Improving the health of communities and individuals around the world is one of the great challenges of this densely connected global era which finds itself rife with disparity. In order to make the best use of our limited resources, spatially-resolved and time-specific estimates of health indicators are required to make well-informed decisions regarding resource allocation and policy implementation. The availability of the health data used to make these estimates is often too sparse, as compared to the spatial and temporal heterogeneity of the population of interest, for traditional methods to produce reliable estimates across the population. To address these difficulties, different data types and sources are frequently combined to achieve reliable high-fidelity estimates and their associated uncertainty. This thesis describes computation and inference for Bayesian spatial-temporal smoothing models which leverage various sources of sparse health data to make granular predictions across the space-time domain of interest. In particular, we provide the first statistical description of Template Model Builder (TMB), a flexible inferential software package for mixed-effects model estimation, and use a suite of extensive continuous and discrete spatial simulations to demonstrate that it is well-suited to spatial-temporal applications. In the following chapter, we use TMB to perform inference on a model developed to jointly estimate European breast cancer incidence and mortality using data from both local registries and national databases. The joint model uses structured random effects across age, space, and time to account for heterogeneity across 18 age groups, 39 countries and 18 years. This is joint work with collaborators at the WHO International Agency for Research on Cancer, and for the 39 European countries in this study one of the primary motivations was to improve incidence estimates and predictions in times and places where incidence data is unavailable. We do this by using the joint model, which relies on a nonlinear relationship between the incidence rates and unconditional mortality rates, to estimate incidence across the age-space-time cube, including backcasting incidence to recent years where only mortality data is available. We conclude with a discussion on the limitations of interpreting the frequentist properties of mixed-effects models uncertainty intervals. To aid in dissemination and interpretation of the results from these models, we propose that the coverage probability of such intervals be estimated and published alongside the uncertainty interval. We demonstrate this approach by developing a novel unbiased coverage probability estimator for Gaussian sampling models and apply it to the uncertainty intervals of estimates of county-level mean household radon concentrations.