Yes, most of the problems (with conditional probability, statistical tests, significance, etc) disappear once you express it in a Bayesian way (it's not only Bayes' formula - it explicitly creating a Bayesian model).
Basically, most of them boil down to:
- mistakes that can be tackled if you write it down explicitly
- hidden assumptions that can be discovered (and made explicit or modified)
While there is some philosophical difference between frequentist and Bayesian probability (and for some reason, I know people moving only one way).
"Frequentist probability is Bayesian probability, where priors are flat, hidden, and considered taboo".
BTW: Frequentists vs. Bayesians https://xkcd.com/1132/ (there is never too much of xkcd!)
That is fascinating. Which one is correct? Is Bayesian using more assumptions than Frequentist, namely the fact that repeated queries that haven't been done yet will show that the machine's answer is NO most of the time?
Who is correct about the sun exploding actually is irrelevant to that; only the conditional probability of the bet being collectable if the sun has exploded vs. that of it has not exploded is needed here. You would care about the probability that the sun actually had exploded if one of those weren't zero, but it is, so it doesn't matter.
Frequentist statistics is "classical" statistics, the kind most people know. It's often used to control the chance of reporting an effect when there's no real effect (the p-value); in many fields 5% is accepted, which, of course, leaves a lot of false positives among millions of studies.
In the comic, the frequentist is asking: "the machine said yes, what is the probability of that if the Sun hasn't exploded?" Since that probability (the p-value) is less than 0.05, the frequentist concludes the Sun has exploded, which illustrates a common error: mistaking statistical significance for truth.
Bayesians, in contrast, interpret probabilities as beliefs about the world and use experiments to update those beliefs, in accordance with Bayes rule.
In the comic, the Bayesian has presumably started with a strong belief that the Sun has not exploded, and the evidence that the machine says "Yes" one time slightly reduces their certainty but isn't strong enough to convert that belief to "the Sun has (probably) exploded".
Most people (even frequentist statisticians) actually interpret statistics that way, but Bayesian statistics formalizes it mathematically.
The core difference is philosophical, but building on that, Bayesians have built a new set of mathematical tools involving things like conjugate priors, posterior probabilities, credible intervals, and Bayes factors.
The result is a different way of practicing statistics, not merely a difference in interpretation.
Basically, most of them boil down to:
- mistakes that can be tackled if you write it down explicitly
- hidden assumptions that can be discovered (and made explicit or modified)
While there is some philosophical difference between frequentist and Bayesian probability (and for some reason, I know people moving only one way).
"Frequentist probability is Bayesian probability, where priors are flat, hidden, and considered taboo".
BTW: Frequentists vs. Bayesians https://xkcd.com/1132/ (there is never too much of xkcd!)