Multinomial with non-integer data

Modeling
1

Suppose we wanted to implement this in Stan:

http://www.stat.columbia.edu/~gelman/research/published/misterp.pdf
deff

But multinomial_lpmf(int[] y, vector theta) doesn’t allow non-integer data. What do you recommend?

Thanks!!

2 
vector[D] theta[N];
vector[D] y_real[N];
for(i in 1:N)
  target += sum(y_real[i] .* (theta[i] - log_sum_exp(theta[i])));

where N number of observations, D dimension of simplex, y_real …

But the paper talks about binomial.

Updated. Added sum(). Should work without too.
same: target += sum(y_real[i] .* log_softmax(theta[i])));

3

thanks so much!! the log-likelihood for the Multinomial is a bit different from your expression, I think? See e.g. p.272 of Agresti’s Categorical Data Analysis:

21%20AM
Screen Shot 2019-06-13 at 11.16.21 AM.png1048×596 92.9 KB

4

That’s just the J - 1 expression. Look at the PMF given in

https://en.wikipedia.org/wiki/Multinomial_distribution

image

p_1 + ... + p_k =1, the softmax.
log(p_i) = theta[i] - log\_sum\_exp(theta[i]))

The factor with n! / x1! … /x_k! can be omitted for calc. the likelihood, it
contains only constants.

For identifiable reasons your theta’s should sum-to-zero or contain one element being 0,
the reference element. Ref. Stan manual.

5

thanks so much!! got it.

(I didn’t catch at first that you switched theta from probabilities to logit scale)

6

Hi all – sorry to resurrect this thread, I was trying to implement this and I’m getting strange results so I thought I’d check in. I’m implementing as follows:

data {
    int<lower=1> N; // number of observations
    int<lower=2> J; // number of choices
    vector[J] y_real[N]; // number of supporters for each choice
}

parameters {
    vector[J] beta; // baseline rate of support
    real<lower = 0> beta_sd;
}

transformed parameters {
    matrix[N,J] mu_s; 
    vector[J] beta_s = beta * beta_sd;

    for(j in 1:J) mu_s[1:N,j] = beta_s[j] ;
}

model {
  beta ~ std_normal();
  beta_sd ~ std_normal();

  for (i in 1:N) target += sum(y_real[i] .* log_softmax(mu_s[i,1:J]'));
}

Does this look vaguely correct ? There are more parameters in the actual model but I’ve slimmed it down to the bare minimum.

Thank you !