\chapter*{Abstract}
This thesis is motivated by estimating the cause specific mortality fraction (CSMF) for children deaths in Mozambique. In countries where many deaths are not assigned a cause of death, CSMF estimation is often performed by performing a verbal autopsy (VA) for a large number of deaths. A cause for each VA is then assigned via one or more computer coded verbal autopsy (CCVA) algorithms, and these cause assignments are aggregated to estimate the CSMF. We show that CSMF estimation from CCVAs is poor if there is substantial misclassification due to CCVAs being informed by non-local data. We develop a parsimonious Bayesian hierarchical model that uses a small set of labeled data that includes deaths with both a VA and a gold-standard cause of death. The labeled data is used to learn the misclassification rates from one or multiple CCVAs, and in-turn these estimated rates are used to produce a calibrated CSMF estimate. A shrinkage prior ensures that the CSMF estimate from our Bayesian model coincides with that from a CCVA in the case of no labeled data. To handle probabilistic CCVA predictions and labels, we develop an estimating equations approach that uses the Kullback-Liebler loss-function for transformation-free regression with a compositional outcome and predictor. We then use Bayesian updating of this loss function, which allows for calibrated CSMF estimation from probabilistic predictions and labels. This method is not limited to CSMF estimation and can be used for general quantification learning, which is prevalence estimation for a test population using predictions from a classifier derived from training data. Finally, we obtain CSMF estimates for child deaths in Mozambique by applying all of the developed methods to VA data collected from the Countrywide Mortality Surveillance for Action (COMSA)-Mozambique and VA and gold-standard COD data collected from the Child Health and Mortality Prevention project
