Frequentists vs Bayesians
infoproc.blogspot.com
infoproc.blogspot.com
I know what a Bayesian is. I'm not sure that even frequentists know what a frequentist is.
Like, suppose someone hands me a biased coin. A frequentist can flip that coin 10 times, then make a bound on the true bias of the coin that holds with the promised degree of confidence. (E.g. they can make a bound that will hold on 95% of such experiments.)
Bayesians can't do this. There are lots of other advantages to the Bayesian approach, but to frequentists, nothing could be worth giving up "coverage" (confidence intervals obeying their guarantees).
P.S. When looking at those ten coin flips, and trying to estimate the true bias, the frequentist and bayesian will have the following argument:
Bayesian: The probability that the true bias is b is __some formula___
Frequentist: Are you insane? The true bias of the coin is what it is! It's a number! Just because you don't know what that number is doesn't mean anything probabilistic is going on.
Bayesian: Well it is extremely useful to be allowed to make such statements.
As the saying goes, there are a million lies but only one reality. I know what it means to be a Bayesian; it means you acknowledge the sovereignty of probability theory as the law governing uncertainty. I'm not sure that even frequentists know what the rules are for being a "frequentist".
I take no particular side on this debate, but that seems very unfair. The point of frequentism is this: Suppose we want to determine, say, the constant G of gravity. Suppose that 1000 groups around the world perform experiments to measure G, and then use frequentist statistics to compute 95% confidence intervals for G. ~950 of the groups will compute intervals that contain the true constant.
There are many advantages to bayesian methods, but none of them give you coverage.
Your point about not being able to say anything about the true value is technically true. The reason is that frequentists believe that one should avoid making such statements.
Bayesian (subjective) and Frequentist (objective) are two schools of thought about how statistics should operate. One of the best ways to think about the difference is to call Bayesians pragmatic and Frequentists rigorous.
In that face of trying to quantify something you don't know, Bayesians take the stance that if you just say something — even if it's incorrect — and then keep adjusting it as more data comes in then you'll eventually have a valid statement. The idea is that even if your model is only so correct, at least you have one. Unfortunately, no one can actually prove that sort of thing actually works all of the time. It just seems to.
Frequentists faced with this situation instead try to understand the reason why something is happening and then model it from the beginning. By considering these rigorous models and testing them against data they eventually build a resilient model for the unknown which validates. That is, unless they don't, in which case Frequentists are kind of out of luck.
So when you're talking about statistics, which is all about trying to model things you don't understand, Bayesians and Frequentists get up in arms all the time because they each have something to call foolish about one another.
The coolest part is that this sort of schism is being reflected in the world of physics as well where Frequentists are in the Newtonian/Einsteinian school but, bit by bit, that worldview is being shaken up by Quantum.
Bayesians claim to be more rigorous than frequentists, living or dying by the theorems of probability theory, rather than using a toolbox of ad-hoc tools. For example. And quantumness has very little to do with it.
:-)
A Bayesian interpretation of probability is "degree of belief". What is the probability that tomorrow there will be rain ? Well, a priori I believe it is about 33%. However the sky is clouded today, which has 50% probability to cause rain the next day, etc. etc. So Bayes rule kicks in and I can make a decision whether or not to carry umbrella.
For a frequentist, probability of a single event is meaningless: 33% of all days in the past have been rainy. 50% of all days following cloudy days have been rainy, etc. . . If there were a large number of parallel worlds, in x % of them there would be rain, etc. etc.
Bummer.
I'm not sure how to clarify. The idea that reality is deterministic is so completely unsupported by evidence, I don't even know where to start. Between uncertainty principles forbidding you from knowing the precise state of the universe even in principle and the randomness of the universe's behavior even in situations you have full knowledge of (which may yet be traced back to mathematical undecidability, per a recent, interesting paper on arxiv), determinism is dead. If they can track it back to mathematical undecidability, no conceivable heroic effort can bring it back, either. (I suspect there's some fruitful work to be done there.)
It's deterministic. QM objections are based on bad pop sci.
Undecidability in the context of physics sounds interesting, care to add detail?
"it is easy to account for the interference pattern that results by assuming that electrons that travel in pairs are interfering with each other because they arrive at the screen at the same time, but when a laboratory apparatus was developed that could reliably fire single electrons at the screen, the emergence of an interference pattern suggested that each electron was interfering with itself; and, therefore, in some sense the electron had to be going through both slits"
PS: "it has been shown that large molecular structures like fullerene (C60) also produce interference patterns." which suggest at some level you could get the same result when firing planet's though a double slit experiment.