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:

```
# Possible values that were rounded to R
possible <- function(rounded) {
if(rounded == 0) {
options <- c(0, 1, 2)
} else {
options <- c(rounded - 2, rounded - 1, rounded,
rounded + 1, rounded + 2)
}
return(options)
}
# Probability mass function of numbers rounding to R
# given theta
prior_y <- function(options, theta) {
p <- dbinom(options, 15, prob = theta)
return(p)
}
# Likelihood of rounding
like_round3 <- function(options) {
if(length(options) == 3) {
like <- c(1, 2/3, 1/3) }
else {
like <- c(1/3, 2/3, 1, 2/3, 1/3)
}
return(like)
}
# Estimating posterior mass function and drawing a
# random value of it
posterior_sample_y <- function(R, theta) {
po <- possible(R)
pr <- prior_y(po, theta)
li <- like_round3(po)
post <- li*pr/sum(li*pr)
samp <- sample(po, 1, prob = post)
return(samp)
}
```

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 \(mbox{Beta}(alpha + sum Y_i, eta + 12*15 - sum Y_i)\).

```
## Function to sample from the posterior Pr(theta | Y, R)
posterior_sample_theta <- function(alpha, beta, Y) {
theta <- rbeta(1, alpha + sum(Y), beta + 12*15 - sum(Y))
return(theta)
}
```

I then implemented the sampler as:

```
## Data
R <- c(0, 0, 3, 9, 3, 0, 6, 3, 0, 6, 0, 3)
nsim <- 10000
burnin <- 1000
alpha <- 1
beta <- 1
store <- matrix(0, nrow = nsim, ncol = length(R) + 1)
starting.values <- c(R, 0.1)
## Sampling
store[1,] <- starting.values
for(draw in 2:nsim){
current <- store[draw - 1,]
for(obs in 1:length(R)) {
y <- posterior_sample_y(R[obs], current[length(R) + 1])
# Jump or not still missing
current[obs] <- y
}
theta <- posterior_sample_theta(alpha, beta, current[1:length(R)])
# Jump or not still missing
current[length(R) + 1] <- theta
store[draw,] <- current
}
```

And plotted the results as:

```
plot((burnin+1):nsim, store[(burnin+1):nsim,13], type = 'l')
library(ggplot2)
ggplot(data.frame(theta = store[(burnin+1):nsim,13]), aes(x = theta)) +
geom_density(fill = 'blue', alpha = 0.5)
```

```
multiple_plot <- data.frame(Y = matrix(store[(burnin+1):nsim, 1:12],
nrow = (nsim - burnin)*12,
ncol = 1))
multiple_plot$obs <- factor(rep(1:12, each = (nsim - burnin)))
ggplot(multiple_plot, aes(x = Y)) + geom_histogram() + facet_grid(~obs)
```

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