Multivariate logistic likelihood

[jysullivan]

Create better documentation about this likelihood Generate posterior predictive interval

[jysullivan]

Potential code examples: https://github.com/smartell/CatchMSY/commit/85313528d94f790d95fffe3448970cdccf3ebc70 https://github.com/Craig44/LogisticNormalLL

Papers

Appendix B of Richards LJ, Schnute JT, Olsen N. Visualizing catch–age analysis: a case study. Canadian Journal of Fisheries and Aquatic Sciences. 1997 Jul 1;54(7):1646-58.

Francis RC. Replacing the multinomial in stock assessment models: A first step. Fisheries Research. 2014 Mar 1;151:70-84.

A logistic-normal distribution is formed by applying a logistic transformation to a multivariate normal vector.

Martell, S., 2011. iSCAM Users Guide Version 1.0, Available from https://sites.google.com/site/iscamproject/.

A nice feature of the multivariate logistic distribution is that the age-proportion data can be weighted based on the conditional maximum likelihood estimate of the variance in the age-proportions. Therefore, the contribution of the age-composition data to the overall objective function is “self-weighting” and is conditional on other components in the model.

Schnute, J. T. and Haigh, R. 2007. Compositional analysis of catch curve data, with an application to Sebastes maliger. – ICES Journal of Marine Science, 64: 218–233

Status of the Pacific Hake (Whiting) stock in U.S. and Canadian Waters in 2011 Joint U.S. and Canadian Hake Technical Working Group

The multivariate logistic likelihood function (Richards et al. 1997) uses the conditional maximum likelihood estimate of the variance to weight the age composition data.

In general, the multivariate logistic likelihood is more robust to weighting problems, although it does assume a single variance across all years, which may produce overly large residuals in some years.

[jysullivan]

Talked with S. Martell today, he made a couple points:

• Variance not estimated for multivariate logistic because it's derived analytically (pointed me to Appendix of Richards et al. 1997)
• Main difference between so-called multivariate logistic and logistic normal is that only a single variance is estimated for the multivariate logistic, while logistic normal can have variance changing over time (several options for that discussed in Francis 2014, for example as an AR(1) process).