[https://en.wikipedia.org/wiki/Lindley%27s_paradox](https://en.wikipedia.org/wiki/Lindley%27s_paradox)
In: Mathematics
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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