Subjective Likelihood for the Assessment of Trends in the Ocean’s Mixed-Layer Depth Ana Grohovac RAPPOLD, Michael LAVINE, and Susan LOZIER This article describes a Bayesian statistical analysis of long-term changes in the depth of the ocean’s mixed layer. The data are thermal profiles recorded by ships. For these data, there is no good sampling model and thus no obvious likelihood function. Our approach is to elicit posterior distributions for training data directly from the expert. We then infer the likelihood function and use it on large datasets. KEY WORDS: Bayes; Climate change; Elicitation; Foundations; Likelihood; Oceanography. 1. INTRODUCTION The typical Bayesian analysis posits data from a parametric family of sampling distributions, as in y ∼ p(y | θ). (1) After y has been observed, it is treated as fixed, and the likeli- hood function is defined to be ℓ(θ) ≡ p(y | θ), a function of θ . The interpretation is that likelihood ratios ℓ(θ 1 )/ℓ(θ 2 ) quan- tify y’s evidence for θ 1 as opposed to θ 2 . For our dataset, there is no believable sampling model p(y | θ), so we cannot assign ℓ(θ) ≡ p(y | θ). We take a different approach, wherein lies the statistical novelty of this article. For several values of i (about a dozen), we show the expert y i and directly elicit his or her posterior distribution p(θ | y i ). Elici- tation is done under conditions in which the expert has an ap- proximately uniform prior for θ . After elicitation, we know the prior and posterior and thus can infer the likelihood function. After examining the dozen or so elicited posteriors and con- ferring with the expert, we constructed an algorithm that ac- cepts a y as input and yields a likelihood function ℓ(θ) as out- put. After constructing the algorithm, we checked that it gave sensible results on several hundred more y’s. We then applied the algorithm to our full collection of data {y i } T i =1 , which, when combined with our real prior, yields the posterior that we use for inference. We call ℓ a likelihood function because it approxi- mately summarizes the expert’s weight of evidence and, when multiplied by the uniform prior, yields the posterior. The data arise in a study of the ocean’s climate. The situation is more complicated than described in this introduction because the data are a time series and our model must account for an annual cycle. Section 2 provides the scientific background and Section 3 describes the data. Section 4 describes our subjective likelihood, how it was elicited, and how it is modeled. It con- tains whatever statistical novelty is in this article. Section 5 de- scribes our full model, prior, and posterior inference, account- ing for the annual cycle, year-to-year variation, heteroscedas- ticity, and a possible secular trend. Finally, Section 6 presents a discussion. Ana Grohovac Rappold is National Research Council Associate at US EPA, Duke University, Durham, NC 27708 (E-mail: ana@stat.duke.edu) (currently employed by the EPA). Michael Lavine is Professor, Depart- ment of Statistical Science, Duke University, Durham, NC 27708 (E-mail: michael@stat.duke.edu). Susan Lozier is Professor and Chair, Earth and Ocean Sciences, Nicholas School of the Environment and Earth Sciences, Duke Uni- versity, Durham, NC 27708 (E-mail: mslozier@duke.edu). The authors grate- fully acknowledge the support of the National Science Foundation’s Collabo- ration in Mathematical Geosciences program. 2. THE OCEAN’S MIXED LAYER Recent evidence that the world’s oceans have warmed over the past 50 years (Levitus, Antonov, Boyer, and Stephens 2000) and that the attendant increase in the ocean’s heat content is an order of magnitude larger than the increase in the atmospheric and cryospheric heat content (Levitus et al. 2001) have made it abundantly clear that a determination of how our global cli- mate is changing in response to long-term natural and/or an- thropogenic forcing depends on the effectiveness of the ocean as a heat reservoir. However, the effectiveness of the ocean as a reservoir is curtailed by increasing thermal stratification, which limits the extent to which surface signals can be transmitted to depth. Thus interest has focused on the upper ocean. To a first approximation, oceanographers regard the ocean as having two layers, a mixed layer from the surface down to as much as several hundred meters, and a stratified layer be- neath. The mixed layer is that part of the surface ocean that displays uniformity in such properties as temperature, salinity, and density. The mixed layer forms because the upper waters of the ocean are mixed through waves and wind and also through thermal convection when the surface waters overturn on losing heat, and thus buoyancy, to the atmosphere. Such overturning creates a mixed layer. The depth M of the mixed layer evolves through an annual cycle and depends on geographic location. Because M depends crucially on heat exchange with the at- mosphere, long-term changes in heat exchange may result in long-term changes in M. Essentially, in this application we ad- dress the question as to whether or not there has been a secular trend in M in the North Atlantic subtropical gyre. Such an eval- uation will increase our understanding of potential physical and biological consequences of global warming. 3. DATA Hydrographic data such as temperature salinity and pressure, are collected from ships, sent to the National Oceanographic Data Center (NODC) where it is quality controlled, and then made publicly available (www.nodc.noaa.gov). This article re- ports on an analysis of NODC historical hydrographic data recorded over a small spatial region near Bermuda. We chose a sufficiently small region for our study so we can safely ignore spatial variability. We will give an analysis of data from a wide region of the North Atlantic elsewhere. A map of the data is provided in Figure 1; the data’s temporal distribution is shown in Figure 2. © 2007 American Statistical Association Journal of the American Statistical Association September 2007, Vol. 102, No. 479, Applications and Case Studies DOI 10.1198/016214507000000761 771