First, good for you for joining the fray on this issue. I hope it doesn’t consume your entire life, but judging by this comment thread, it may just be too late

Anyway, I do have a reason for posting, and it’s this: If you’re not familiar with it, I’d like to recommend Ed Jaynes’ “Probability Theory: The Logic of Science” (Cambridge, 2003). Jaynes speaks more sensibly on the meaning and interpretation of probability than anyone else I’m aware of. The first three chapters of the book are available online, if you don’t want to shell out for the hardcover. Here are a couple of the lessons I’ve taken away from the book:

Jaynes’ basic approach to probability is that it is simply an extension of Aristotelian logic. That is, it should be an honest and consistent method for conducting plausible reasoning based on incomplete information. “Consistent” here means that the basic mathematical rules of probability theory apply (and incidentally, those rules do simplify to the rules of logic in cases of certainty). “Honest” means that the reasoning should include all the information known (including the uncertainty of that information), and should be as non-committal as possible about what is not known. Much of the book is dedicated to explicating just what that means, both mathematically and philosophically.

Jaynes is a “subjectivist”, insofar as he considers probabilities to represent states of information. That information need not be just some person’s WAG at whether it’s going to rain tomorrow, as subjectivism is often caricatured. In fact, a running example in Jaynes’ book is what he calls “the robot.” The robot is just a hypothetical computer, and the problem is to make the robot reason as well as possible about propositions. How, exactly, should we program this robot? Jaynes argues convincingly that, if the robot is to satisfy some pretty basic desiderata of consistency, then the programming must be equivalent to the basic rules of probability.

An important consequence of the “probability as extended logic” interpretation is that probability is not just a method for calculating frequencies of random variables; it can be applied to plausible reasoning of any kind. The frequentists’ hang-up with defining “randomness” as the sole domain of application is a distraction; probability is the only consistent method of reasoning under uncertainty, whether that uncertainty is about something that is unknown, or something that can’t be known, or something that is truly “random,” if truly random phenomena even exist.

And Jaynes has plenty to say about randomness, finding that in most cases the idea of randomness is an instance of what he calls the “mind-projection fallacy”, an assertion that one’s own internal mental states are a property of external reality. Calling something random is like saying “I don’t know what causes this phenomenon to occur differently in different trials, therefore Nature doesn’t know either!”

Anyway, I’ve gone on too long. Check out Jaynes; he changed the way I think, probably as much as anyone ever has.

Mike

Here is why judgments of value (economic or not) are not always necessarily subjective. Libertarians are always bragging about how their theories are always purely deducted from true premises. Well, here is one of my own syllogisms that basically states plainly and explicitly that the choice to use reason to make inductive generalizations is itself a value judgment, the only way to avoid the truth is to reject reason or reject reality or reject volition. Take your pick.

“If discovering the truth of what is requires the choice to observe and infer, then the ought is established with the very first observation.”

Your inference that causality is subjective (by which I think you mean relative or reference-frame dependent) as a consequence of Special Relativity is incorrect. While observers in relative motion may observe a given event at different times (on initially synchronized clocks), the time difference between two observed events (e.g. a cause followed by its effect) will always be positive, as the proper time for each observer is always moving forward. Einstein would have considered his theory worthless if it meant that causality was relative.

You are right to say that practitioners of probability today often completely overlook the definition of probability and delve right into the mathematics. To think that this solves the definitional issues I am discussing, however, is t completely misunderstand the debate.

I have no idea what you mean when you say that my description of the classical definition is “wrong.” Are you saying that this was not the method that was utilized prior to the frequentist revolution? If you are saying this, then you are completely ignorant of the history of probability prior to Venn et al.

That’s just wrong. That’s basically low-down Laplace distribution. Today, everyone uses:

http://en.wikipedia.org/wiki/Probability_axioms (Ta-daaa!)

“The relative frequency method does not rely on the assumption that all outcomes are equally likely, which is extremely helpful for most applications outside of the casino.”

