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PhilGoetz comments on Bayesian Flame - Less Wrong
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Being a frequentist who hangs out on a Bayesian forum, I've thought about the difference between the two perspectives. I think the dichotomy is analogous to bottom-up verses top-down thinking; neither one is superior to the other but the usefulness of each waxes and wanes depending upon the current state of a scientific field. I think we need both to develop any field fully.
Possibly my understanding of the difference between a frequentist and Bayesian perspective is different than yours (I am a frequentist after all) so I will describe what I think the difference is here. I think the two POVs can definitely come to the same (true) conclusions, but the algorithm/thought-process feels different.
Consider tossing a fair-coin. Everyone observes that on average, heads comes up 50% of the time. A frequentist sees the coin-tossing as a realization of the abstract Platonic truth that the coin has a 50% chance of coming up heads. A Bayesian, in contrast, believes that the realization is the primary thing ... the flipping of the coin yields the property of having 50% probability of coming up heads as you flip it. So both perspectives require the observation of many flips to ascertain that the coin is indeed fair, but the only difference between the two views is that the frequentist sees the "50% probability of being heads" as something that exists independently of the flips. It's something you discover rather than something you create.
Seen this way, it sounds like frequentists are Platonists and Bayesians are non-Platonists. Abstract mathematicians tend to be Platonists (but not always) and they've lent their bias to the field. Smart Bayesians, on the other hand, tend to be more practical and become experimentalists.
There's definitely a certain rankle between Platonists and non-Platonists. Non-platonists think that Platonists are nuts, and Platonists think that the non-Platonists are too literal.
May we consider the hypothesis that this difference is just a difference in brain hard-wiring? When a Platonist thinks about a coin flipping and the probability of getting heads, they really do perceive this "probability" as existing independently. However, what do they mean by "existing independently"? We learn what words mean from experience. A Platonist has experience of this type of perception and knows what they mean. A non-Platonist doesn't know what is meant and thinks the same thing is meant as what everyone means when they say "a table exists". These types of existence are different, but how can a Bayesian understand the Platonic meaning without the Platonic experience?
A Bayesian should just observe what does exist, and what words the Platonist uses, and redefine the words to match the experience. This translation must be done similarly with all frequentist mathematics, if you are a Bayesian.
"A Bayesian, in contrast, believes that the realization is the primary thing ... the flipping of the coin yields the property of having 50% probability of coming up heads as you flip it."
Thanks for trying to explain the difference, but I have no idea what this means.
What I was thinking about was this: Bayesians and frequentists both agree that if a fair coin is tossed n times (where n is very large) then a string of heads and tails will result and the probability of heads is .5 in some way related to the fact that the number of heads divided by n will approach .5 for large n.
In my mind, the frequentist perspective is that the .5 probability of getting heads exists first, and then the string of heads and tails realize (i.e., make a physical manifestation of) this abstract probability lurking in the background. As though there is a bin of heads and tails somewhere with exactly a 1:1 ratio and each flip picks randomly from this bin. The Bayesian perspective is that there is nothing but the string of heads and tails -- only the string exists, there's no abstract probability that the string is a realization of. No picking from a bin in the sky. Inspecting the string, a Bayesian can calculate the 0.5 probability ... so the 0.5 probability results from the string. So according to me, the philosophical debate boils down to: what comes first, the probability or the string?
I definitely get the impression that the Bayesians in this thread are skeptical of this description of the difference, and seem to prefer describing the difference of the Bayesian view as considering probability a measure of your uncertainty. However, probability is also taught as a measure of uncertainty in classical probability, so I'm skeptical of this dichotomy. (In favor of my view, the name "frequentist" comes from the observation that they believe in a notion of "frequency" -- i.e., that there's a hypothetical distribution "out there" that observed data is being sampled from.)
Perhaps the difference in whether the correct approach is subjective or objective better gets to the heart of the difference. I am leaning towards this hypothesis because I can see how a frequentist can confuse something being objective with that something having an independent "existence".
I have a little difficulty with the notion that the probable outcome of a coin toss is the result of the toss, rather like the collapse of a quantum probability into reality when observed. Looking at the coin before the toss, surely three probabilities may be objectively observed - H, T or E, and the likelihood of the coin coming to rest on its edge dismissed.
Since the coin MUST then end up H or T ; the sum of both probabilities is 1, both outcomes are a priori equally likely and have the value1/2 before the toss. Whether one chooses to believe that the a priori probabilities have actual existence is a metaphysical issue.