doing test with c=1 at p=0.3

Test is worth doing if (1.5+c)/87 < ps < (8.5-c)/13


Unit 1 - 22 -



• For a test with cost c:– E[U|FollowTest] = 98.5 – 13π - c– NoTreat is best when π < (1.5+c)/87– FollowTest is best when (1.5+c)/87 < π < (8.5-c)/13

• Probability range where testing is optimal depends on cost of test– If 0.018<π<0.029 then test if c= 0.1 but do nothing if c= 1– If 0.577<π<0.646 then test if c=0.1 but treat if c=1

• Information collection is optimal when EVSI is greater than cost of test

EVSI and Costly Test

88"

90"

92"

94"

96"

98"

100"

0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" 0.8" 0.9" 1"

Expected(U*lity(of(Op*mal(Strategy(with(Costly(Test(

c=0"

c=1"

c=4"

c≥7.2"

0"

1"

2"

3"

4"

5"

6"

7"

8"

0" 0.2" 0.4" 0.6" 0.8" 1"

EVSI%as%a%Func,on%of%Prior%Probability%

Range of optimality of test with c=1


Unit 1 - 23 -



Value of Information: Summary

• Collecting information may have value if it might change your decision

– Expected value of perfect information (EVPI) is utility gain from knowing true value of uncertain variable

– Expected value of sample information (EVSI) is utility gain from available information

• In our example, EVSI is positive for 0.017 < π < 0.654– If 0.017 ≤ π ≤ 0.1 EVSI is 87π - 1.5– If 0.1 ≤ π ≤ 0.654 EVSI is 8.5 - 13π– If π = 0.3 EVSI is 8.5 - 13π = 4.6 (testing is optimal)

• Costly information has value when EVSI is greater than cost of information

• In our example:– If 0.017 ≤ π ≤ 0.1 Test if 87π - 1.5 > c (where c is cost of test)– If 0.1 ≤ π ≤ 0.654 Test if 8.5 - 13π > c– If p = 0.3 Test if 4.6 > c (test if c is less than 4.6)


Unit 1 - 24 -



When Parameters of a Model are Unknown• Our disease model depends on several parameters

– Prior probability of disease– Sensitivity of test– Specificity of test

• Usually these probabilities are estimated from data and/or expert judgment

– �Randomized clinical trials have established that Test T has sensitivity 0.95 and specificity 0.85 for Disease D�

– �Given the presenting symptoms and my clinical judgment, I estimate a 30% probability that the patient has Disease D.�

• How does a Bayesian combine data and expert judgment?– Use clinical judgment to quantify uncertainty about the parameter as a

probability distribution– Collect data and use Bayes rule to obtain posterior distribution for the

parameters given the data– If appropriate, use clinical judgment to adjust results of studies to apply to a

particular patient


Unit 1 - 25 -



Learning from a Random Sample drawn from a Parametric Model

• Many statistical models assume observations are �drawn at random� from a common probability distribution

• Data X1, …, Xn are �drawn at random� from distribution with probability mass function f(x|q) if:

– P(Xi=x|q) = f(x|q) for all i– Xi is independent of Xj givenq for i≠j

• Example: – �Middle-aged male patients who complain of symptom S are drawn at random

from a population with a proportion q who have disease D�– Data X1, …, Xn are independent and identically distributed (iid) given q = P(sD)=(Sick)

• If the value q is unknown we can express our uncertainty about it by defining a probability distribution for its value

– Xi=1 (disease) or Xi=0 (no disease)– Pr(Xi = 1 | Q) = Q (this is called a Bernoulli distribution)– Pr(Q = q) = g(q)

• We now use Bayes rule to find a posterior distribution g(q | X1, …, Xn )

Notation convention: uppercase letters for unknowns; lowercase letters for particular values the unknowns can take on

Xi iid means that if q is known, learning the condition of some patients does not affect our beliefs about the conditions of the remaining patients


Unit 1 - 26 -



Random Sample: Graphical Representation

• We can use a graph to represent conditional independence

– Arc from Θ to Xi means the distribution of Xi depends on Θ

– No arc from Xi to Xj means that Xi and Xjare independent given Θ

• A �plate� represents repeated structure – Because all the Xi are inside the same plate,

they all have the same probability distribution

X1

Θ

X2X3

Θ

Xii=1,…,3

Θ ~ g(q)

Xi | θ~ Bernoulli(θ)


