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:

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# 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 \(\mbox{Beta}(\alpha, \beta)\), with \(\alpha\) and \(\beta\) equal to 1, as prior density function for \(\theta\), so the posterior density is a \(\mbox{Beta}(\alpha + \sum Y_i, \beta + 12*15 – \sum Y_i)\).

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## 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:

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## 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:

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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) |

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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.