Bayesian Statistics. Please help me to find an example where posterior variance is greater than prior variance Suppose we observe y successes in n trials where the probability of success in each trial is θ. If we choose a Beta(1, 1) (Uniform) prior then the posterior variance will be smaller than the prior variance. Show that the above isn’t necessarily the case if we choose a general Beta(a, b) prior. That is, find set of values for a, b, n, y where the above is not true.
I have tried many random examples and still have yet found a case where posterior variance is greater than prior variance. Please help me to find a better direction. 
 A: Gamma-Poisson: Suppose your prior for Poisson data is $\lambda \sim \mathsf{Gamma}(\text{shape}=4, \text{rate}=1/4).$
This distribution has mean 16 and variance 64. It's 95th percentile is about 31.
qgamma(.95, 4, .25)
[1] 31.01463

But your first Poisson observation is $x = 500.$ Then your posterior
distribution is $\mathsf{Gamma}(\text{shape}=4+500, \text{rate}=1/4+1) = 
\mathsf{Gamma}(\text{shape}=504, \text{rate}=1.25),$ which has mean $504/1.25 = 403.2$ and variance $504/1.25^2 =  322.56 > 64.$

Beta-binomial: For a beta-binomial example (along lines suggested by @whuber), tossing a coin. Suppose your prior for $P(\text{Head})=\theta$ is $\mathsf{Beta}(10,1),$ for a coin heavily biased in favor of Heads. Then on four tosses you get two Heads
and two tails. Find the variances of the prior and posterior distributions.

Note: In both examples, the idea is to have an informative prior and then a small amount of data that doesn't match the prior. With a large amount of data, the
data can overwhelm the prior, yielding a posterior with small variance. Now I hope you can find examples of your own.
