#StackBounty: #estimation #binomial #beta-distribution #measurement-error How to model errors around the estimation of proportions – wi…

Bounty: 100

I have a situation I’m trying to model. I would appreciate any ideas on how to model this, or if there are known names for such a situation.

Background:

Let’s assume we have a large number of movies (M). For each movie, I’d like to know the proportion of people in the population who enjoy watching these movies. So for movie $m_1$ we’d say that $p_1$ proportion of the population would say "yes" to "did you enjoy watching this movie?" question. And the same for movie $m_j$, we’d have proportion $p_j$ (up to movie $m_M$).

We sample $n$ people, and ask each of them to say if they enjoyed watching movies $m_1, m_2, …, m_M$ of the movies. We can now easily build estimations for $p_1, …, p_M$ using standard point estimates, and build confidence intervals for these estimations using the standard methods (ref).

But there is a problem.

Problem: measurement error

Some of the people in the sample do not bother to answer truthfully. They instead just answer yes/no to the question regardless of their true preference. Luckily, for some sample of the M movies, we know the true proportion of people who like the movies. So let’s assume that M is very large, but that for the first 100 movies (of some indexing) we know the real proportion.
So we know the real values of $p_1, p_2, …, p_{100}$, and we have their estimations $hat p_1 , hat p_2, …, hat p_{100}$. While we still want to know the confidence intervals that takes this measurement error into account for $p_{101} , p_{102}, …, p_M$, using our estimators $hat p_{101} , hat p_{102}, …, hat p_M$.

I could imagine some simple model such as:

$$hat p_i sim N(p_i, epsilon^2 + eta^2 )$$

Where $eta^2$ is for the measurement error.

Questions:

  1. Are there other reasonable models for this type of situation?
  2. What are good ways to estimate $eta^2$ (for the purpose of building confidence interval)? For example, would using $hat eta^2 = frac{1}{n-1}sum (p_i – hat p_i)^2$ make sense? Or, for example, it makes sense to first take some transformation of the $p_i$ and $hat p_i$ values (using logit, probit or some other transformation from the 0 to 1, to an -inf to inf scale)?


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#StackBounty: #binomial #beta-distribution #inverse-problem Distribution of population size $n$ given binomial sampled quantity $k$ and…

Bounty: 50

Given a drawn (without replacement) sample size $k$ from a binomial distribution with known probability parameter $pi$, is there a function which gives distribution of likely population size $n$ from which these $k$ were sampled? For instance, let’s say we have $k=315$ items randomly selected with known probability $pi=0.34$ from a population of $n$ items. Here most likely value is $hat{n}=926$ but what is probability distribution for $n$. Is there a distribution which gives $p(n)$?

I know that $p(pi | k,n)$ is given by the beta distribution and that $p(k |pi, n)$ is the binomial distribution. I’m looking for that third creature, $p(n |pi, k)$, properly normalized of course such that $sum_{n=k}^{infty} p(n)=1$

first "attempt" at this, given the normal approximation to binomial distribution is $p(k|pi, n)=mathcal{N}(k/pi,kpi(1-pi))$, is that $p(n|pi,k)approxmathcal{N}(k/pi,kpi(1-pi))$?


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#StackBounty: #mcmc #beta-distribution #stan #finite-mixture-model Finite Beta mixture model in stan — mixture components not identified

Bounty: 50

I’m trying to model data $0 < Y_i < 1$ with a finite mixture of Beta components. To do this, I’ve adapted the code given in section 5.3 of the Stan manual. Instead of (log)normal priors, I am using $mathrm{Exponential}(1)$ priors for the $alpha$ and $beta$ parameters. Thus, as I understand it, my model is as follows:

begin{align}
alpha_k, beta_k &overset{iid}{sim} mathrm{Exponential}(1) \
Z_i &sim mathrm{Categorical}(1, ldots, K) \
Y_i mid left(Z_i = kright) &sim mathrm{Beta}_{alpha_k, beta_k}
end{align
}


Now, for my implementation in stan, I have the following two code chunks:

# fit.R
y <- c(rbeta(100, 1, 5), rbeta(100, 2, 2))
stan(file = "mixture-beta.stan", data = list(y = y, K = 2, N = 200))

and

// mixture-beta.stan

data {
  int<lower=1> K;
  int<lower=1> N;
  real y[N];
}

parameters {
  simplex[K] theta;
  vector<lower=0>[K] alpha;
  vector<lower=0>[K] beta;
}

model {
  vector[K] log_theta = log(theta);

  // priors
  alpha ~ exponential(1);
  beta ~ exponential(1);
  
  for (n in 1:N) {
    vector[K] lps = log_theta;

    for (k in 1:K) {
      lps[k] += beta_lpdf(y[n] | alpha[k], beta[k]);
    }

    target += log_sum_exp(lps);
  }
}


After running the code above (defaults to 4 chains of 2000 iterations, with 1000 warmup) I find that all the posterior components are essentially the same:

> print(fit)
Inference for Stan model: mixture-beta.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

          mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
theta[1]  0.50    0.01 0.13  0.26  0.42  0.50  0.58  0.75   259 1.01
theta[2]  0.50    0.01 0.13  0.25  0.42  0.50  0.58  0.74   259 1.01
alpha[1]  2.40    0.38 1.73  0.70  0.94  1.20  3.89  6.01    21 1.16
alpha[2]  2.57    0.37 1.74  0.70  0.96  2.29  4.01  6.05    22 1.16
beta[1]   3.54    0.11 1.10  1.84  2.66  3.46  4.26  5.81    93 1.04
beta[2]   3.58    0.12 1.07  1.88  2.77  3.49  4.26  5.89    82 1.05
lp__     30.80    0.05 1.74 26.47 29.92 31.21 32.08 33.02  1068 1.00

Samples were drawn using NUTS(diag_e) at Thu Sep 17 12:16:13 2020.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

I read the warning about label switching, but I can’t see how to use the trick of ordered[K] alpha since I also need to integrate the constraint of $alpha$ and $beta$ being positive.

Could someone help explain what’s going on here?


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