Bayesian probability (update)
Blimey! Look ye here! The lady doth protest too much, methinks. I will try to respond succinctly.
I will continue to write in a fairly informal style in this blog, and to point to the relevant literature for the more discerning readers. I made this decision to embrace a wider readership at the cost of annoying a few readers. I can see why one might uncharitably compare this style of writing to that of postmodern literature criticism, but I will just have to live with that. I have a more "scholarly" (in places) blog here, but the blog is currently dormant because of problems with uploading images.
My main interest here is how we do inference about complicated systems consisting of many interacting parts. Note that there are two levels here: the system's behaviour itself (e.g. physics) and the reasoning about the system (e.g. inference). I am mainly talking about the second of these two levels. Note that this second level is where we all operate when we reason about "the world", because all we are doing is manipulating knowledge about things, rather than manipulating the things themselves.
The above paragraph must sound rather postmodern, but it's not! See the next paragraph.
Let's start by citing the literature yet again: Cox R T, "Probability, frequency and reasonable expectation", Am. J. Phys., 1946, 14(1), 1-13. This paper gives a neat derivation of the Bayesian approach to inference, by deriving everything from some elementary axioms which demand that the inference process must be internally consistent. Loosely speaking, if x is the state of a system, and Pr(x) is the joint probability of the components of x, then all inferences can be done by Bayesian manipulations of Pr(x) not x.
Bayesian inference is about manipulating joint probabilities (i.e. inference) rather than about defining joint probabilities in the first place (i.e. prior probabilities). These prior probabilities can be constructed in any way that you please, as long as they satisfy the usual properties (i.e. non-negativity, summing to unity), and then the Bayesian inference machinery can make use of them.
The freedom to choose a prior probability is an advantage, not a disadvantage, because it allows you freedom in your choice of model (or ensemble of alternative models). Bayesian inference then uses any relevant data to convert this prior probability into a posterior probability, which effectively updates the model (or ensemble of alternative models) in the light of the data.
Here is my original posting on the anthropic principle and Bayesian inference, so you can see for yourself how it has been selectively quoted here. In particular, check out the penultimate paragraph (the one starting "If the properties of the universe are correctly described by string theory (this may indeed be the case)...") for what I say about science, philosophy, and string theorists.
One can aspire to relate one's conjectured scientific theory to the real world, but the longer you are unable to demonstrate a strong connection between the two, the more your activity can be credibly labelled as "philosophy" rather than "science". I too would like to see the laws of physics derived from first principles, but I would not go so far as to assert that the laws of physics had to be derivable in this way. Whether we like it or not, the landscape is still a logical possibility, and we should at least be aware of what science would look like in that type of universe.
I also live dangerously in my own research activities, where I follow up some fairly wild ideas for long periods of time, but I always have "bread and butter" threads of research running alongside, where I dive for cover when the going gets tough. I never put all of my eggs in one basket no matter how elegant (or even how promising) it looks.