At its surface, Bayesianism seems to be very compatible with Objectivism. The establishment of priors is the equivalent of asking yourself "What do I already know about this?", and the subsequent adjustment of probability seems equivalent to adjusting one's understanding based on new knowledge. There are, however, a few issues I can see with the idea, and I'd like to get the views of someone more knowledgeable on the topic. The Bayesianism article on Wikipedia seems solid, and the website "Less Wrong" is devoted to the application of Bayesianism.
Here two of are my main issues, as Bayesianism seems to be practiced:
1) Unknown systems are not treated as unknown. The prior for a truly unknown system with n possible results is to assign each possible result a probability of 1/n. However, since you known nothing about the system there's nothing to justify this conclusion. It's arbitrary.
2) Bayes' theorum can at best give probabilities. The real world, however, is binary--something either happens, or it doesn't. In terms of choosing actions (where Less Wrong focuses), you can either act or not--the concept of acting 70% is absurd. What this means is that either Bayesianism offers no guidance for action, or that it forces one to set up arbitrary limits (say, "If it's above a 75% chance I'll act as if it's certain"). This again introduces arbitrariness into the system.
(Priors are not something I think needs criticized--except in cases where they are arbitrary, they actually amount to sitting down and systematically thinking about what's known about the system already, which is a useful habit to get into regardless of the validity of Bayesianism.)
asked Jan 05 at 12:26
The question didn't provide an exact reference in Wikipedia. I found several Wikipedia articles dealing with the general topic of Bayesian statistics, but no article titled, "Bayesianism." One apparently very good article that I found is "Bayes' theorem," which describes the theorem mathematically as follows:
To understand what these terms mean and why the relation is valid, the article presents an analysis essentially the same as the following. Consider a set of n items. Some of the items meet criterion 'A', others meet 'B', some meet both 'A' and 'B', and some fail to meet either 'A' or 'B'. The set can be represented by the following table, where an apostrophe (') denotes the property of not meeting the associated criterion:
Let w = number of items that meet both criterion A and criterion B.
Then the probabilities of an item meeting criterion A or B or both or either, relative to n, are:
P(A) = (w+x)/n
P(A&B) = w/n
(When used with A and B, 'v' means inclusive logical 'or,' and '&' means logical 'and'.)
Also, "probability" can be understood in terms of a set of items randomly mixed together, with one item at a time being drawn randomly from the set, recorded, and then replaced randomly in the set before performing the next random draw. Each item then has an equal chance of being drawn on any drawing. The average or expected number of drawn items found to meet some criterion C will then be expected to match the percentage of such items existing in the set, with a statistical variation that depends on the number of items drawn. The larger the number of "trials" (drawn items), the closer the average number of C's will approach the actual number of C items that exist in the set.
Now define two important "subset" probabilities for the wxyz set described above:
P(A|B) = w/(w+y) = relative rate of occurrence of A within subset B
Consider some key mathematical relations ('*' denotes multiplication):
P(A|B) * P(B) = [w/(w+y)] * [(w+y)/n] = w/n = P(A&B)
Hence, P(A|B) = [P(B|A) * P(A)] / P(B).
Bayes' relation can also be written as:
Note that it is possible for P(A|B) to be greater than P(A); compare w/(w+y) vs. (w+x)/n. Bayesian analysis seems to be especially interested in cases where the fact that an item meets B raises the probability that the item also meets A, relative to the original probability, P(A), in the total set n.
According to Wikipedia, "Bayesian statistics" is basically just an application of Bayes' Theorem. To the extent that the application is performed in a mathematically valid manner, with well defined meanings of the terms in the analysis, the results should be valid and applicable to the real world (assuming that the definitions of the terms and assigned estimates are tied to reality). I do not know enough about the mathematics of Bayesian analysis myself to comment further on its mathematical validity, but I note that the Wikipedia articles mention criticisms of some of the methodologies used by some Bayesian analysts. Before an Objectivist would be able to comment further in terms of Objectivist epistemology, any criticisms on mathematical grounds normally would need to be understood and evaluated first. I suspect that the number participants on this website who know enough about both Bayesian analysis and Objectivism to comment beyond this is very probably zero, but I'm open to the possibility of statistical "outliers" (exceptions to the "main" statistical distribution).
If this Answer seems too mathematical for an Objectivist Answer, let it serve as a demonstration of the difference between a mathematical discussion and an Objectivist one, and of where the two modes of discussion may intersect.
Update: Less Wrong
In a comment, the questioner seeks additional evaluation of the lesswrong.com website. The FAQ page on lesswrong.com provides the following high-level overview of what the website is all about:
Less Wrong is an online community for discussion of rationality. Topics of interest include decision theory, philosophy, self-improvement, cognitive science, psychology, artificial intelligence, game theory, metamathematics, logic, evolutionary psychology, economics, and the far future.
Rationality is certainly of great importance to Objectivism (including rational philosophy and economics), but I do not know enough about lesswrong.com myself to comment on how well they do or do not succeed in their mission to promote rationality. The questioner's own comments in the formulation of the question provide grounds for some concern. Without a more exhaustive study of lesswrong.com myself, I cannot readily judge whether or not the questioner's concerns are well placed. I would be especially interested to know whether or not lesswrong.com delves into the realm of moral values in relation to rationality, and whether or not they see reason as applicable to ethics at all (as against so many other, more conventional philosophical perspectives on reason and ethics). Unless or until I complete a more detailed study of lesswrong.com myself, I must rely on others' evaluations of what lesswrong.com most fundamentally stands for, compared to Objectivist (and/or Aristotelian) philosophical principles.
There are many other websites today that also claim to be promoting some aspects of reason, individualism and capitalism. It's not clear to me (so far) whether lesswrong.com is any more deserving of Objectivists' attention than other alternatives, although lesswrong.com does seem (on the surface) to be more mathematical in their main focus areas.
I also checked the discussion titled, "What Do We Mean by Rationality?" (http://lesswrong.com/lw/31/what_do_we_mean_by_rationality/). I found that discussion to be meandering and groping for wisdom amidst chaos, apparently without ever having heard of Ayn Rand or Objectivism.