model in the form () as follows. The sample space, , of our model comprises the set of all possible pairs (age, height). Each possible value of = (b0, b1, σ2) determines a distribution on ; denote that distribution by . If is the set of all possible values of , then . (The parameterization is identifiable, and this is easy to check.) In this example, the model is determined by (1) specifying and (2) making some assumptions relevant to . There are two assumptions: that height can be approximated by a linear function of age; that errors in the approximation are distributed as i.i.d. Gaussian. The assumptions are sufficient to specify —as they are required to do. General remarks A statistical model is a special class of mathematical model. What distinguishes a statistical model from other mathematical models is that a statistical model is non-deterministic. Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values, but instead have probability distributions; i.e. some of the variables are stochastic. In the above example with children's heights, ε is a stochastic variable; without that stochastic variable, the model would be deterministic. Statistical models are often used even when the data-generating process being modeled is deterministic. For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process). Choosing an appropriate statistical model to represent a given data-generating process is sometimes extremely difficult, and may require knowledge of both the process and relevant statistical analyses. Relatedly, the statistician Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis". There are three purposes for a statistical model, according to Konishi & Kitagawa. Predictions Extraction of information Description of stochastic structures Those three purposes are essentially the same as the three purposes indicated by Friendly & Meyer: prediction, estimation, description. The three purposes correspond with the three kinds of logical reasoning: deductive reasoning, inductive reasoning, abductive reasoning. Dimension of a model Suppose that we have a statistical model () with . The model is said to be parametric if has a finite dimension. In notation, we write that where is a positive integer ( denotes the real numbers; other sets can be used, in principle). Here, is called the dimension of the model.