Probability Essay, Research Paper
Probabilities are not readily available in the world around us. Expressing uncertainty, probability represents precisely what is epistemically unavailable to us. Also the concepts chaos and free choice indicate a lack of predictability of the world. Probability is distinct from chaos and free will in that it presupposes some type of long run regularity. In this section we shall deal with questions such as how probabilities can be assessed and evaluated and to what extent long run regularities are relevant to this issue. Is every long run relative frequency a probability? What is the probability of a single event?
There are several methods to assess a probability and they can be distinguished broadly in four different methods, i.e., the frequency theory, the propensity theory, the logical theory and the personal theory. Each of these methods has its own criteria of assessment and evaluation. We shall advance an eclectic mixture of all four theories. It seems to us that the wide semantic range of chance should be reflected in an equally rich interpretative approach to probability. The quantum mechanical behavior of subatomic wave-particles is generally given a propensity interpretation, whereas a die, if not suspected of being biased, exhibits probabilistically logical behavior. The traffic in New York City has been modeled with a probabilistic, frequentist model. Decisions by drivers have been replaced by impersonal, randomized events. A doctor interprets the posterior probability of having breast-cancer given a positive test-result of the mammogram as the level of epistemic certainty she has on the basis of the test alone.
In the following sections we shall touch upon three separate issues of probability, its interpretations, methodologies and structure. Besides an eclectic interpretation of probability, it is our claim that any probability, whatever interpretation suits best, possesses a frequency structure, i.e., it is always possible to express a probability in terms of a long-run relative frequencies. One important methodological aspect of probability follows from our observations concerning coincidence. Any Platonic image of probability, namely as the world as a list of probabilities out there, is misguided. This idea is defended by some personalists, logicists and frequentists, but is most frequently found among the former. Platonic ideas about probabilities can be found among logicists and personalists with respect to the issue of a uniform prior. In the following sections we shall examine the personal theory of probability and the frequency theory of probability.
Some things are thought to be more probable than others. It is more probable that the sun will rise tomorrow than that it will not. One holds this belief quite strongly — and legitimately so. Probability captures, in some sense, the strength of belief, i.e., of the confidence that the event will be repeated. Personalist theory postulated that probability not only expresses the strength of belief, but that it is actually defined as such. Personalists associate probability with the subjective magnitude of seeming probable. The sun-rise example is particularly interesting, because it has been a focus of controversy between personalists and defenders of other theories. We shall return to this example shortly.
Personalists make a distinction between the construction and the evaluation of a probability. According to personalists, “the question when a probability statement is correctly made has two different meanings: (1) How should we make or construct well justified probability judgments? For example, when is a surgeon justified in saying that an operation will succeed with probability 0.90? (2) How should we evaluate such judgments after we know the truth of the matter? For example, how would we evaluate the surgeon’s probability judgment if the operation succeeds, or if it fails?” In what follows, we shall concentrate on the construction issue of a personal probability. Methods of evaluation, such as calibration, are discussed only briefly.
Personalists come in several flavors. After ground-breaking work by Definetti and Savage, different schools of personalist thought have developed. All personalists have in common that they define probability as a numerical measure of the strength of a belief in a certain event. Their picture of belief bears strong affinity with that of the British empiricists. John Locke argued that every belief is held with a certain “strength” in the human mind. The personalist theory of probability interpreted this strength as the individual’s idea of the likelihood of the event. The current mainstream personal theory is the Bayesian theory, and we shall use the terms interchangeably. Bayesians believe that it is possible to make probability assessments even in the absence of frequency information. However, a personal probability is not a mere opinion. It is an orderly opinion. The personal theory specifies consistency rules. One of the concerns of the theory is with the revision of a probability in the light of new evidence. Bayesian theory developed a calculus of beliefs that specifically deals with this issue. The original degree of belief is replaced by a new degree of belief when new evidence is obtained.
The personal theory of probability raises a number of issues. The great advantage of the theory, viz., the completeness of probability judgments over the set of all events, is also its great weakness. The initial prior probability, i.e., the initial strength of belief, is based on a mixture of the available information (which includes feelings, knowledge or pressure from some authority) of the individual. Prior beliefs are subject to the individual’s bias. Personalists defend a Peircean stance to truth, i.e., they believe that the true probability is the point of convergence of the scientific process. The possibility of wide variation of prior personal probability assessments has been recognized by the personalists as their Achilles’ heel, and it has been become the aim of serious theory to show mathematically that in the light of new evidence different personalist probability inferences will converge to the same numerical value. Several convergence theorems have been proven.
In the following sections we shall focus on some important aspects of Bayesian theory. Bayesians have recognized that acting under uncertainty is essentially the same as making a bet. In constructing probabilities Bayesians have historically invoked several additional assumptions that are not directly related to any frequency ideas: the idea of risk neutrality and the principle of insufficient reason. We shall briefly discuss some Bayesian calculations. Further mathematical details are placed in the appendix. We shall then return to the Sunrise example and show that the world cannot be considered as a list of probabilities. This will call upon the personalist — but not only on her — to show moderation in `finding’ probabilities.