Once again about the science-policy interface

1. Once again about the science-policy interface

2. Q R A Open risk management: overview

3. Human-human interface • There are really interesting new interfaces for transmitting information from person to person: • Facebook: How are you? • Wikipedia: What is thing X? • Opasnet: What should we do? • A universal interface for communication about decisions and decision support is urgently needed. • When that problem is solved, the communication problem between science and policy is solved as well. There is no need for a separate science-policy interface.

4. Introduction to probability theory Jouni Tuomisto THL

5. Probability of a red ball • P(x|K) = R/N, • x=event that a red ball is picked • K=your knowledge about the situation

6. Probability of an event x Prize p Red Red ball 100 € • If you are indifferent between decisions 1 and 2, then your probability of x is p=R/N. Decision 1 White ball 0 € 1-p x happens 100 € Decision 2 0 € x does not happen

7. The meaning of uncertainty • Uncertainty is that which disappears when we become certain. • We become certain of a declarative sentence when (a) truth conditions exist and (b) the conditions for the value ‘true’ hold. • (Bedford and Cooke 2001) • Truth conditions: • It is possible to design a setting where it can be observed whether the truth conditions are met or not.

8. Different kinds of uncertainty • Aleatory (variability, irreducible) • Epistemic (reducible), actually the difference only depends on its purpose in a model. • Weights of individuals is aleatory if we are interested in each person, but epistemic if we are interested in a random person in the population. • Parameter (in a model): should be observable! • Model: several models can be treated as parameters in a meta-model

9. Different kinds of uncertainty: not really uncertainty • Ambiguity: not uncertainty but fuzziness of description • Volitional uncertainty: “The probability that I will clean up the basement next weekend.” • Uncertainties about own actions cannot be measured by probabilities.

10. What is probability? • 1. Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.[1] • 2. Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, or an objective degree of rational belief, given the evidence. • Source: Wikipedia

11. Significance and confidence • Significance level: • The probability of some aspect of the data, given H is true. • Probability: • Your probability of H, given data. • Confidence: • Probability that the interval includes θ (θ is given). • Probability: • Probability that θ is included in the interval (data is given).

12. Positions of Bayesian approach • Statistics is the study of uncertainty. • Uncertainty should be measured by probability. • Data uncertainty is so measured, conditional on the parameters. • Parameter uncertainty is similarly measured by probability. • Inference is performed within the probability calculus, mainly by Bayesian rule.

13. Probability and conditional probabilities • The totality of possible states of the world P(A) P(B)

14. Probability rules • Rule 1 (convexity): • For all A and B, 0 ≤ P(A|B) ≤ 1 and P(A|A)=1. • Cromwell’s rule P(A|B)=1 if and only if A is a logical consequence of B. • Rule 2 (addition): if A and B are exclusive, given C, • P(A U B|C) = P(A|C) + P(B|C). • P(A U B|C) = P(A|C) + P(B|C) – P(A ∩ B|C) if not exclusive. • Rule 3 (multiplication): for all A, B, and C, • P(AB|C) = P(A|BC) P(B|C) • Rule 4 (conglomerability): if {Bn} is a partition, possibly infinite, of C and P(A|BnC)=k, the same value for all n, then P(A|C)=k.

15. Binomial distribution • You make n trials with success probability p. The number of successful trials k follows the binomial distribution. • Like drawing n balls (with replacement) from an urn and k being red. • P(n,k|p) = n!/k!/(n-k)! pk (1-p)(n-k)

16. Example • You draw randomly 3 balls (with replacement) from an urn with 40 red and 60 white balls. What is the probability distribution for the number of red balls?

