Bayesian criticism of frequentist p-value I recently started reading the Bayesian criticism of the p-value and it seems that there is a lot of discussion around the fact that a frequentist approach is not that good when the Null Hypothesis is true.
For instance in this paper the authors write that "p-values overstate the evidence against the null [...] this does not have to do with type-I or type-II errors; it is an “independent” property of p-value."
To illustrate this point, the authors show that when the null is true, the p-value has a uniform distribution.
What I do not get is that even when the null is true, a frequentist approach, thanks to the Central Limit Theorem, is still able to construct confidence intervals that includes 0 (non-significance) at the appropriate $\alpha$ level.
I do not get why the fact that the p-value is uniform when the null is true shows that a frequentist approach is biased. And what does it mean "independent property of p-value"?

library(tidyverse)
library(broom)

n=1000
x = rnorm(n,100,30)
d = 0
y = x*d + rnorm(n,0,20)
df = data.frame(y,x)
plot(x,y)
abline(lm(y~x), col = 'red')

r = replicate(1000, sample_n(df, size = 50), simplify = F)
m = r %>% map(~ lm(y~x,data = .)) %>% map(tidy)

# Central Limit Theorem
bind_rows(.id = 'sample', m) %>% filter(term =='x') %>% ggplot(aes(estimate)) + facet_grid(~term) + geom_histogram()

s = bind_rows(.id = 'sample', m) %>% filter(term =='x')
s$false_positive = ifelse(s$p.value < 0.05, 1, 0)
prop.table(table(s$false_positive))

# uniform
hist(s$p.value, breaks = 50)


The idea that p-values overstate the evidence against the null hypothesis is partly due to a misunderstanding about the nature of p-values (as others have mentioned). A p-value can be regarded as a measure to address the hypothesis testing question: Is the data consistent with the null hypothesis? Bayesian posterior probabilities are Bayesian answers to a different question: Given the data, what is the relative plausibility of each hypothesis? The key word here is relative. It should hardly be surprising then that answers to such different questions can differ, even within the one school. In short, P-values are about consistency with one hypothesis, and not about the relative plausibility of two hypotheses.
