Font Valverde, Marti; Puig Oriol, Xavier; Ginebra Molins, Josep Journal of statistical computation and simulation Vol. 83, num. 2, p. 229-246 DOI: 10.1080/00949655.2011.600311 Date of publication: 2013-02 Journal article
The zero truncated inverse Gaussian–Poisson model, obtained by first mixing the Poisson model assuming its expected value has an inverse Gaussian distribution and then truncating the model at zero, is very useful when modelling frequency count data. A Bayesian analysis based on this statistical model is implemented on the word frequency counts of various texts, and its validity is checked by exploring the posterior distribution of the Pearson errors and by implementing posterior predictive consistency checks. The analysis based on this model is useful because it allows one to use the posterior distribution of the model mixing density as an approximation of the posterior distribution of the density of the word frequencies of the vocabulary of the author, which is useful to characterize the style of that author. The posterior distribution of the expectation and of measures of the variability of that mixing distribution can be used to assess the size and diversity of his vocabulary. An alternative analysis is proposed based on the inverse Gaussian-zero truncated Poisson mixture model, which is obtained by switching the order of the mixing and the truncation stages. Even though this second model fits some of the word frequency data sets more accurately than the first model, in practice the analysis based on it is not as useful because it does not allow one to estimate the word frequency distribution of the vocabulary.
The analysis of word frequency count data can be very useful in authorship attribution problems. Zerotruncated
generalized inverse Gaussian–Poisson mixture models are very helpful in the analysis of these
kinds of data because their model-mixing density estimates can be used as estimates of the density of the
word frequencies of the vocabulary. It is found that this model provides excellent fits for theword frequency
counts of very long texts, where the truncated inverse Gaussian–Poisson special case fails because it does
not allow for the large degree of over-dispersion in the data. The role played by the three parameters of
this truncated GIG-Poisson model is also explored. Our second goal is to compare the fit of the truncated
GIG-Poisson mixture model with the fit of the model that results from switching the order of the mixing
and truncation stages. A heuristic interpretation of the mixing distribution estimates obtained under this
alternative GIG-truncated Poisson mixture model is also provided.