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Belief, probability and exchangeability

Belief, probability and exchangeability - 2 Belief...

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2 Belief, probability and exchangeability We first discuss what properties a reasonable belief function should have, and show that probabilities have these properties. Then, we review the basic ma- chinery of discrete and continuous random variables and probability distribu- tions. Finally, we explore the link between independence and exchangeability. 2.1 Belief functions and probabilities At the beginning of the last chapter we claimed that probabilities are a way to numerically express rational beliefs. We do not prove this claim here (see Chapter 2 of Jaynes (2003) or Chapters 2 and 3 of Savage (1972) for details), but we do show that several properties we would want our numerical beliefs to have are also properties of probabilities. Belief functions Let F , G , and H be three possibly overlapping statements about the world. For example: F = { a person votes for a left-of-center candidate } G = { a person’s income is in the lowest 10% of the population } H = { a person lives in a large city } Let Be() be a belief function , that is, a function that assigns numbers to statements such that the larger the number, the higher the degree of belief. Some philosophers have tried to make this more concrete by relating beliefs to preferences over bets: Be( F ) > Be( G ) means we would prefer to bet F is true than G is true. We also want Be() to describe our beliefs under certain conditions: Be( F | H ) > Be( G | H ) means that if we knew that H were true, then we would prefer to bet that F is also true than bet G is also true. P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/978-0-387-92407-6 2, c ± Springer Science+Business Media, LLC 2009
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14 2 Belief, probability and exchangeability Be( F | G ) > Be( F | H ) means that if we were forced to bet on F , we would prefer to do it under the condition that G is true rather than H is true. Axioms of beliefs It has been argued by many that any function that is to numerically represent our beliefs should have the following properties: B1 Be(not H | H ) Be( F | H ) Be( H | H ) B2 Be( F or G | H ) max { Be( F | H ) , Be( G | H ) } B3 Be( F and G | H ) can be derived from Be( G | H ) and Be( F | G and H ) How should we interpret these properties? Are they reasonable? B1 says that the number we assign to Be( F | H ), our conditional belief in F given H , is bounded below and above by the numbers we assign to complete disbelief (Be(not H | H )) and complete belief (Be( H | H )). B2 says that our belief that the truth lies in a given set of possibilities should not decrease as we add to the set of possibilities. B3 is a bit trickier. To see why it makes sense, imagine you have to decide whether or not F and G are true, knowing that H is true. You could do this by first deciding whether or not G is true given H , and if so, then deciding whether or not F is true given G and H .
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