[https://en.wikipedia.org/wiki/Lindley%27s_paradox](https://en.wikipedia.org/wiki/Lindley%27s_paradox)
In: Mathematics
Ok, this is challenging, I’ll explain the example on the wiki. We want to know if the probability of a newborn being male is 50% or not, we can do this in two ways.
First approach (Bayesian or “a priori”, i.e. before knowing the reality): I assume that the probability is in fact 50%, then I check the official numbers and see how well they fit this this assumption.
Second approach (frequentist or “a posteriori”, i.e. already knowing the real outcome): I take the official numbers and try to find out if they are better explained by one hypothesis or the other.
The point is that the first approach will answer “yes, it’s likely to be 50%” because it will find out that our numbers are a plausible result of that probability; the second approach will answer “no way that it’s exactly 50%” because you can’t make such a precise assumption (and, in our example, if you’re giving such a bold explanation the better one would be that the probability is exactly k/n), it could easily be 51% or 49%.
This is because Bayesian penalizes the “wider” hypothesis (if we say the probability is 50% is a very good hypothesis vs saying that it can be anywhere between 0% and 100%) while frequentist prefers it (it’s infinitely more likely that the percentage is close to 50% than exactly 50%).
Edit: god, this took 40 mins to ely5, I had to sacrifice precision a bit
The two approaches are asking/answering different questions.
For example, for birth ratios, Frequentist tests ask “If the birth ratio were exactly 0.5, would we see the number of male and female births that we do?”
There are two things that might happen with a Bayesian analysis. One question would be “What’s a reasonable range for the birth ratio?” and one could then check whether that includes 0.5. Another question you could ask/answer with a Bayesian approach is “which of these two possible estimates or ranges for the birth ratio better match the numbers we see?”
Frequentist tests are sometimes hard to interpret. We don’t expect that the birth ratio is exactly 0.5, and if we tallied enough births, we’d definitely (and correctly) reject that hypothesis of “exactly 0.5”. But it wouldn’t tell us what a good estimate of the ratio actually is.
That can also be a problem with the Bayesian approach, but it depends somewhat on what question is being asked/answered. If we’re using Bayesian methods to compare two bad theories, e.g., we might want to check if the birth ratio is one of 0.25 or 0.75, it will also give somewhat absurd answers. If we use it for ratio estimation, though, it’s likely to do better.
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