Dear Rich, This is probably much more than you bargained for, but you asked for it! :-)
The Oxford American Dictionary defines objective as "having real existence outside a person's mind." Strict subjectivists such as de Finetti and Savage regard probabilities as entirely subjective except for the requirement that they conform to coherence constraints. Other scientists take the position that there are phenomena in nature that follow intrinsically probabilistic laws, and that the probabilities associated with these phenomena are objective properties of the world, reflecting an aspect of reality outside the minds of human scientists. As you know, this is an issue that arouses great passions and profound disagreement. You undoubtedly know of de Finetti's exchangeability theorem, which states that if You regard a sequence of events as infinitely exchangeable, then You assign a probability distribution whose mathematical form is that of a sequence of independent and identically distributed events conditional on an unknown parameter which is distributed according to a probability density function. If you show this theorem to a strict subjectivist, she will say: "You see, I TOLD you! There is no need for a theory with 'true but unknown' probabilities! We can do everything we need by sticking strictly to subjective probabilities and exchangeability assumptions." If you show this theorem to a believer in objective probabilities, he will say: "AHA!! You subjectivists DO believe in objective probabilities! You just won't admit it. But there they are, right there in the theorem, a mathematical consequence of your very own subjective theory. That parameter you are mixing over in the equation is the true but unknown probability!" The two of them agree on the mathematics but disagree on the words to attach to the mathematics. Which is correct? Phil Dawid (in a wonderful paper called "The Well Calibrated Bayesian") went beyond de Finetti's theorem to arbitrary sequences of events, not just ones that frequentists would model as independent events with a given probability. He proved something similar to but more general than de Finetti's result. If You are a coherent Bayesian who assigns probabilities to a sequence of events, and Your assessments reflect all Your knowledge prior to occurrence of the event, and You receive feedback on the actual outcomes of the events, then You believe, with probability 1, that after a long enough sequence of trials, Your assessed frequencies will match actual observed frequencies to arbitrary accuracy. Now of course, You could be wrong in your belief that Your assessments will become calibrated against frequencies. But if any two assessors are calibrated in this sense, the numerical difference between their probability assessments will become arbitrarily small over long periods of time. Now, someone whose assessed probabilities reflect lack of knowledge of a deterministic process won't be calibrated in this sense, because an assessor who had knowledge of the deterministic process would assign probabilities of zero or one, and wouldn't agree in the infinite limit with any probabilistic forecast. A counter-argument to this is that a process might be deterministic but not knowable by any forecaster -- but what exactly would that mean? Now let's consider quantum theory. Quantum theory is the best-validated scientific theory that has been discovered to date. It makes probabilistic predictions of the form: "If a system has been prepared according to experimental procedure E, and measurement M is performed on it, then result R will occur with probability P." There are very strict constraints on quantum probabilities. Specifically, the experimental procedures covered by the theory are ones that reliably produce a system in a given quantum state, where a state is represented by a mathematical object called a density matrix, which is an operator on a Hilbert space (you can think of the quantum state as a square matrix with complex entries with positive eigenvalues that sum to one). A measurement is represented by a projection operator that projects the density matrix onto a subspace (i.e., the state is replaced randomly by one of a set of a matrices of lower rank). The probability that the result will be a given random lower-rank matrix is given by a rule called the Born rule (you sum the eigenvalues for the subspace spanned by the lower rank matrix). In all experiments to date for which we have sufficient knowledge and control to produce a system in a definite state and subject it to a definite measurement, i.e., where we can make accurate tests of the predictions of quantum theory, the predictions of the theory have been borne out to stunning accuracy -- and in many cases these predictions profoundly violated the intuitions of the best scientific minds of the time. No one has ever found a violation of the predictions of quantum theory. All physical phenomena we know of appear to be consistent with quantum theory. So are quantum probabilities objective? Einstein didn't think so. Jaynes didn't think so. They both believed that quantum probabilities reflect ignorance of some more fundamental deterministic theory. Despite considerable effort, no one has yet found such a theory. But people haven't stopped trying. Carlton Caves has asserted that quantum probabilities must be subjective because an observer with full knowledge cannot have a probability other than zero or one, given that full knowledge would include the outcome that actually occurred. But a meaningful definition of "full