of Review
Interpretations Probabilty
Probability and Forecasts
Eco331: Subjective Probability as a Rational Forecast
Marciano Siniscalchi
September 27, 2009
Review
Interpretations of Probabilty
Probability and Forecasts
Probability, Random Variables, Expectation
Denition A probability over a nite set S of states is a function P : P (S ) R such that:
1 2 3
P (E ) 0; P () = 0; P (S ) = 1. If E F = , then P (E F ) = P (E ) + P (F ) [additivity].
Denition Consider a nite state space S . A (real-valued) random variable is a function X : S R that associates a real number X (s ) to each state s S . Its expectation (or expected value) is E[X ] =
s S
X (s )P ({s }).
Review
Interpretations of Probabilty
Probability and Forecasts
Three Views of Probability
Classical: Probability describes the physical chance that an experiment produces some outcome. P (event) = #favorable outcomes . #possible outcomes
Self-referential, because it assumes equally likely outcomes. Frequentist: Probability is the long-run frequency of an event in an innite sequence of repetitions of the experiment. What if experiment cannot be repeated? Subjectivist / Bayesian: Probability is a numerical representation of the individuals beliefs, as revealed by her behavior.
Review
Interpretations of Probabilty
Probability and Forecasts
Subjective Probability
The individual is whoever is making the decision. Maybe you! Thats why its called subjective... Actually, (subjective) probability only makes sense in a decision context. Subjective Probability is based on behavior. Belief is a psychological concept. We need an operational denition. Probability is just a function, but we must be able to derive (elicit) it from the individuals choices in suitable settings.
Review
Interpretations of Probabilty
Probability and Forecasts
Decision-Theoretic Foundations (1)
A simple idea that requires a lot of work to get right. Coin ip; S = {H , T }. Consider the following two bets: B1: if H, Ill pay you $1; if T , youll pay me $1. B2: if T , Ill pay you $1; if H, youll pay me $1. If you think P ({H }) = 1 , B1 and B2 are equally good for you. 2 Turn this on its head:
1 Dene P ({H }) = 2 to mean B1 and B2 equally good for you.
Review
Interpretations of Probabilty
Probability and Forecasts
Decision-Theoretic Foundations (2)
Potential issues: What if B1 is deemed strictly better than B2? Clearly P ({H }) > 1 , but can we pin down P ({H })? 2 Must make sure more is better... Could the size of the bets matter ($1 vs. $1000)? How is P used in more complex decision problems? These issues can all be addressed: Savage, 1954. Will get back to it at the end of this course. For now, simpler idea: Probability as rational forecast (de Finetti, 1937).
Review
Interpretations of Probabilty
Probability and Forecasts
A Penalty Scheme
Fix a state space S and an event E S . Contract: You provide a forecast R. If E occurs, you must pay (1 )2 ; if E does not occur, you must pay 2 . To ensure participation, regardless of whether or not E occurs, you get a xed, large forecasters fee . Indicator function of E : 1E (s ) = 1 if s E , = 0 if s E . Payo in state s S : 1E (s )(1 )2 [1 1E (s )] 2 .
Review
Interpretations of Probabilty
Probability and Forecasts
Eliciting true beliefs
Useful to dene the random variable L (s ) = 1E (s )(1 )2 + [1 1E (s )] 2
Remark Fix a probability P on S. The forecast maximizes E[ L ], or equivalently minimizes E[L ], if and only if = P (E ). Interpretation: if P is true belief, this contracts elicits P (E ). Proof : E[L ] = P (E )(1 )2 + [1 P (E )] 2 ; strictly convex in . FOC is 2P (E )(1 ) + 2[1 P (E )] = 0. Clearly, = P (E ) is the unique minimizer.
Review
Interpretations of Probabilty
Probability and Forecasts
Basic Properties of Forecasts
Even without xing P , forecasts must still be sensible. Remark Suppose is there no R such that L (s ) < L (s ) for all s S. Then [0, 1]. Interpretation: if there is no obvious way to improve upon , then looks like the probability of an event. Proof : If < 0, then (1 )2 > 1, 2 > 0, so, for every s S , L0 (s ) = 1E (s ) < 1E (s )(1 )2 + [1 1E (s )] 2 = L (s ) (think about s E and s E separately). If > 1, then again (1 )2 > 0, 2 > 1, so, for every s S , L1 (s ) = 1 1E (s ) < 1E (s )(1 )2 + [1 1E (s )] 2 = L (s ).
