Statistical Methods Bayesian methods 3. Daniel Thorburn Stockholm University 2012-04-03. Slides presented previous time. Rational behaviour – one person. Axiomatic foundation of probability. Type:
Axiomatic foundation of probability. Type:
For any two events A and B exactly one of the following must hold A < B, A > B or A v B (pronounce A as less likely than B, B less likely than A, equally likely)
If A1, A2, B1 and B2 are four events such that A1A2 = B1B2 is empty and A1> B1 and A2> B2 then A1 U A2> B1 U B2. If further either A1 > B1 or A2 > B2 then A1 U A2 > B1 U B2
…(see next page)
If these axioms hold all events can be assigned probabilities, which obey Kolmogorovs axioms (Villegas, Annals Math Stat, 1964),
Axioms for behaviour. Type …
If you prefer A to B, and B to C then you must also prefer A to C
If you want to behave rationally, then you must behave as if all events were assigned probabilities (Anscombe and Aumann, Annals Math Stat, 1963)
Axioms for probability (these six are enough to prove that a probability following Kolmogorovs axioms can be defined plus the definition of conditional probability)
Further one needs some axioms about comparing outcomes, (utilities) in order to be able to prove rationality
Classical inference has a mess of different types of numbers e.g.
Latent variables like in factor analysis
Independent (explaining) variables
Inference – intervals and proving scientific results
Definiton of confidence intervals:
An interval constructed in this way will in the long run cover the true values in 1-a of all cases if it is repeated many many many times.
Like a person throwing rings (or horse-shoes) around a peg. If he is skilful he will get the ring around the peg in 95% of all cases. It’s a property of the person not a particular throw
Definition of probability intervals.
The true value lies with probability 1-a in the interval in this case (given what is known)
The interval is fixed and known. The probability statement refers to the unknown quantity
Synonyms (roughly): credibility intervals, prediction intervals
An interval (a,b) such that
Synonyms: Prediction intervals, Credibility intervals
Highest Posterior Density (HPD)-interval
Note that this interval may be empty if the distribution is wide spread i.e.
inf (P(q|X,K); K reasonable) > 1 –a for some a.
You must check for many possible priors that might be reasonable. Try priors that assess small probabilitis to the statement that you intend to prove.
Your results (correct sun shined, OMX30 lower, Sweden beat Canada, No medal, ?, Oslo more to the south, Swedish GDP less than 4 Billions
Abrar Raza Khan was best according to both measures.
L = P(pi) = 0.081 resp MSE = 0.82.
The results were not quite as spread as last time, but still the average was not better than chance. (5 resp 6) worse than chance.)
Parameters a=47 b=129 c=-3 ABC 0,87 -0,59 -1,2