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Decision theory and Bayesian statistics. More repetition . Tron Anders Moger 22.11.2006. Overview. Statistical desicion theory Bayesian theory and research in health economics Review of previous slides. Statistical decision theory.

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Decision theory and Bayesian statistics. More repetition 

Tron Anders Moger



  • Statistical desicion theory

  • Bayesian theory and research in health economics

  • Review of previous slides

Statistical decision theory

  • Statistics in this course often focus on estimating parameters and testing hypotheses.

  • The real issue is often how to choose between actions, so that the outcome is likely to be as good as possible, in situations with uncertainty

  • In such situations, the interpretation of probability as describing uncertain knowledge (i.e., Bayesian probability) is central.

Decision theory: Setup

  • The unknown future is classified into H possible states of nature: s1, s2, …, sH.

  • We can choose one of K actions: a1, a2, …, aK.

  • For each combination of action i and state j, we get a ”payoff” (or opposite: ”loss”) Mij.

  • To get the (simple) theory to work, all ”payoffs” must be measured on the same (monetary) scale.

  • We would like to choose an action so to maximize the payoff.

  • Each state si has an associated probability pi.

Desicion theory: Concepts

  • If action a1 never can give a worse payoff, but may give a better payoff, than action a2, then a1 dominates a2.

  • a2 is then inadmissible

  • The maximin criterion for choosing actions

  • The minimax regret criterion for choosing actions

  • The expected monetary value criterion for choosing actions




Maximin and minimax

  • Maximin: Maximize the minimum payoff:

  • For each row, compute the minimum

  • Maximize over the actions

  • Minimax regret: Minimize the maximum regret possible

  • Compute the regrets in each column, by finding differences to max numbers

  • Maximize over the rows

  • Find action that minimizes these maxima.


Find that action C is preferred under the maximin criterion

Regret table:



Action C is also preferred under the minimax criterion

Expected monetary value criterion

  • Need probabilities for each state

  • Assume P(no outbreak)=P1=95%, P(small outbreak)=P2=4.5%, P(pandemic)=P3=0.5%

  • EMV(A)=P1*M11+P2*M12+P3*M13=

    0*0.95-500*0.045-100000*0.005= -522.5

  • EMV(B)=-55.45

  • EMV(C)=-1000

  • Should choose action B

Decision trees

  • Contains node (square junction) for each choice of action

  • Contains node (circular junction) for each selection of states

  • Generally contains several layers of choices and outcomes

  • Can be used to illustrate decision theoretic computations

  • Computations go from bottom to top (or left to right in the book) of tree


No outbreak (0.95)


Action A

Small outbreak (0.045)


Pandemic (0.005)



No outbreak (0.95)



*Action B

Small outbreak (0.045)


Pandemic (0.005)


No outbreak (0.95)



Small outbreak (0.045)


Action C

Pandemic (0.005)


Updating probabilities by aquired information

  • To improve the predictions about the true states of the future, new information may be aquired, and used to update the probabilities, using Bayes theorem.

  • If the resulting posterior probabilities give a different optimal action than the prior probabilities, then the value of that particular information equals the change in the expected monetary value

  • But what is the expected value of new information, before we get it?


  • Prior probabilities: P(no outbreak)=95%, P(small outbreak)=4.5%, P(pandemic)=0.5%.

  • Assume the probabilities are based on whether the virus has a low or high mutation rate.

  • A scientific study can update the probabilities of the virus mutation rate.

  • As a result, the probabilities for no birdflu, some birdflu, or a pandemic, are updated to posterior probabilities: We might get, for example:

The new information might affect what action we would take

  • But not in this example:

    • If we find out that birdflu virus has high mutation rate, we would still choose action B!

    • EMV(A)=-5075, EMV(B)=-515.8, EMV(C)=-1000

    • If we find out that birdflu virus has low mutation rate, we would still choose action B!

    • EMV(A)=-104.5, EMV(B)=-11.9, EMV(C)=-1000

Expected value of perfect information

  • If we know the true (or future) state of nature, it is easy to choose optimal action, it will give a certain payoff

  • For each state, find the difference between this payoff and the payoff under the action found using the expected value criterion

  • The expectation of this difference, under the prior probabilities, is the expected value of perfect information


  • Found that action B was best using the prior probabilities

  • However, if there is no outbreak, action A is one unit better than B

  • Similarily, if there is a pandemic, action C is 9000 units better than B

  • The expected value of perfect information is then

  • EVPI=0.95*1+0.045*0+0.005*9000=45.95

Expected value of sample information

  • What is the expected value of obtaining updated probabilities using a sample?

    • Find the probability for each possible sample

    • For each possible sample, find the posterior probabilities for the states, the optimal action, and the difference in payoff compared to original optimal action

    • Find the expectation of this difference, using the probabilities of obtaining the different samples.


  • When all outcomes are measured in monetary value, computations like those above are easy to implement and use

  • Central problem: Translating all ”values” to the same scale

  • In health economics: How do we translate different health outcomes, and different costs, to same scale?

  • General concept: Utility

  • Utility may be non-linear function of money value

Risk and (health) insurance

  • When utility is rising slower than monetary value, we talk about risk aversion

  • When utility is rising faster than monetary value, we talk about risk preference

  • If you buy any insurance policy, you should expect to lose money in the long run

  • But the negative utility of, say, an accident, more than outweigh the small negative utility of a policy payment.