Unit 1 - 27 -



Example: Bayesian Inference about a Parameter(with a very small sample)

• Uninformative prior distribution Pr(Q = q) = g(q)– Pretend that Q must take one of 20 values:

0.025, 0.075, …, 0.975– Assume these are equally likely– If we have prior knowledge we can assign

some values of Q a greater probability than others• Observe 5 iid cases: X1, X2, X3, X4, X5• Case 2 has disease; cases 1, 3, 4 and 5 do not• Likelihood function (probability of observations as function of !):

– Pr(X1, X2, X3, X4, X5 | Q = q) = q (1-q)4

– The likelihood function depends only on how many cases have the disease– The number of cases having the disease and the total number of cases are sufficient for inference about Q

• Use Bayes rule to calculate posterior distribution for Q:

– Because the prior distribution is uniform, the posterior distribution is proportional to the likelihood

Θ actually has a continuous range of values. We will treat continuous parameters later. For now we approximate with a finite set of values.

g(θ | x) = g(θ ) f (x |θ )g(θ ') f (x |θ ')

θ '∑=

120θ(1−θ )4

120θ '(1−θ ')4

θ '∑=

θ(1−θ )4

θ '(1−θ ')4θ '∑

0.025 0.125 0.225 0.325 0.425 0.525 0.625 0.725 0.825 0.925

Prior Distribution for Theta

Theta

Prio

r Pro

babi

lity

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.025 0.125 0.225 0.325 0.425 0.525 0.625 0.725 0.825 0.925

Posterior Distribution for Theta

Theta

Pos

terio

r Pro

babi

lity

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Underscore indicates a vector: x=(x1, x2, x3, x4, x5)


Unit 1 - 28 -



R Computing Environment

• R (http://www.r-project.org) is a free, open source statistical computing language and environment that includes:

– data handling and storage– matrix and array operations– tools for data analysis– graphical facilities for data analysis and display– programming language which includes conditionals, loops, user-defined

recursive functions and input and output facilities• Vibrant and active user community contributes new functionality

– Users can contribute packages to extend functionality• Many resources exist to help users at a variety of levels

– http://cran.r-project.org/doc/manuals/R-intro.pdf• RStudio (http://www.rstudio.com) is a free, open-source integrated

development environment for R• We will use R heavily in this course• R assignments can usually be done by modifying sample code

provided to students

http://cran.r-project.org/doc/manuals/R-intro.pdf


Unit 1 - 29 -



Bayesian Inference Example: R Code

Horizontal axis is θ = P(Sick); height of bar is probability that Θ=θ


Unit 1 - 30 -



Bayesian Learning and Sample Size• When the sample size is very large:

– The posterior distribution will be concentrated around the maximum likelihood estimate and is relatively insensitive to the prior distribution

– We won�t go too far wrong if we act as if the parameter is equal to the maximum likelihood estimate

• When the sample size is very small:– The posterior distribution is highly dependent on the prior distribution– Reasonable people may disagree on the value of the parameter

• When the sample size is moderate, Bayesian learning can be a big improvement on either expert judgment alone or data alone

– Achieving the benefit requires careful modeling– This course will teach methods for constructing Bayesian models

• A powerful characteristic of the Bayesian approach is the flexibility to tailor results to moderate-sized sub-populations

– Bayesian estimate �shrinks� estimates of sub-population parameters toward population average

– Amount of shrinkage depends on sample size and similarity of sub-population to overall population

– Shrinkage improves estimates for small to moderate sized sub-populations


Unit 1 - 31 -



Effect of Sample Size on Posterior Distribution

These plots show the posterior distribution for Θ when:

– Prior distribution is uniform– 20% of patients in sample have

the diseasePosterior distribution becomes more concentrated around 1/5 as sample size gets larger

0"

0.05"

0.1"

0.15"

0.2"

0.25"

0.3"

0.35"

0.4"

0.025" 0.075" 0.125" 0.175" 0.225" 0.275" 0.325" 0.375" 0.425" 0.475" 0.525" 0.575" 0.625" 0.675" 0.725" 0.775" 0.825" 0.875" 0.925" 0.975"

0"

0.05"

0.1"

0.15"

0.2"

0.25"

0.3"

0.35"

0.4"

