------------------------------------- What is the meaning of probabilities? ------------------------------------- To say that "The probability that someone in risk group A will die of cancer is 1/3" does _not_ mean that "10 out of 30 people in risk group A will die of cancer". It only means that, "on the average, 10 out of 30 randomly chosen people in risk group A will die of cancer". (Indeed, a probability is nothing but an averaged relative frequency, the average being taken in the sense of the mathematical expectation, i.e., the mean of the complete ensemble of interest.) Given a reliable stochastic model, such a statement can be checked (in the limit) by many repeated simulations, or (directly) by a theoretical computation; both require that the complete ensemble is available. On the other hand, the reliability of the model can be checked only by monitoring a large group of people and by counting how many will die of cancer. Of course, in using probabilities for predictive purposes, an insurance company tacitly assumes (without any guarantee) that the group of 30 people of interest is actually well approximated by a random sample, so that one can expect 10 out of the 30 to die of cancer. But this tacit assumption may well turn out to be wrong. Statements about ensembles are in principle exactly checkable: Operationally, to say that "The probability that someone in risk group A will die of cancer is 1/3" means nothing more or less than that exactly 1/3 of _all_ people in risk group A will die of cancer. (This assumes that risk group A is finite. For infinite ensembles, to define the precise meaning of '1/3 of all', one needs to go into technicalities leading to measure theory. Indeed, measures are the mathematically rigorous versions of 'classical ensembles' in general. For quantum ensembles, see quant-ph/0303047.) Of course, we cannot check this before we have information about how _all_ people in risk group A died, but once we have this information, we can check and verify or falsify the statement. In terms of precise mathematics: A classical ensemble is the set of elementary events underlying the sigma algebra over which the measure is defined. For example, in any finite sigma algebra containing random variables representing a fair coin (realizations 0,1; 1=head) with probability 50%), one has a finite ensemble of elementary events, and exactly half of them come out heads. For an infinite sigma algebra, the ensemble is infinite; but with the natural weighting, again exactly half of them come out head. Usually, however, we only have incomplete knowledge about the ensemble. For example, 'Tossing 10 fair coins' is just a sloppy way of saying 'Selecting a sample of size 10 from the total ensemble'. The sigma algebra for modeling this must contain at least 10 indepemdent random variables representing fair coins. This is the case, e.g., in the direct product of N>=10 sigma algebras isomorphic to 2^{0,1}. For N>10, it is obvious that here the number of heads is 5 (=50%) only on average over many random samples; and it is impossible to infer the exact probability from a single sample. This is why statisticians say that they _estimate_ probabilities based on _incomplete_ knowledge, collected from a sample. The resulting estimated probabilities are known to be inherently inaccurate; but they can be checked approximately by independent data (cross-validation) providing confidence levels indicating how much the predictions can be trusted. On the other hand, they _compute_ probabilities from _assumed_complete knowledge about the ensemble, namely the theoretical probability distribution. Thus if complete information goes in, exact information comes out, while computations based on incomplete information naturally only gives approximate results inheriting some uncertainty from the input. Computed probabilities are powerful, but only if the assumed stochastic model is correct. Empirical estimates are usually inaccurate but useful. The two approaches are not contradictory; indeed, they are combined in practice without difficulties at all. The only subjective aspect in the whole thing is the choice of a stochastic model when making theoretical predictions; and even this is made almost objective by the standard rules of statistical inference and model building. Indeed, the choice of ensemble is _always_ a subjective act that determines what the probabilities mean. It encodes what the user is prepared to assume about the given situation. Once the ensemble is chosen - either a theoretical, exactly known ensemble, defined by specifying a distribution, or as a real life ensemble of which only a (perhaps growing) sample is available, all probabilities have an objective meaning. A chosen ensemble is knowledge precisely if it is close to the correct ensemble, and we have a good idea of how close it is. That's why we value highly scientists such as Gibbs who guessed the right ensembles for statistical mechanics, which turned out to be a highly accurate description of equilibrium situations. Only good choices are knowledge. And what is good is found out only through proper checking, and not through the principle of insufficient reason. In case of tossing a coin we know that the fairness assumption is usually reasonable, being consistent with experience. In case of taking an exam at a newly appointed professor about whom no one knows anything, reasoning from the two possible outcomes (pass or fail) and the principle of insufficient reason to assign a probability of 50% failure is ridiculous, and dangerous for those who are not prepared.