Desicion theory and Bayesian theory in health economics research

  • As health economics is often about making optimal desicions under uncertainty, decision theory is increasingly used.

  • The central problem is to translate both costs and health results to the same scale:

    • All health results are translated into ”quality adjusted life years”

    • The ”price” for one ”quality adjusted life year” is a parameter called ”willingness to pay”.

Curves for probability of cost effectiveness given willingness to pay

  • One widely used way of presenting a cost-effectiveness analysis is through the Cost-Effectiveness Acceptability Curve (CEAC)

  • Introduced by van Hout et al (1994).

  • For each value of the threshold willingness to pay λ, the CEAC plots the probability that one treatment is more cost-effective than another.

Repetition: What is relevant for the exam

  • Probability theory

  • Expected values and variance

  • Distributions

  • Tests, regression, one-way ANOVA and at least an understanding of two-way ANOVA are all relevant (obviously)

  • Interpretation of a time-series regression model might also show up

  • Do not forget how to interpret SPSS output (including graphs and figures)!!

  • Also, do not forget the chi-square test!!

Conditional probability

  • If the event B already has occurred, the conditional probability of A given B is:

  • Can be interpreted as follows: The knowledge that B has occurred, limit the sample space to B. The relative probabilities are the same, but they are scaled up so that they sum to 1.

Probability postulates 3

  • Multiplication rule: For general outcomes A and B:


  • Indepedence: A and B are statistically independent if P(AB)=P(A)P(B)

    • Implies that

The law of total probability - twins

  • A= Twins have the same gender

  • B= Twins are monozygotic

  • = Twins are heterozygotic

  • What is P(A)?

  • The law of total probability

    P(A)=P(A|B)P(B)+P(A| )P( )

    For twins: P(B)=1/3 P( )=2/3

    P(A)=1 · 1/3+1/2 · 2/3=2/3

Bayes theorem

  • Frequently used to estimate the probability that a patient is ill on the basis of a diagnostic

  • Uncorrect diagnoses are common for rare diseases

Example: Cervical cancer

  • B=Cervical cancer

  • A=Positive test

  • P(B)=0.0001P(A|B)=0.9 P(A| )=0.001

  • Only 8% of women with positive tests are ill

Probability postulates 4

  • Assume that the events

    A1, A2 ,..., An are independent. Then P(A1A2....An)=P(A1)·P(A2) ·.... ·P(An)

    This rule is very handy when all P(Ai) are equal

  • The complement rule: P(A)+P( )=1

Example: Doping tests

  • Let’s say a doping test has 0.2% probability of being positive when the athlete is not using steroids

  • The athlete is tested 50 times

  • What is the probability that at least one test is positive, even though the athlete is clean?

  • Define A=at least one test is positive

Complement rule Rule of independence 50 terms

Expected values and variance

  • Remember the formulas E(aX+b) = aE(X)+b and

  • How do you calculate expectation and variance for a categorical variable?

  • For a continuous variable?

  • How do you construct a standard normal variable from a general normal variable?

  • Finding probabilities for a general normal variable?


  • Distributions we’ve talked about in detail

  • Binomial

  • Poisson

  • Normal

  • Approximations to normal distributions?

  • Other distributions are there just to allow us to make test statistics, but you need to know how to use them

Remember this slide? (This was difficult)

  • The probabilities for

    • A: Rain tomorrow

    • B: Wind tomorrow

      are given in the following table:

Some wind

Strong wind


No wind

No rain

Light rain

Heavy rain

And this one?

  • Marginal probability of no rain: 0.1+0.2+0.05+0.01=0.36

  • Similarily, marg. prob. of light and heavy rain: 0.34 and 0.3. Hence marginal dist. of rain is a PDF!

  • Conditional probability of no rain given storm: 0.01/(0.01+0.04+0.05)=0.1

  • Similarily, cond. prob. of light and heavy rain given storm: 0.4 and 0.5. Hence conditional dist. of rain given storm is a PDF!

  • Are rain and wind independent? Marg. prob. of no wind: 0.1+0.05+0.05=0.2

    P(no rain,no wind)=0.36*0.2=0.072≠0.1

Think wheat fields!

  • Wheat field was a bivariate distribution of wheat and fertilizer

  • Only: Continuous outcome instead of categorical

  • Calculations on previous incomprehensible slide is exactly the same as we did for the wheat field!

  • Mean wheat crop for wheat 1 regardless of fertilizer->Marginal mean!!

  • Mean crop for wheat 1 given that you use fertilizer ->Conditional mean!!

    (corresponds to mean for a single cell in our field)

Chi-square test:

  • Expected cell values: Abortion/op.nurses: 13*36/70=6.7

    Abortion/other nurses: 13*34/70=6.3

    No abortion/op.nurses: 57*36/70=29.3

    No abortion/other nurses: 57*34/70=27.7

  • Can be easily extendend to more groups of nurses

  • As long as you have only two possible outcomes, this is equal to comparing proportions in more than two groups (think one-way ANOVA)

We get:

  • This has a chi-square distribution with (2-1)*(2-1)=1 d.f.

  • Want to test H0: No association between abortions and type of nurse at 5%-level

  • Find from table 7, p. 869, that the 95%-percentile is 3.84

  • This gives you a two-sided test!

  • Reject H0: No association

  • Same result as the test for different proportions in Lecture 4!


Check Expected under Cells, Chi-square under statistics, and

Display clustered bar charts!

Next time:

  • Find some topics you don’t understand, and we can talk about them

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