1. Introduction Paleontologists often need to model complex systems with many variables and complex relationships. In such models, information is often characterized by high-dimensional statistical distributions that are difficult to analyze mathematically. In this poster, we describe the use of Markov Chain Monte Carlo (MCMC) in generating an approximate sample from any desired distribution. In so doing, the sample provides important insights into the nature of the distribution at hand. MCMC is an iterative approach towards simulating a sample, creating a sequence of values (or vector of values) whose distribution more closely approximates the desired distribution the longer the chain is allowed to run. More specifically, we present the Metropolis-Hastings Algorithm, a particular implementation of MCMC, which easily adapts to high-dimensional problems. In this poster, we use MCMC to gain insight into the primary causes of the Permian mass extinction. We use a computer model to simulate secondary extinctions in a Late Permian food web. Then, using MCMC methods and Bayesian modeling, we infer the level of primary producer shutdown needed to cause observed levels of secondary extinction in Late Permian Karoo basin fauna. Mysterious Mass Extinction The end-Permian or Permian-Triassic extinction event occurred approximately 251 million years ago and nearly wiped out all life on Earth. Possibly as few as ten percent of all species then extant survived the event. In contrast, approximately half of the species present 65 million years ago survived the end-Cretaceous extinction. However, the differences do not end there. The causes of the end- Cretaceous event are well-explained by clear physical evidence. Conversely, less direct physical evidence remains of the causes (primary extinctions) leading up to the end- Permian event. There are numerous theories to account for the mass extinction, including massive volcanism (Figure 1) and a meteor impact scenario similar to the end-Cretaceous event (Figure 2), Markov Chain Monte Carlo: New Tools for Bayesian Modeling in Paleontology Michael D. Karcher ’07 (1), Steve C. Wang (1), Peter Roopnarine (2), Kenneth Angielczyk (3) (1) Department of Mathematics and Statistics, Swarthmore College (2) Department of Invertebrate Zoology & Geology, California Academy of Sciences (3) Department of Earth Sciences, University of Bristol For further information: e-mail mkarche1@swarthmore.edu or scwang@swarthmore.edu Food Web Collapse During any ecological disturbance, repercussions propagate throughout the entire food web. It is possible to simulate these effects computationally. The necessary framework to create these simulations has already been explored (Roopnarine 2006, Paleobiology 32:1, 1-19). Using a network graph to describe the food web in terms of “guilds” (for example, “Large Herbivores”, “Small Amphibians”, etc.), the function ceg(x) (Cascading Extinction on Graphs) takes a vector of primary extinctions—values in [0,1] representing extinctions as a direct result of an extinction event—and outputs a vector of secondary extinctions—extinctions as a result of food web collapse. For example, in a system with three guilds, ceg(0.45, 0.05, 0.0) = (0.85, 0.75, 0.8), shows that moderate extinction in one guild can cause major effects in other guilds, mirroring reality. Metropolis-Hastings Algorithm One particular MCMC technique that adapts well to high- dimensional problems is the Metropolis-Hastings Algorithm. It takes a Random-Walk (Figure 3) approach towards exploring the given distribution. Suppose we wish to sample from a given distribution with density function f(x). First, we choose a “jumping” distribution g(x|θ) that is easy to sample from and depends only on one parameter (e.g. Normal with mean θ). We choose x 1 from any point where f(x) is positive, and at each step x t of the chain, we sample a new point x * from the jumping distribution and calculate: Example: MCMC in Action The plots in Figure 4 were created using the Metropolis- Hastings Algorithm in one dimension. It was run on the food web networks from the Karoo basin dataset and its objective was to find the true value of the primary extinction. Paleontological Uses Using the data from the simulations of food web collapse, it is possible to treat the potential primary extinction scenarios in the end-Permian event as a statistical distribution and analyze them using MCMC. In order to do so, we set up a Metropolis- Hastings chain with points from the possible primary extinctions. We apply the CEG function (transforming primary extinction into secondary extinction) to each point many times to generate a likelihood function by counting how many of those outputs fall within a predetermined n-cube around the observed secondary extinction. Once the chain has sufficiently explored the distribution, we can apply Bayes’ Theorem with a flat prior to calculate the posterior distribution for the primary extinction scenarios. Using simple statistical techniques, we can then construct estimates and confidence intervals about which primary extinction scenario caused the end-Permian event. Markov Chain Monte Carlo Statisticians often need to explore statistical distributions that are difficult to analyze directly. Inconvenient distributions often have pathological or even no mathematical formulae, domain spaces that are too large to fully exhaust, or other traits that make direct analysis nearly impossible. One class of tools statisticians possess to work through this is Markov Chain Monte Carlo (MCMC). What binds the tools of MCMC together is that they are all iterative and asymptotic processes; they all proceed step-by-step and get more accurate the longer they are run. The Markov property plays a large role in the utility of the techniques. As a given MCMC process iterates, each step depends only on the last step and no other information. It can be shown that as the length of a chain with the Markov property increases, it converges to some distribution, and methods exist to ensure that it converges to any desired distribution, even inconvenient distributions, allowing analysis to continue. Future The simulations produce a great deal of data, and much work still remains in its interpretation. Running the Algorithm on the real world data produces 24-dimensional results. Figure 5 is a collection of histograms from one of our runs summarizing the data marginally. More useful inferences can be gathered from analyzing the variables and their correlations. However, this is no trivial task in 24 dimensions. A great deal of time could be spent in future projects analyzing and determining how to analyze the data that is produced. Fig. 1Fig. 2 Fig. 3: Metropolis-Hastings Random-Walk Markov Chain Fig. 4 And then we set: There are several techniques to improve the sample, including discarding the first part of the chain to reduce the effect of choosing x 1 arbitrarily and discarding all but every k th step to reduce correlation. Fig. 5