In a handwaving sense, yes. But you cannot actually work mathematically with this, as the frequentist approach tries to define the probability by using it in its definition (the probability of event e is the relative occurrence of the event over a large number of trials, whereby the actual number obtained lies – with high probability – near the correct value (i.e. “converges to the correct, value”)

Apparently Quantum Mechanics can be seen as a “straightforward” extension (if you are at home with linear algebra on complex vector spaces) of Classical Probability Theory. This is detailed in Streater’s work where he also explains why Schrödinger’s cat decoheres quickly as it becomes larger as well as why you naturally can have strong correlations between random variables measured at non-causally-connected points in QM.

Lucien Hardy has another approach. I’m still working on understanding this in detail though.

Free Reading: Lucien Hardy – Quantum Theory From Five Reasonable Axioms http://arxiv.org/abs/quant-ph/0101012

Free Reading: Lucien Hardy – Why Quantum Theory?
http://arxiv.org/abs/quant-ph/0111068

Free Reading: R.F. Streater – Classical and Quantum Probability
http://arxiv.org/abs/math-ph/0002049

Costly reading: R.F. Streater – Lost Causes in and beyond Physics http://books.google.com/books?id=xbfNVmrfonIC

[Streater takes no prisoners and rolls over frequentists while discussion elementary probability on page 25. Definition of classical probability not based on a mathematical Kolmogorov Model (to be properly tested against the real world) are to be avoided.]

If a person has 10 shuffled cards of the numbers 1 to 10, everyone agrees that the probability of my blindly drawing the card number 6 is exactly 10%.When the weather person says there is a 10% chance of rain tomorrow, or your father says there is a 10% chance of finding oil on a particular property, these numbers are not at all precise. For example, another weather person may say there is a 20% chance of rain tomorrow, and someone other than your father may say there is a 25% chance of finding oil on the property in question.

If you want to call the probability figures stated by these different people “subjective probabilities”, you, of course, are free to do so. But since such subjective probabilities of particular events vary with the person making them, clearly such numbers are essentially indicators of the strength of belief particular individuals have (given their expertise, data and analytic ability) in an event occuring rather than not occuring. For example, the second weather person has a stronger belief that it will rain tomorrow, while still believing, as the first weather person does, that it is most likely not going to rain tomorrow.

I can understand your concern that my defense of subjective probability is a bit thin here. I encourage you to read my papers at Libertarian Papers for my more thorough argument. I can make a few comments, However.

First, I agree that it is important to address the metaphysical issues that I bring up here, and I do address them in my first paper at Libertarian Papers. My claim there is that the use of the relative frequency method for generating numerical probabilities assumes that the world is governed by time-invariant causal laws. This is so because the relative frequency method is predicated on being able to identify cases that are virtually identical in every way in order to conceptually classify them as a “class.” And this means that one must assume that each case is affected by the causal laws of the world in exactly the same way, or else it would not be possible to construct conceptual classes.

What this means is that the scientist who would try to prove that the world is inderterministic at the micro level with relative frequency-based methods would be contradicting himself, because the very methods he would be using assume the world is causally deterministic.

Second, your question regarding what you call the “Bayesian interpretation” is a bit complex. Often, when people refer to Bayes in this context, (instead of Savage and de Finetti, for example), they are referring merely to “conditional probabilities.” This is an important idea, but it is not equivalent to what I am describing in my articles. My position is simply that probability is a measure of human belief or uncertainty, like Savage, de Finetti and especially Good.

Third, I think you are assuming that I do not approve of the relative frequency method, the classical method, or any other methods that are not “subjective.” Actually, however, my position is that these methods are perfectly fine to use. Indeed, I think that if one has the data, one should use it. I still say that the numbers generated by these methods are measures of human belief or uncertainty, but I am all for using the relative frequency method when one can, because it has proved itself to be extremely useful in practice. Saying that all probabilities are measures of uncertainty by no means forces us to say that ONLY subjective methods are acceptable, however.

My father spent 30 years using “subjective methods” to generate probabilities about oil and gas resources, because no data was available to use the relative frequency methods. He always says: “If you have the data, of course you are going to use the relative frequency approach. But, if the data doesn’t exist, that doesn’t mean we can’t generate probabilities.”

