# Why is the beta distribution so flat when a, b=1?

If the beta distribution is a prior of a Bernoulli distribution (i.e. a rate of success for a binary outcome), then it is completely counterintuitive to me that the beta distribution should be equivalent to the uniform distribution when a, b = 1.

Unless I'm mistaken, you can interpret a and b to be the number of measured successes and failures of your outcome (e.g. heads or tails of a flipped coin). If that's true, then the likelihood of your posterior Bernoulli parameter being either 0 or 1 should be much closer to 0 (i.e. and NOT equally likely as values closer to 0.5).

If the underlying Bernoulli really was as likely to be 0 or 1 as something closer to 0.5, you would NOT expect to get both a 0 and a 1 out of two samples, right?

What am I missing that would make this more intuitive?

• "If the underlying Bernoulli really was as likely to be 0 or 1 as something closer to 0.5, you would NOT expect to get both a 0 and a 1 out of two samples, right?" This suggests a beta-binomial model for $a=b=1$, which is a discrete uniform variable. There doesn't seem to be a contradiction.
– Sycorax
Commented Jul 9, 2021 at 21:05
• You seek intuition. One place is the discussion at stats.stackexchange.com/questions/4659. Another is to study Bayes' billiard table experiment (Google it). The case $a=b=1$ corresponds to the situation after a single ball has been rolled on the table: it could be anywhere, with constant probability density. That's the uniform distribution.
– whuber
Commented Jul 9, 2021 at 21:35
• Non-informative priors do not exist! Commented Jul 10, 2021 at 12:37
• The interpretation of $a$ and $b$ being the number of (virtual) successes and failures observed a priori is a way to calibrate and understand the prior, no a real thing, especially when $a$ or $b$ are less than one. Commented Jul 10, 2021 at 12:40

Every conjugate prior distribution to an exponential family has some set of parameters $$\eta_0$$ that result in a uniform distribution over the space. You can see this here by considering what happens to the natural parameters $$\eta'$$ of the conjugate prior when the number of pseudo-observations $$n$$ equals $$0$$.

Unless I'm mistaken, you can interpret a and b to be the number of measured successes and failures of your outcome (e.g. heads or tails of a flipped coin). If that's true, then the likelihood of your posterior Bernoulli parameter being either 0 or 1 should be much closer to 0 (i.e. and NOT equally likely as values closer to 0.5).

That is true, but beta$$(1, 1)$$ corresponds to $$a=b=0$$. The natural parametrization of the beta distribution is $$(\alpha-1, \beta-1)$$ compared the "source parametrization" (or common parametrization) of $$(\alpha, \beta)$$.

In Bayesian statistics one uses a 'flat' prior distribution for a parameter in the absence of knowledge or opinion about about the parameter value. When the parameter is binomial success probability $$p$$ it may seem natural to use either a uniform prior $$\mathsf{Beta}(\alpha=1,\beta=1)\equiv\mathsf{Unif}(0,1)$$ or even the "bathtub shaped" prior $$\mathsf{Beta}(.5, .5).$$

par(mfrow=c(1,2))
hdr1 = "BETA(1,1)"
curve(dbeta(x,1,1), -.05,1.05, ylab="PDF",
col="blue", lwd=2, xaxs="i", n=10001, main=hdr1)
abline(v=0, col="green2"); abline(h=0, col="green2")
hdr2 = "BETA(0.5,0.5)"
curve(dbeta(x,.5,.5), 0,1, ylab="PDF", col="blue", lwd=2,
ylim=c(0,10), n=1001, main=hdr2)
abline(v=0, col="green2"); abline(h=0, col="green2")
par(mfrow=c(1,1))


The purpose of using a flat prior distribution may be for the posterior distribution on the parameter to be mainly due to the data. For example, if the prior is $$\mathsf{Beta}(1,1)$$ and the data show $$x=23$$ successes in $$n=50$$ trials, then the likelihood is proportional to $$p^x(1-p)^{n-x} = p^{23}(1-p)^{27}.$$

Thus the posterior distribution is $$\mathsf{Beta}(24, 28)$$ and a 95% Bayesian credible interval for $$p$$ is $$(0.33,\,0.600,$$ which agrees numerically with a frequentist 95% frequentist Agresti-Coull confidence interval $$(0.33,\,0.60)$$ to two decimal places.

qbeta(c(.025,.975),24,28)
[1] 0.3293001 0.5965812
p.est = 25/54
p.est + qnorm(c(.025,.975))*sqrt( p.est*(1-p.est)/54 )
[1] 0.3299707 0.5959553


Note: The distribution $$\mathsf{Beta}(.5,.5)$$ is called a Jeffreys prior. It can be argued that it is less informative as a prior than is $$\mathsf{Beta}(1,1).$$ The interval estimate $$(0.33, 0.60)$$ from this prior distribution is sometimes used as a frequentist CI:

qbeta(c(.025,.975),23.5,27.5)
[1] 0.3273505 0.5971336

• "In Bayesian statistics one uses a 'flat' prior distribution for a parameter in the absence of knowledge or opinion about about the parameter value." Sorry, but I think it's a common mistake to consider a flat prior to be uninformative. Any prior to some distribution $D$ can be made flat by simply reparametrizing $D$. Commented Jul 10, 2021 at 7:03
• I agree that a flat likelihood is uninformative though. Commented Jul 10, 2021 at 7:09