A tutorial on bridge sampling.

The marginal likelihood plays an important role in many areas of Bayesian statistics such as parameter estimation, model comparison, and model averaging. In most applications, however, the marginal likelihood is not analytically tractable and must be approximated using numerical methods. Here we provide a tutorial on bridge sampling (Bennett, 1976; Meng & Wong, 1996), a reliable and relatively straightforward sampling method that allows researchers to obtain the marginal likelihood for models of varying complexity. First, we introduce bridge sampling and three related sampling methods using the beta-binomial model as a running example. We then apply bridge sampling to estimate the marginal likelihood for the Expectancy Valence (EV) model-a popular model for reinforcement learning. Our results indicate that bridge sampling provides accurate estimates for both a single participant and a hierarchical version of the EV model. We conclude that bridge sampling is an attractive method for mathematical psychologists who typically aim to approximate the marginal likelihood for a limited set of possibly high-dimensional models.

Integrated modelling of age and sex patterns of European migration.

Age and sex patterns of migration are essential for understanding drivers of population change and heterogeneity of migrant groups. We develop a hierarchical Bayesian model to estimate such patterns for international migration in the European Union and European Free Trade Association from 2002 to 2008, which was a period of time when the number of members expanded from 19 to 31 countries. Our model corrects for the inadequacies and inconsistencies in the available data and estimates the missing patterns. The posterior distributions of the age and sex profiles are then combined with a matrix of origin-destination flows, resulting in a synthetic database with measures of uncertainty for migration flows and other model parameters.

Bayesian Population Forecasting: Extending the Lee-Carter Method.

In this article, we develop a fully integrated and dynamic Bayesian approach to forecast populations by age and sex. The approach embeds the Lee-Carter type models for forecasting the age patterns, with associated measures of uncertainty, of fertility, mortality, immigration, and emigration within a cohort projection model. The methodology may be adapted to handle different data types and sources of information. To illustrate, we analyze time series data for the United Kingdom and forecast the components of population change to the year 2024. We also compare the results obtained from different forecast models for age-specific fertility, mortality, and migration. In doing so, we demonstrate the flexibility and advantages of adopting the Bayesian approach for population forecasting and highlight areas where this work could be extended.

Letter to the Editor: Probabilistic population forecasts for informed decision making.

Demographic forecasts are inherently uncertain. Nevertheless, an appropriate description of this uncertainty is a key underpinning of informed decision making. In recent decades various methods have been developed to describe the uncertainty of future populations and their structures, but the uptake of such tools amongst the practitioners of official population statistics has been lagging behind. In this letter we revisit the arguments for the practical uses of uncertainty assessments in official population forecasts, and address their implications for decision making. We discuss essential challenges, both for the forecasters and forecast users, and make recommendations for the official statistics community.