First of all, thank you for the interesting paper. However, I do have some notes:

1. In the future it may be more prove useful for you to explain the the metaphysical assumptions you make before delving straight into epistemology. It seems to me that you adopt a deterministic framework of some sort. In such a paradigm the probability of every event that happens or will happen is 1, and therefore probability can only have meaning as an epistemological concept reflecting the subjective ability of humans to predict events. While some may not agree with your assumptions it would save a lot of arguing.

2. I do have a problem with your definition of causality though. Why do you assume that causality is an objective property, rather than a subjective framework you place on reality in order to make sense of it. You say:

“In the natural world, an event occurs because some force or forces caused it to happen. Leaves do not simply fall off of trees for no reason, and neither does any other event in the world occur for no reason whatsoever.”

But according to our current model of understanding the world on a macroscopic level (which was also verified experimentally) which is the theory of relativity, given two events you cannot say which one preceded the other. This property is dependent on your point of view or reference frame to be more precise. The same two events may occur simultaneously to a different observer. So, from the onset I fail to see why it is so clear to you why causality is objective and probability is subjective, and not the other way around for example.

3. Is there a difference between what you call the ‘subjective approach’ and the more commonly used name ‘Bayesian interpretation’? I will assume that the answer is no until you reply. It is true that the Bayesian interpretation offers many benefits for anyone dealing with statistics: it allows for direct comparison between hypotheses, it allows for iterative changes of probability based on experience, it enables one to apply probability to events he has never encountered or has prior knowledge of, and its most important property is that it gives a meaningful interpretation to the the concept of probability for a singular event in a deterministic framework.

However, this approach also has problems in my opinion, mainly in dealing with the basic central theorems of probability theory. How can you explain the strong law of large numbers subjectively? It seems to work without fail (hence the word law) for large numbers which would make it objective rather than subjective.

A subjective approach also has problems explaining our intuitive ability to calculate probability and expectation correctly. If probability is subjective how come I can reach the correct probability of results within a margin of error arbitrarily small for large enough numbers before the I commence the experiment (in certain cases)? Bayesian interpretation has a problem justifying its own a priori probability if it is correct from the onset.

Therefore; whilst objective probability is a viable concept for events that are not subject to the non-linear effects of complexity, for the most part, we can only approach probability from a subjective angle because we live in a world fraut with complexity

http://www.youtube.com/watch?v=SFu5BlJClYI

And yes, NOT all human action has a determinable causation – the drunkard drinks not because he ideally would have preferred or liked doing so, but simply because he or she cannot help it… Often than note, humans behave not from their best self-interest, as classical Praxeology tries to convince us to be the case…

Praxeology makes very few claims about the structure of utility functions. As we use it, “self interest” means performing actions that rank highly on the utility function, even if that function tells them to set themselves on fire and jump off a cliff.

Wow, it’s been a while since I read such harsh criticism of Bohr. Granted, it’s been a while since I’ve read anything written by Bohr other than the phrase in Latin on his coat of arms. Anyhow, is it justified to attribute nihilism to him without qualification? After all, did he not claim until the end of his life to know that his own conception, ca. 1912, of electrons contradicted reality?

No argument that nihilism is intellectual degeneracy, however. And it’s very prevalent among cheerleaders for science. Just this past weekend I attended a meeting, my first, of a large group of skeptics that meets monthly in my neighborhood. It was an orientation meeting at which the group’s primary organizer repeated the buzzword “science” many, many times.

She also cast suspicion on absolutes with the text of her PowerPoint presentation. Then she upped her ante by openly disdaining absolutes while speaking. Several days later, the organizers solicited comments about the meeting, so I asked, rhetorically, “if there are no absolutes, then why suppose it the case that ‘existence is’ ? And if existence were not, then from where would your skepticism of absolutes come?” Still no reply from the skeptics.

You might have enjoyed the meeting for its entertainment value, by the way. It was like a scene from the dystopia of Atlas Shrugged. Balph Eubank, Dr. Floyd Ferris, and Bertram Scudder would have been among kindred spirits.