Empirical Bayes method
The Empirical Bayes method is a statistical approach that blends elements of Bayesian inference with frequentist techniques. It operates by estimating prior distributions from the data itself, rather than requiring a subjective input from external knowledge or beliefs, which differentiates it from traditional Bayesian methods. This method is particularly valuable in situations where data is limited or when integrating prior information is challenging. By formulating a prior distribution based on observed data, the Empirical Bayes method updates this prior using new data to improve inference about unknown parameters.
One significant application of the Empirical Bayes method is in climate change studies, where it aids in modeling uncertainties associated with climate variables such as temperature and sea level. Researchers can use this method to refine projections over time, ultimately enhancing predictive accuracy as new data becomes available. However, the method has faced criticism for its reliance on subjective prior distributions and for the way people often fail to update beliefs according to Bayesian principles. Despite these critiques, the Empirical Bayes method remains a powerful tool in various scientific fields, allowing for a more informed analysis of complex uncertainties.
Empirical Bayes method
Definition
In statistics, there are two very different approaches to making inferences about unknown parameters: the frequentist, or non-Bayesian, and the Bayesian. One of the key differences between these two approaches is the notion of probability employed. The frequentist approach defines probability as the relative frequency of an event occurring in repeated trials; probability under this definition is also termed “objective probability.” The Bayesian approach regards probability as a measure of the uncertainty inherent in a researcher’s rational belief about the values of parameters or unknown quantities of concern. Another important difference is related to the specification of unknown parameters. The frequentist approach considers parameters as unknown but nonrandom values. By contrast, the Bayesian method regards parameters as random variables and uses probability distribution to specify possible values of those parameters.
In application, the Bayesian method is basically a way of learning from data. A Bayesian application refers to a three-step process employed to update a researcher’s rational belief about unknown parameters or about the validity of a proposition, given the data observed. The first step concerns the formulation of a prior distribution for an unknown parameter in a statistical model of concern. The prior distribution reflects knowledge or results from past studies. Next, data or observations are collected to incorporate information about the parameter that generates those data. In the final step, the prior distributions are updated with the new data to create a new distribution. This method follows from a theorem formulated by and named after the Reverend Thomas Bayes, a British mathematician (1702-1761).
Applications of the Bayesian method and its implications for rational decisions are especially useful in studies for which the researcher has just a few data points available or for which uncertainty over some parameters needs to be resolved in the light of new data or observations. The method has become very popular over the past few decades in part because of the drastic growth of computer power, which renders much more feasible the calculations necessary to resolve simulations. There are various applications of the Bayesian method in such sciences as biostatistics, health outcomes, and global climate, to name just a few.
Significance for Climate Change
Climate is the long-term average of weather events occurring in a region. The weather on a day in January in Chicago may be mild or sunny, but the winter climate in the city is on average cold, snowy, and rainy. Climate change reflects a change in long-term trends of the aggregate of these weather events. For example, annual precipitation can increase or decrease, and the climate can become warmer or colder. On a global scale, global warming refers to an increasing trend of Earth’s temperature, which in turn causes changes in rainfall patterns, a rise in sea level, and a wide range of impacts on ecological systems and human life. The prediction of these changes and their impacts is difficult, because many uncertainties are associated with various relations and parameters present in the climate system. However, there has been significant study of these uncertainties under different methodologies.
As an approach to analyze uncertainty that allows incorporating expert knowledge and empirical observations into the analysis of updated data, the Bayesian method provides a powerful tool to study climate change. Recent studies have employed the Bayesian method to construct statistical models characterizing climate change. In those models, typical climate variables include, but are not limited to, surface air temperatures, precipitation, sea level, and ocean heat contents, on either a global or a regional scale. These models are built to serve many purposes, such as attribution, estimation, detection, and prediction of climate change.
In order to illustrate how the Bayesian approach can be applied to climate change, take the case of sea-level rise as an example. A recent study in this area tries to develop a Bayesian model for using evidence to update probability distributions for a climate model’s parameters, which reflect the unknown states of nature, including sea-level rise. Once developed, the model and its updated probability distributions can be used to make projections of sea-level rise.
The steps to build such a model are as follows: First, define a prior probability distribution over the parameters of a model of sea-level rise as formulated based on expert knowledge or past studies. Second, draw a certain number of samples of those parameters at random from the prior distribution, then feed these samples into the model to calculate projected sea levels. Third, observe the actual sea levels and use the data to update the model’s projections using the Bayes theorem. The updated projections are then translated into a new probability distribution. In the next cycle, with new data on sea levels obtained, the second and third steps are repeated, and the probability distribution is updated once more. Thus, each new observation of sea levels is incorporated into the model, providing better data and refining the probability distributions to increase the predictive accuracy of the system. The model parameters are partially resolved over time.
Although the Bayesian method provides an attractive approach for analyzing climate change, it is subject to some criticism. First, the idea of subjective judgments of prior probabilities, which influence the inferences drawn from models, is not accepted by many scientists. Critics of the Bayesian method argue that subjectivity prevents observers from viewing data objectively, so inferences should be based on observed data alone. Another problem is that people do not actually think like Bayesians. There is ample empirical evidence that people fail to update their prior beliefs using Bayes’ law and that they act differently from the assumptions of Bayesian analysis would predict. It should be noted that there are alternative approaches to the Bayesian method to analyze uncertainty involving climate change, such as fuzzy set theory.
Bibliography
Bolstad, William M. Introduction to Bayesian Statistics. Hoboken, N.J.: John Wiley & Sons, 2007.
Gelman, Andrew, et al. Bayesian Data Analysis. New York: Chapman & Hall, 2004.
Hobbs, Benjamin F. “Bayesian Methods for Analyzing Climate Change and Water Resource Uncertainties.” Journal of Environmental Management 49, no. 1 (January 1997): 53-72.
Massoud, Elias C., et al. "Bayesian Weighting of Climate Models Based on Climate Sensitivity." Communications Earth & the Environment, vol. 4, no. 364, 20 Oct. 2023, doi.org/10.1038/s43247-023-01009-8. Accessed 26 Dec. 2024.