Probability theory and decision theory

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This page about probability theory and decision theory will deal about Bayesian, that is, subjective probability, and its use in normative decision analysis.

• Probability theory [frequentist and subjective (Bayesian) approach] Marja-Leenalla sopivia kirjoja? Uncertainty 220-230. Jouni
• Decision theory (utility, decision tree, utilitarian, egalitarian and other decision rules) Decision theory

Two approaches in probability

• Frequentist: Probability is the frequency distribution of outcomes, when a large number of similar events is repeated and obseverd, such tossing a coin, throwing a dice, or taking a blood test from a random sample of individuals in a population.
• Bayesian: When you are indifferent about choosing between A and B, your probability for U₁ given A is E.

Basic rules in probability

The probability of A happening = p(A)

p(A)+p(not A)=1

p(A or B) = p(A)+p(B)-p(A and B)

If A and B are mutually exclusive: p(A)+p(B)

p(A|B) = p(A given B) = p(A and B)/p(B)

p(A and B) = p(A|B)p(B)

If A and B not dependent (uncorrelated): p(A)p(B)

The probability of causation (POC)

POC = P(risk from the cause of interest)/P(risk due to all causes)

What is the POC for PM_2.5 causing cardiovasular deaths in the EU?

What is the POC for vinylchloride causing angiosarcoma in liver?

POC(PM_2.5 causing cardiovasular deaths in the EU):
350 000 deaths/a / 2 000 000 deaths/a = 0.175

"Internationally: Worldwide incidence also is low. Reports show angiosarcoma of the liver with an annual incidence of 2 cases per 10 million population, rising to 2.5 cases per 1000 population in workers exposed to vinyl chloride." [1]

POC(vinylchloride causing angiosarcoma in liver):
POC = (2.5*10^-3 - 2*10^-7)/(2.5*10^-3) = 0.99992

Bayes' rule

en:Bayes' theorem

en:Bayes' theorem#Examples

en:Utility

(en:Risk aversion en:Risk premium)

Decision rules:

• Utilitarian: max_d(E(∑(u_d,i)))
• Egalitarian: max_d(E(min(u_d,i)))
• Elitist: max_d(E(max(u_d,i)))

u=utility, d=decision, i=individual

• Cost benetit analysis
• Cost effectiveness analysis