Review
Interpretations of Probabilty
Probability and Forecasts
Forecasts: General Denitions
Note that 1E (s )(1 )2 + [1 1E (s )] 2 = [1E (s ) ]2 . Hence: Denition Fix a nite state space S . A forecast system is a map : P (S ) R. The penalty associated with a forecast system is the random variable L : S R such that s S , L (s ) =
E S
[1E (s ) (E )]2 .
Provide forecast for all events, get penalized for all mistakes.
Review
Interpretations of Probabilty
Probability and Forecasts
Main Result
Denition A forecast system on a nite state space S is admissible i there is no forecast system such that L (s ) < L (s ) for all s S . Admissible = not obviously improvable. View as mild rationality requirement. Theorem Let S be a nite state space. A forecast system is admissible if and only if it is a probability. We can dene probability to be an admissible forecast system.
Review
Interpretations of Probabilty
Probability and Forecasts
Admissible Probability: Basic Properties
To show (E ) 0 for all E S , argue roughly as before. To show () = 0, argue by contradiction. Suppose () > 0; let (G ) = (G ) for G = , () = 0. L (s ) L (s ) =
G S 2
[1G (s ) (G )]2
G S
[1G (s ) (G )]2 =
= [1 (s ) ()] [1 (s ) ()]2 = = 0 ()2 < 0, which contradicts admissibility of . Thus, () = 0. Argument for (S ) = 1 is similar.
Review
Interpretations of Probabilty
Probability and Forecasts
Admissibility Probability: Additivity (1)
To show E F = implies (E F ) = (E ) + (F ): Suppose E F = , (E F ) = (E ) + (F ). Dene (G ) = (G ) for G {E , F , E F }.
1 Dene = 3 [ (E F ) (E ) (F )]; then let (E F ) = (E F ) , (E ) = (E ) + , and (F ) = (F ) + .
Now compute: [1E F (s ) (E F )]2 [1E F (s ) (E F )]2 = =[1E F (s ) (E F ) + ]2 [1E F (s ) (E F )]2 = =1E F (s ) + (E F )2 + 2 2 1E F (s ) (E F ) + 2 1E F (s ) 2 (E F ) 1E F (s ) (E F )2 + 2 1E F (s ) (E F ) = =2 + 2 1E F (s ) 2 (E F ),
Review
Interpretations of Probabilty
Probability and Forecasts
Admissibility Probability: Additivity (2)
However: [1E (s ) (E )]2 [1E (s ) (E )]2 = =[1E (s ) (E ) ]2 [1E (s ) (E )]2 = =1E (s ) + (E )2 + 2 2 1E (s ) (E ) 2 1E (s )+ + 2 (E ) 1E (s ) (E )2 ) + 2 1E (s ) (E ) = =2 2 1E (s ) + 2 (E ) and similarly [1F (s ) (F )]2 [1F (s ) (F )]2 = 2 2 1F (s ) + 2 (F ). Summing up these quantities yields, for every state s S , L (s ) L (s ) =32 2[ (E F ) (E ) (F )] = =32 62 = 32 < 0, which contradicts the assumption that is admissible. Done!
Review
Interpretations of Probabilty
Probability and Forecasts
Probability Admissibility: Sketch
Now assume is a probability and another forecast system. Compute expectation of L L w.r.to Show that it is positive Conclude that there must be s with L (s ) > L (s ) Since was arbitrary, is admissible. See details in lecture notes. Done!
Review
Interpretations of Probabilty
Probability and Forecasts
Extensions and Problems
Expectation: extend () to random variables (redene L ). Conditional Probability: allow for (F |E ), require conditional admissibility. However: Suppose penalties and fee are multiplied by 1000. Would you provide the same forecast? Would Bill Gates? Main issue is risk aversion. Only full-edged decision-theoretic analysis can disentangle subjective probability and risk attitudes.