EXPLORING REALISTIC BAYESIAN MODELING SITUATIONS Per Nilsson 1 , Per Blomberg 2 and Jonas Bergman Ärlebäck 3 1 Örebro University, Sweden 2 Linnaeus University, Sweden 3 Linköping University, Sweden per.nilsson@oru.se The study reported in the present paper is part of a larger project, which aims to explore possibilities and challenges in developing a teaching practice that supports students’ ability to model random dependent situations by a Bayesian approach. A central premise is that modeling should be based on situations that appear realistic to the students. Given this premise, the specific purpose of the present study is to identify and characterize uncertain situations that are realistic and suitable for a Bayesian treatment. The study involves reviewing some of the literature related to Bayesian applications. Based on that review we distinguish detecting (test) situations and construction composition situations as two general types of Bayesian modeling situations. INTRODUCTION Models are essential to mathematics. We develop mathematical models to understand the world and to predict the behavior of phenomena we encounter in the world (Blum, Galbraith, & Niss, 2007). Roughly, we can classify mathematical models into deterministic and stochastic models. Deterministic models include a number of elements and relations that completely determines a system, i.e. we can make certain predictions of how a system behaves. Stochastic models include elements that make it impossible for us to predict the behavior of the system with certainty. There is an uncertainty in how the results of a stochastic system will occur, an uncertainty we cannot trace to causal factors. Many phenomena are not suitable to deal with by a deterministic model. We need to develop models that take into account the random behavior of phenomena. A frequentist modeling approach is based on the assumption that we can repeat a random experiment a large number of times under exactly the same conditions each time. However, in practice it is often impossible to repeat an experiment a very large number of times and to achieve exactly the same conditions in each trial. Think, for instance, of the simple experiment of throwing one die. Is it possible to throw the die from exactly the same angle and height each time? We guess not! Moreover, many situations involve the assessment of a probability when we only have data from a single, or a short termed, sample. Consider, for example, the situation that you doubt whether you turned off the coffee maker before leaving your house this morning. Although there may be frequency information on how common it is that people forget to turn their coffee maker off before leaving their house, this general frequency offers limited information about the probability that you would have done so, exactly this morning. A way to handle these situations, where we cannot meet the objective requirements of the frequentist interpretation, is to apply to the situation a probability model that is subjective in nature. A subjective model is relative to the information available and specified by the modeler of the random situation in question (Goldstein, 2006). The main objection raised against a subjective interpretation of probability concerns the scientific status of results, which is based on and varies with the observer and the information available (Batanero, Henry, & Parzysz, 2005). To meet the objections we need to organize and formalize the modeling in a scientific way. The Bayesian rule offers a way of structuring and strengthening a subjective rationality of probability modeling (Goldstein, 2006). The current paper constitutes the initial step of a larger project, which aims at exploring and developing a framework for structuring teaching and learning of Bayesian modeling in school. A modeling sequence takes departure from some concrete, realistic situation that begs to be modeled (Freudenthal, 1983). This precondition means that we should have a rather good picture of what kind of situations that are to be modeled by a Bayesian approach. On account of that purpose, the particular aim of the present study is to identify and distinguish from a literature review various situations typical for Bayesian modeling. The analysis is motivated and guided by the idea of the ICOTS9 (2014) Invited Paper Nilsson, Blomberg & Ärlebäck In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9, July, 2014), Flagstaff, Arizona, USA. Voorburg, The Netherlands: International Statistical Institute. iase-web.org [© 2014 ISI/IASE]