17. Answer • P(n,k|p) = n!/k!/(n-k)! pk (1-p)(n-k) • 0 red: 3!/0!/3! *0.40*(1-0.4)3-0 • = 1*1*0.63 = 0.216 • 1 red: 3!/1!/2! *0.41*(1-0.4)3-1 = 0.432 • 2 red: 3!/2!/2! *0.42*(1-0.4)3-2 = 0.288 • 3 red: 3!/3!/0! *0.43*(1-0.4)3-3 = 0.064

18. Binomial distribution (n=6) P(•|p,n)

19. Binomial distribution: likelihoods P(k|•,n)

20. Bayes’ theorem • P(θ|x) = P(x|θ) P(θ) / P(x) • Proof: • P(θ,x) = P(θ|x)P(x) = P(x|θ) P(θ) • P(x) is often difficult to determine, but it is independent of θ and thus a constant over θ. Therefore: • P(θ|x) ~ P(x|θ) P(θ) (proportionality)

21. Bayes with words • Likelihood: P(x|θ) • Prior: P(θ) • Posterior: P(θ|x) • Posterior ~ Prior*Likelihood

22. Getting rid of nuisance factors • P(θ,α|x) = P(x ,α |θ) P(θ ,α) / P(x ,α) • P(θ|x) = ∫P(θ ,α |x) dα

23. Example of Bayes’ rule with balls • You draw randomly 3 balls (with replacement) from an urn with 40 red and 60 white balls. What is the probability distribution for the number of red balls? • How do you update your prior if you draw three red balls in row? • Probability of 3 red given prior = likelihood =0.43 = 0.064 • Prior = 0.4 • Posterior=P(θ|R) = P(R|θ) P(θ) / P(R) =0.064*0.4/0.064=0.4 !!

24. Why doesn’t the probability change with new data? • Because the prior is not uncertain, although it is a probability. • P(p=0.4)=1, P(p<>0.4)=0 • Therefore, it is unaffected by any data, even if you get, say, five red balls in row P=0.45=0.010. • You talk about your unlikely results with the guy who sold the urn to you. He replies: ”Did I say it has less red balls? Maybe it was more red balls. I really don’t remember, but the ratio is 40:60 for sure.” • How does this change your model?

25. P(Y|R)=P(R|Y) P(Y)/P(R) =0.4*0.5/(0.4*0.5+0.6*0.5) =0.2/0.5 = 0.4 New Bayes model with uncertain prior: red ball drawn

26. P(Y|R)=P(R|Y) P(Y)/P(R) =0.4*0.4/(0.4*0.4+0.6*0.6)=0.16/0.52=4/13≈0.308 New Bayes model with uncertain prior: second ball is red

27. P(Y|R)=P(R|Y) P(Y)/P(R) =0.4*(4/13)/((1.6+5.4)/13) = 1.6/7 = 8/35≈0.229 New Bayes model with uncertain prior: third ball is red

28. Conclusion from the red ball study • We are not modelling the reality directly; we are modeling our understanding of reality. • It might be useful to think of the Bayes rule as a 2*2 table. • The principle is the same, even if there are more than 2 rows or columns. • The principle is the same, even if there are more than two dimensions in the table.

29. Bayes’ rule in diagnostics • Imagine there is a clinical test for narcolepsy with 0.99 sensitivity and 0.99 specificity. • A man was worried about his 6-year-old daughter who got the swine flu vaccination. He took the daughter to a private laboratory for the test. • Now he comes to you with the daughter. The test result is positive. • Does the daughter have narcolepsy?

30. Narcolepsy diagnostics? • What is sensitivity? • N(true positive)/N(disease) =P(test+|disease) • What is specificity? • N(true negative)/N(healthy) =P(test-|healthy)

31. P(N|t+)=P(t+|N) P(N)/P(t+) =0.99*0.001/0.01099 ≈ 0.0901 Narcolepsy diagnostics

32. P(N|t+)=P(t+|N) P(N)/P(t+) =0.95*0.0901/0.1311 ≈ 0.6528 Narcolepsy: importance of anamnesis. sensitivity=specificity=0.95

33. P(N|t-)=P(t-|N) P(N)/P(t-) =0.05*0.0901/0.8689 ≈ 0.0052 Narcolepsy: importance of negative anamnesis. Sens.=spec.=0.95