Functionality/option to combine estimates

[prabasaj]

Suggesting an option to combine multiple estimates of the same parameter set (shape and scale of gamma distribution fitted to incubation period for Ebola, for example, across 5 studies). Here's some basic code to start with:

Weighted combination of multivariate estimates and population covariance
User supplied weights may be some combination: (weight) = (effective sample size) X (quality weight) X (relevance weight)
Neglects sample size corrections and is based on definition
Includes option of weighting by inverse radius of covariance matrix
estimate set is a list of estimate-lists: list(est1 = list(mean, covariance, weight), ...)

weighted_estimates = function(estimate_set, inverse_variance_weighting=FALSE) {
  
  if (inverse_variance_weighting) {
    wts = proportions(sapply(estimate_set, function(ll) 1/max(eigen(ll$covariance)$values) ))
  } else {
    wts = proportions(sapply(estimate_set, function(ll) ll$weight))
  }

  mu =  c(sapply(estimate_set, function(ll) ll$mean) %*% wts)
  sigma = matrix(sapply(estimate_set, function(ll) (ll$mean - mu) %*% t(ll$mean - mu)) %*% wts, nrow=length(mu)) +
          matrix(rowSums(mapply(`*`, wts, lapply(estimate_set, function(ll) ll$covariance))), nrow=length(mu))
  list(mean = mu, covariance = sigma)
}

Example: combining bivariate parameter estimates (shape and scale of gamma distributions mean=c(shape, scale), for example)

estimate_set = list(
                 list(mean=c(1, 2), covariance=cbind(c(0.25, 0.1), c(0.1, 0.36)), weight=2),
                 list(mean=c(1.2, 2.5), covariance=cbind(c(0.25, 0.1), c(0.1, 0.36)), weight=3) 
               )
weighted_estimates(estimate_set, inverse_variance_weighting=TRUE)
weighted_estimates(estimate_set)

[joshwlambert]

Thanks for contributing this @prabasaj. Similar functionality to combine epidemiological parameter distributions was added to the package in #388 using mixture distributions to combine individual distributions.

I will leave this issue open as we can explore this method of combining estimates in the next development cycle (v0.4.0).