0.025" 0.075" 0.125" 0.175" 0.225" 0.275" 0.325" 0.375" 0.425" 0.475" 0.525" 0.575" 0.625" 0.675" 0.725" 0.775" 0.825" 0.875" 0.925" 0.975"

Sample size 5: 1 with, 4 without


0"

0.05"

0.1"

0.15"

0.2"

0.25"

0.3"

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Horizontal axis is q = P(sD); height of bar is probability that Θ =q


Unit 1 - 32 -



Sample Size and Impact of the Prior Distribution

Prior distribution favors low probabilities:

Prior distribution favors high probabilities:

• Bayesian inference �shrinks� posterior distribution toward prior expectations – Posterior distribution for smaller sample is more sensitive to prior distribution– Posterior distribution for larger sample is less sensitive to prior distribution

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Unit 1 - 33 -



The Bayesian Resurgence

• Bayesian inference is as old as probability• Bayesian view fell into disfavor in nineteenth and early

twentieth centuries– Positivism, empiricism, and quest for objectivity in science– �Paradoxes� and systematic deviation of human judgment from

Bayesian �norm�• There has been a recent resurgence

– Computational advances make calculation possible for complex models– Bayesian models can coherently integrate many different kinds of

information» Physical cause and effect» Logical implication» Informed expert judgment» Empirical observation

– Unified theory and methods for data-rich and data-poor problems– Clear connection to decision making


Unit 1 - 34 -



Probability really is none of these things.Probability can represent all of these things.

Some Concepts of Probability

• Classical - Probability is a ratio of favorable cases to total (equipossible) cases

• Frequency - Probability is the limiting value as the number of trials becomes infinite of the frequency of occurrence of some event

• Logical - Probability is a logical property of one�s state of information about a phenomenon

• Propensity - Probability is a propensity for certain kinds of event to occur and is a property of physical systems

• Subjectivist - Probability is an ideal rational agent�s degree of belief about an uncertain event

• Algorithmic - The algorithmic probability of a finite sequence is the probability that a universal computer fed a random input will give the sequence as output (related to Kolmogorov complexity)

• Game Theoretic - Probability is an agent�s optimal �announced certainty�for an event in a multi-agent game in which agents receive rewards that depend on both forecasts and outcomes


Unit 1 - 35 -



Historical Notes• People have long noticed that some events are imperfectly predictable• Mathematical probability first arose to describe regularities in games of

chance• The subjectivist interpretation dates from the 18th century but fell out of

favor because of the positivitist orientation of Western 19th and 20th century science

• In the twentieth century it became clear that probability theory provided a good model for a much broader class of problems:

– Physical (thermodynamics; quantum mechanics)– Social (actuarial tables; sample surveys)– Industrial (equipment failures)

• Von Mises formulated a rigorous (and much-debated) frequency theory in the mid-twentieth century

• Subjectivist probability is enjoying a resurgence• Hierarchy of generality:

– Classical interpretation is restricted to �equipossible cases�– Frequency interpretation is restricted to repeatable, random phenomena– Subjectivist interpretation applies to any event about which the agent is uncertain– Game theoretic interpretation applies even when probabilities are not �true beliefs�


Unit 1 - 36 -



The Frequentist• A frequentist believes:

– Probability can be legitimately applied only to repeatable problems– Probability is an objective property in the real world – Probability applies only to random processes– Probabilities are associated only with collectives not individual events

• Frequentist Inference– Data are drawn from a distribution of known form but with an unknown parameter

(this includes �nonparametric� statistics in which the unknown parameter is the distribution itself)

– Often this distribution arises from explicit randomization (when not, statistician argues that the procedure was �close enough� to randomized that inferences apply)

– Inferences regard the data as random and the parameter as fixed (even though the data are known and the parameter is unknown)

– For example: A sample X1,…XN is drawn from a normal distribution with mean Θ . A 95% confidence interval is constructed. The interpretation is:

If an experiment like this were performed many times we would expect in 95% of the cases that an interval calculated by the procedure we applied would include the true value of Θ .

• A frequentist can say nothing about any individual experiment or about what we should believe about q!


Unit 1 - 37 -



The Subjectivist

• A subjectivist believes:– Probability as an expression of a rational agent�s degrees of belief about

uncertain propositions.

– Rational agents may disagree. There is no �one correct probability.�

– If the agent receives feedback her assessed probabilities will in the limit

converge to observed frequencies

• Subjectivist Inference:– Probability distributions are assigned to the unknown parameters and to

the observations given the unknown parameters.

