La morfología urbana

The Bayesian approach to scientific reasoning has become a popular one, due to its response to the problem of induction: that is, how to determine whether or not a particular theory is the right one (or the likely one), given that all evidence for such theories is empirical evidence. It proposes a probabilistic induction, where a given theory or hypothesis is deemed more or less probable, based on the scientists’ degrees of belief about them and how these degrees of belief ought to change given some particular evidence that bears on that theory or hypothesis. (Howson Urbach 9) I will need to go into more detail for all of these elements. What I will turn to first, however, is the theorem that all Bayesians utilize in order to determine the probability of some hypothesis or theory given the empirical data: Bayes’s Theorem.

Bayes’s Theorem can be stated in the following form:

P(a | b) = (P(b | a) P(a))/ P(b), where P(a), P(b) >0 (Howson Urbach 28)

P(a | b) represents the probability of the hypothesis a, given the evidence b, which is equal to the likelihood of the evidence given the hypothesis multiplied by the initial probability of the hypothesis, all divided by the probability of the evidence. (Howson Urbach 28)

So, in order to determine how much some piece of evidence affects (negatively or positively) the probability of a certain hypothesis or theory (being true), the scientist just needs to determine what the relevant prior probabilities were – those before the evidence came to light – and plug them into the theorem.

Of course, this depends on there being some principled way to determine the prior probabilities needed to complete this equation: clearly, the probabilities cannot just be determined by a roll of the dice, or by picking numbers out of the air. If they were so determined, there would be no reason to believe the Bayesian when she said that her theory was a good theory of confirmation.

The standard method of describing the prior probabilities is to speak in terms of fair betting odds. (Howson Urbach 75-76) Given a certain hypothesis, the fair betting odds of that hypothesis would be ones where, if someone were to take those odds and bet for the hypothesis, there would be no expectation of an advantage or disadvantage as opposed to betting against the hypothesis. (Howson Urbach 75) These odds, then, are the prior probability of the hypothesis, or the degree of belief in that hypothesis. P(a) is the bet that the hypothesis is true. P(b | a), then, will be the odds of the evidence, given the truth of the hypothesis. This would be a conditional bet on b, given a. (Howson Urbach 81-82) And these bets, as they do not predict a net advantage for either betting for or against, obey the probability calculus.3 This is important because, as Howson and Urbach claim:

[I]f a set of betting quotients fails to satisfy the probability calculus, then were anybody to bet indifferently on or against the associated hypotheses, at the odds determined by those quotients, he or she could be made to suffer a net loss (or gain) independently of the truth or falsity of those hypotheses. The importance of this

3

Howson and Urbach present a series of arguments for this claim (Howson Urbach 78- 95), which I will defer to, here. The thrust of these sections seems to be that there are so many different arguments, starting in different places, that lead to the same conclusion – the probability calculus. As they say: “The latter [probability calculus] seems, in other words, to be a sort of invariant of different ways of defining uncertainty, or as Lindley puts it, “inevitable”, meaning that the choice of any plausible way of mathematically measuring uncertainty will lead to it. This convergence of arguments has a powerful cumulative effect and increases our conviction that the probability calculus corresponds to some quite objective feature of subjective uncertainty.” (Howson Urbach 95)

result lies in the corollary, that betting quotients that do not satisfy the probability axioms cannot consistently be regarded as determining fair odds. (Howson Urbach 79)

In other words, if these ‘fair odds’ do not satisfy the probability calculus, the probabilities associated with these odds will not represent justified prior probabilities.

There are two elements of this description that are stressed, rightly, by Howson and Urbach. The first is that there need not be a propensity to bet in favor of the

hypothesis if the odds are fair, much less does there need to be an acceptance of a bet on those, or greater, odds. (Howson Urbach 77) There may be many reasons as to why a person would not actually entertain, or make, a bet even if the odds are considered fair. It is enough that there is no perceived advantage or disadvantage to someone that would bet for the hypothesis rather than against it. Secondly, Howson and Urbach stress that this discussion of subjectively fair odds needs only that people sometimes perceive odds as being fair. There do not, in fact, have to be fair odds at all in order for there to be subjectively fair odds and, therefore, justified prior probabilities. (Howson Urbach 77)