knowledge" for a probability assessment would have to include only events occurring at times before the event in question. I suspect Caves doesn't see any non-subjective way to define "full prior knowledge" in a universe where temporal relationships between events are relative. But there is a meaningful, observer-independent definition of "before", according to which we can specify what "full knowledge" for quantum probability assessments should mean. In a universe subject to relativity theory, full prior knowledge means everything in the past light cone of the event. That is, quantum theory predicts a probability for the state of a system immediately after a measurement is performed, conditional on the measurement operator that was applied and the entire past light cone of the system at the instant of measurement. Quantum theory says that the only aspects of the past light cone that are relevant to predicting the probability are: (i) the state of the system just prior to applying the measurement operator; and (ii) the operator that was applied. Quantum theory gives a precise rule -- that depends only on the prior state and the operator applied -- for calculating the probability. In experimental tests, quantum theory's predictions have been confirmed to very high accuracy. Do you think this satisfies the Oxford American Dictionary definition of objective? Do you think quantum probabilities have existence outside a person's mind? Or are quantum probabilities nothing but the opinions of a bunch of subjective assessors who agree in their probability assessments and have thus far been very accurate in predicting frequencies -- but there is nothing objective in the real world that corresponds to those probabilities? I've been having this kind of discussion, and reading and listening to others having this kind of discussion, for a very long time. What I've discovered is that there are many people are sure that the answers to these questions are blindingly obvious, and that anyone who doesn't see the truth must either be stupid or willfully blind. The problem is, some of these people are sure the answer is yes, there are objective probabilities. Others are just as sure the answer is no, there are no objective probabilities. All of these people use the same equations and make the same empirical predictions. There is no observation you could make and no test you could run that could prove one right and the other wrong. Thus we aren't arguing science here. We're talking religion. So are you an objective realist about probabilities? Or are you a strict subjectivist? It's not a matter of the math or the science. It's a matter of metaphysics, religion, the story you tell yourself to make sense of the math and the empirical data. As a teacher, I want to understand people on both sides of the issue think, so that I can help students to understand and to find their own most natural stories to tell themselves to make sense of the math and the empirical data. Best regards, Kathy At 12:32 PM -0700 8/21/06, Doug Morgan wrote: >Rich, > >Today's best models for detailed physical behavior involve the step of >collapsing a wave function upon making any measurement. This step >is modeled as >a draw from a random distribution computed from the wave function. The wave >function may be a Feynman path integral or a solution to the Schrodinger wave >equation or something similar, but in all cases it is a dynamic thing that >updates in a deterministic way and from which a probability >distribution for the >measurement can be computed. The definition of measurement is ultimately left >to one's physics intuition, but always includes any permanent effect >potentially >noticable by a person (few experiments have been proposed or >performed that make >a sharper distinction between what is and what is not a measurement). This is >just the model people use. Some people look for a non-probabalistic >model that >matches observations as well as or better than this one. Such a model hasn't >been found yet. If at some point such a comprehensive non-probabilistic model >should appear, one might be justified in a strong feeling that the universe >could really be deterministic. Until then, the ubiquitous use of probability >models for physics could make you wonder. What connections between questions >like these and philosophy might be, I can't imagine. > >Doug > >Rich wrote: >> My interests have recently been roused again by the question that has >> plagued many of us, at least since Laplace: Is the universe >> deterministic or is there something truly probabilistic going on >> (whatever that means)? Actually it is my inability to resolve this >> matter that has made me suspect humans are relatively unevolved, and >> therefore incapable of understanding much. >> >> In any cases I would be interested in learning of any recent >> philosophical treatise on this matter. I haven't read anything on the >> matter in about 5 years. >> >> Thanks, >> Rich >> >> Richard E. Neapolitan >> Professor and Chair of Computer Science >> Northeastern Illinois University >> 5500 N. St. Louis >> Chicago, Illinois 60625 >> >> _______________________________________________ >> uai mailing list >> uai@ENGR.ORST.EDU >> https://secure.engr.oregonstate.edu/mailman/listinfo/uai >> > >_______________________________________________ >uai mailing list >uai@ENGR.ORST.EDU >https://secure.engr.oregonstate.edu/mailman/listinfo/uai _______________________________________________ uai mailing list uai@ENGR.ORST.EDU https://secure.engr.oregonstate.edu/mailman/listinfo/uai