– Condition on knowns; use probability to express uncertainty about

unknowns

– For example: A sample X1,…XN is drawn from a normal distribution with

mean q having prior distribution g(q). A 95% posterior credible interval is

constructed, and the result is the interval (3.7, 4.9). The interpretation is:

Given the prior distribution for q and the observed data, the probability that q lies between 3.7 and 4.9 is 95%.

• A subjectivist can draw conclusions about what we should believe about ! and about what we should expect on the next trial


Unit 1 - 38 -



Comparison: Understandability, Subjectivity and Honest Reporting

• Often the Bayesian answer is what the decision maker really wants to hear.

• Untrained people often interpret results in the Bayesian way.• Frequentists are disturbed by the dependence of the posterior

interval on the �subjective� prior distribution.It is more important that stochastics provides a means of communication among researchers whose personal beliefs about the phenomena under study may differ. If these beliefs are allowed to contaminate the reporting of results, … how are the results of different researchers to be compared?

- H. Dinges• Bayesians say the prior distribution is not the only subjective

element in an analysis. Assumptions about the sampling distribution are often also subjective.

• Bayesian probability statements are always subjective, but statistical analyses are often done for public consumption. Whose probability distribution should be reported?

– When there are enough data, a good Bayesian analysis and a good frequentistanalysis will typically be in close agreement

– If the results are sensitive to the prior distribution, a Bayesian analyst is obligated to report this sensitivity, and to present the range of results obtained from a range of prior distributions


Unit 1 - 39 -



Comparison: Generality

• Subjectivists can handle problems the frequentist approach cannot (in particular, problems with not enough data for sound frequentist inference).

• Frequentist statisticians say this comes at a price -- when there are not enough data the result will be highly dependent on the prior distribution.

• Subjectivists often apply frequentist techniques but with a Bayesian interpretation

• Frequentists often apply Bayesian methods if they have good frequency properties


Unit 1 - 40 -



Axioms for ProbabilityDe Groot, 1970

There is a qualitative relationship of relative likelihood , that operates on pairs of events, that satisfies the following conditions: SP1. For any two uncertain events A and B, one of the following relations holds: A

B, A B or A ~ B. SP2. If A1, A2, B1, and B2 are four events such that A1∩A2=∅, B1∩B2=∅, and if Ai

Bi, for i = 1,2, then A1∪A2 B1∪B2. If in addition Ai Bi for either i=1 or i=2, then A1∪A2 B1∪B2.

SP3. If A is any event, then ∅ A. Furthermore, there is some event A0 for which ∅ A0.

SP4. If A1⊃A2⊃ … is a decreasing sequence of events, and B is some event such

that Ai B for i=1, 2, …, then Aii=1

∞ B . SP5. There is an experiment, with a numerical outcome between the values of 0 and

1, such that if Ai is the event that the outcome x lies within the interval ai ≤ x ≤ bi, for i=1,2, then A1 A2 if and only if (b1-a1) ≤ (b2-a2).


Unit 1 - 41 -



Axioms for UtilityWatson and Buede, 1987

A reward is a prize the decision maker cares about. A lottery is a situation in which the decision maker will receive one of the possible rewards, where the reward to be received is governed by a probability distribution. There is a qualitative relationship of relative preference * , that operates on lotteries, that satisfies the following conditions: SU1. For any two lotteries L1 and L2, either L1 *L2, L1 * L2, or L1~*L2.

Furthermore, if L1, L2, and L3 are any lotteries such that L1 *L2 and L2 *L3, then L1 *L3.

SU2. If r1, r2 and r3 are rewards such that r1 * r2 * r3, then there exists a probability p such that [r1: p; r3: (1-p)] ~* r2, where [r1:p; r3:(1-p)] is a lottery that pays r1 with probability p and r3 with probability (1-p ).

SU3. If r1 ~* r2 are rewards, then for any probability p and any reward r3, [r1: p; r3: (1-p)] ~* [r2: p; r3: (1-p )]

SU4. If r1 * r2 are rewards, then [r1: p; r2: (1-p)] * [r1: q; r2: (1-q)] if and only if p > q.

SU5. Consider three lotteries, Li = [r1: pi; r2: (1-pi)], i = 1, 2, 3, giving different probabilities of the two rewards r1 and r2. Suppose lottery M gives entry to lottery L2 with probability q and L3 with probability 1-q. Then L1~*M if and only if p1 = qp2 + (1-q)p3.


