Standard Probability Axioms and Assigning Probabilities to Beliefs

In deductive reasoning we assumed that the semantics for the propositions under consideration was either True or False. Then, we considered how the propositions and the complex propositions that could be formed using these logical connectives could be used to derive the consequences of this set of "beliefs".      Although it certainly seems to be the case that we can hold beliefs that are either true or false, it also seems to be the case that some of our knowledge is uncertain, that is, we can not say that it is True or False. The question is, if we have uncertain beliefs, then How is uncertainty represented? How do we reason with uncertain beliefs?      Normally or at times, we think that it is possible or likely or probable that we may possess knowledge that is uncertain, that is, knowledge to which we are unable to assign the value of true or false....but nonetheless think that we can assign varying degrees of certainty to that knowledge. The words italicized above provide some examples of the linguistic hedges that we employ to convey our uncertainty. You may recall that in discussing deduction we spoke of "default inferences." For example, "Rutgers will probably loose the football game on Saturday may have been a default rule acquired by many football fans during the 2001 season." However, since such statements are about the future, we can often imagine that out inference may turn out to be mistaken even though we think it unlikely. There are perhaps many different types of uncertain knowledge. Predicting the outcome of the flip of the coin at the start of the game may seem to some to call into play a different kind of uncertainty.....or perhaps not.        The most carefully formulated ideas about uncertainty date back to the 17th century when games of chance were subjected to mathematical scrutiny. The theory of probabilities is the result, and one proposal is that probability theory provides the appropriate treatment of all of the various types of uncertain knowledge that we may entertain.        On this view the semantics human of uncertainty is equivalent to probabilities; that is, for each uncertain belief that a person holds, there is some "subjective probability" that represents the degree to which the person believes the statement to be true. This raises the issue of how we might acquire these subjective probabilities as well as the question of how they might be updated in light of new information and experience. But, we will leave this issue aside for now and assume that such subjective probabilities are an appropriate way in which to represent uncertainty. This allows us to consider what is required of the human mind if we are to accurately use a probability calculus to guide our reasoning with uncertain knowledge.        The standard probability axioms are shown in the figure below.

Note: a partition is mutually exclusive and exhaustive. These axioms must be satisfied by any assignment of probabilities to a set of propositions.      The next figure shown below illustrates some of these relations between propositions and the assigned probability for a very simple world that involves the toss of two pennies. At the top of the figure is enumerated the set of propositions that can hold in this simple world. Next, the universe of possibilities is shown. This is the set of all possible state descriptions that could possible hold. In this simple world there are only four such states. Since each proposition may either be true or false, all possible worlds is given by 2 to the n where n is the number of propositions.          Notice, that in the above figure we have only stated constraints on the probability assignment. An actual probability assignment has not been made. If the penny is a "fair" coin and the procedure for tossing the coins is "fair," then it is reasonable to assume that the possible events are equiprobable. Each of the two coins can come up heads or tails. Thus, if these events are equiprobable then each has a probability of 1/2 or .5. And, the outcome for one coin has no influence on the outcome for another coin - the events are independent. Then the probability of the joint events can be obtained by multiplying their probabilities. Thus, the likelihood that they both come up heads is .5 x .5 or .25; that the first comes up heads and the second tails is also .25; that the second comes up heads and the first tails is .25; and that they both come up tails if .25. Note that the probability of at least one of the coins coming up heads is .75 since we must add the cases that can yield this outcome.      Note that in this simple case where the outcomes of the coin tosses were independent, we needn't worry about belief revision. Independence implies that the previous history of the outcomes has no influence on the current likelihood of either coming up heads or tails. However, what if we had rather special coins? Perhaps one of the coins, we aren't sure which will come up heads with a somewhat higher probability whenever the other coin has come up tails on 4 or more of the most recent 6 trials. In this case, the outcomes are not independent and we will constantly have to revise our estimates based on prior outcomes of the coin tosses.      Now that we have examined a very simple world involving only two coins we are ready to consider the general case. Note that in the general case we can not assume that the events are equiprobable and independent. Consequently, belief revision will clearly be a computationally expensive task. And, another difficult problem is to determine exactly how to assign probabilities to a particular event.        For example, pretend that we have 30 pennies that are tossed on every trial. And assume that these pennies are very special pennies that can influence the outcome of some of their fellow pennies. For example, penny 1 and penny 9 may be particularly in sympathy with each other and always take on exactly the same value. And, half of the time penny 6 will take on the opposite value of penny 1, and so on. How can we determine, for example, the probability of penny 6 coming up heads?        The next figure below discusses this general problem of assigning probabilities to a set of propositions in a coherent fashion for this general case.          If the propositions are not all independent, then assigning probabilities to propositions in a coherent fashion seems to be one of those intractable problems that we keep encountering. And, it is not entirely clear that the world is populated with domains where the propositions are independent. The contrived world of games of chance may represent the exception rather than the rule.

	Induction,Concepts, Uncertainty