Fitted Dirichlet Distribution

In SHELF: Tools to Support the Sheffield Elicitation Framework

Elicited judgements

knitr::opts_chunk$set(echo=FALSE, warning=FALSE, message=FALSE)

mydf <- data.frame(params$quantiles, params$thetaMatrix)
colnames(mydf) <- c("quantiles", params$categories)
rownames(mydf) <- NULL
knitr::kable(mydf)

Dirichlet density function and parameters

Define the vector of unknown population proportions as $$\theta := (\theta_1,\ldots,\theta_k),$$ with $k = r length(params$categories)$ . We write $$\theta \sim Dirichlet(a_1,\ldots,a_k),$$ with $$ f(\theta)=\frac{\Gamma(a_1+\ldots +a_k)}{\Gamma(a_1)\ldots \Gamma(a_k)}\prod_{i=1}^k \theta_i^{a_i-1}. $$ The fitted parameter values $a_1,\ldots,a_k$ are as follows:

fitDirichlet(params$allFits, categories = params$categories,
             n.fitted = params$n, silent = TRUE, 
             plotBeta = FALSE)

Comparing the elicited marginals with the marginals from the Dirichlet fit

d <- fitDirichlet(params$allFits, categories = params$categories,
             n.fitted = params$n)

SHELF documentation built on June 7, 2023, 5:11 p.m.