Unit 1 - 42 -



Probabilities and Utilities• If your beliefs satisfy SP1-SP5, then there is a probability

distribution Pr(⋅) over events such that for any two events A1 and A2, Pr(A1) ≥ Pr(A2) if and only if A1 A2.

• If your preferences satisfy SU1-SU5, then there is a utility function u(⋅) defined on rewards such that for any two lotteries L1 and L2, L1 * L2 if and only if E[u(L1)] ≥ E[u(L2)], where E[⋅] denotes the expected value with respect to the probability distribution Pr(⋅).

?


Unit 1 - 43 -



Myth of the Cold-Hearted Rationalist• A common criticism of decision theory is that its adherents are cold-hearted

technocrats who care about numbers and not about what really matters– They would put a dollar value on a human life– They would send people to possible death on the basis of utilitarian calculations– And so on …

• The applied decision theorist�s response– These kinds of tradeoffs are unquestionably difficult– Whether we quantify them or not, as a society and as individuals we make them all the time – They will be irrational and capricious if we approach them without a principled methodology– Refusing to think about the tradeoffs will only ensure that they will be addressed haphazardly

and/or by �back door� manipulation by powerful special interests– As a society we need open debate and discussion of the difficult tradeoffs we are forced to make.

Decision theory provides a justifiable, communicable framework for doing so» disagreements about fact are clearly separated from disagreements about value» inconsistencies can be spotted, discussed, and resolved» commonly recurring problems need not be revisited once consensus has been reached

• Decision theory can be misused if models are sloppily built and leave out important elements

• When a group or society has not reached consensus there is no clear �best choice�• Explicitly modeling subjective elements of a problem provides a framework for

informed debate


Unit 1 - 44 -



Unit 1: Summary and Synthesis

• The inventors of probability theory thought of it as a logic of enlightened rational reasoning. In the nineteenth century this was replaced by a view of probability as measuring �objective�propensities of �intrinsically random� phenomena

• The twentieth century has seen a resurgence of interest in subjective probability and an increased understanding of the appropriate role of subjectivity in science

• Bayesian methods often require more computational power than traditional frequentist methods

• The computer revolution has enabled the Bayesian resurgence• Most statistics texts and courses take a frequentist approach but this

is changing• Bayesian decision theory provides methodology for rational choice

under uncertainty• Bayesian statistics is a theory of rational belief dynamics• We took a broad-brush tour of Bayesian methodology • We applied Bayesian thinking to a simple example that illustrates

many of the concepts we will be learning this semester


Unit 1 - 45 -



References for Unit 1

• Bayes, Thomas. An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370- 418, 1763.

• Bashir, S.A., Getting Started in R, http://www.luchsinger-mathematics.ch/Bashir.pdf• Dawid, A.P. and Vovk, V.G. (1999), Prequential Probability: Principles and Properties,

Bernoulli, 5: 125-162.• de Finetti, Bruno. Theory of Probability: A Critical Introductory Treatment. New York:

Wiley, 1974.• Gelman, A., Carlin, J., Stern, H. and Rubin, D., Bayesian Data Analysis (2nd edition),

Chapman & Hall, 2004. Chapter 1• Hájek, Alan, "Interpretations of Probability", The Stanford Encyclopedia of Philosophy

(Summer 2003 Edition), Edward N. Zaltaﾊ(ed.), URL = <http://plato.stanford.edu/archives/sum2003/entries/probability-interpret/>.

• Lee, P. Bayesian Statistics: An Introduction, 4th ed. Springer, 2012. Chapter 1• Li, Ming and Vitanyi, Paul. An Introduction to Kolmogorov Complexity and Its

Applications. (2nd ed) Springer-Verlag, 2005.• Nau, Robert F. (1999), Arbitrage, Incomplete Models, And Interactive Rationality,

working paper, Fuqua School of Business, Duke University.• Neapolitan, R. Learning Bayesian Networks, Prentice Hall, 2003.• Jaynes, E., Probability Theory: The Logic of Science, Cambridge University Press,

2003)• Savage, L.J., The Foundations of Statistics. Dover, 1972.• Shafer, G. Probability and Finance: It�s Only a Game, Wiley, 2001.• von Mises R., 1957, Probability, Statistics and Truth, revised English edition, New

York: Macmillan

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