# Cute Gibbs sampling for rounded observations

I was attending a course of Bayesian Statistics where this problem showed up:

There is a number of individuals, say 12, who take a pass/fail test 15 times. For each individual we have recorded the number of passes, which can go from 0 to 15. Because of confidentiality issues, we are presented with rounded-to-the-closest-multiple-of-3 data ($$\mathbf{R}$$). We are interested on estimating $$\theta$$ of the Binomial distribution behind the data.

Rounding is probabilistic, with probability 2/3 if you are one count away from a multiple of 3 and probability 1/3 if the count is you are two counts away. Multiples of 3 are not rounded.

We can use Gibbs sampling to alternate between sampling the posterior for the unrounded $$\mathbf{Y}$$ and $$\theta$$. In the case of $$\mathbf{Y}$$ I used:

While for $$theta$$ we are assuming a vague $$mbox{Beta}(alpha, eta)$$, with $$alpha$$ and $$eta$$ equal to 1, as prior density function for $$theta$$, so the posterior density is a $$mbox{Beta}(alpha + sum Y_i, eta + 12*15 – sum Y_i)$$.

I then implemented the sampler as:

And plotted the results as:

I thought it was a nice, cute example of simultaneously estimating a latent variable and, based on that, estimating the parameter behind it.