Models the counts across two or more mutually exclusive categories from a
fixed number of trials, in long format: one row per category per
trial, with group identifying which rows belong to the same trial. All
rows sharing a group must have the same size, and their prob values
must sum to 1.
Arguments
- x
A non-negative whole numeric vector of the category counts.
- size
A non-negative whole numeric vector of the number of trials.
- prob
A numeric vector of the probability of the category. Must sum to 1 across the rows sharing the same
group.NAinsizeorprobfor any row of a trial makes the log-likelihoodNAfor every row of that trial, since a trial's categories are scored jointly.- group
A vector identifying which rows belong to the same multinomial trial (whose
xvalues sum tosizeandprobvalues sum to 1). Every group must have at least 2 rows and the same number of rows as the rest of the data (a fixed set of categories, as in multinomial logistic regression), and must not containNA.
Details
A trial's log-likelihood doesn't split evenly across its rows, since the
multinomial coefficient belongs to the whole trial. log_lik_multinom()
uses the multinomial-as-independent-Poissons identity: each row's value
is the Poisson log-likelihood of x given mu = size * prob, minus an
even share of the trial's normalizing constant, so summing over a
group recovers the trial's exact multinomial log-likelihood.
The even split of the normalizing constant is a bookkeeping convention:
only the sum over a group is exact, so an individual row's value is not
a valid pointwise log-likelihood. Cross-validation and information
criteria should therefore treat the trial, not the row, as the
leave-one-out unit, which is achieved by summing the values within each
group to get the trial's exact log-likelihood.
References
Baker, S.G. 1994. The multinomial-Poisson transformation. The Statistician 43(4): 495-504. doi:10.2307/2348134 .
McCullagh, P., and Nelder, J.A. 1989. Generalized Linear Models. 2nd edition. Chapman and Hall, London.
Forster, J.J. 2010. Bayesian inference for Poisson and multinomial log-linear models. Statistical Methodology 7(3): 210-224. doi:10.1016/j.stamet.2009.12.004 .
Vehtari, A., Gelman, A., and Gabry, J. 2017. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing 27(5): 1413-1432. doi:10.1007/s11222-016-9696-3 .
See also
Other log_lik_dist:
log_lik_bern(),
log_lik_beta(),
log_lik_beta_binom(),
log_lik_binom(),
log_lik_exp(),
log_lik_gamma(),
log_lik_gamma_pois(),
log_lik_gamma_pois_zi(),
log_lik_lnorm(),
log_lik_neg_binom(),
log_lik_norm(),
log_lik_pois(),
log_lik_pois_zi(),
log_lik_skewlnorm(),
log_lik_skewnorm(),
log_lik_student(),
log_lik_unif()