is the set of possible observations, i.e. the sample space, and is a set of probability distributions on . The intuition behind this definition is as follows. It is assumed that there is a "true" probability distribution induced by the process that generates the observed data. We choose to represent a set (of distributions) which contains a distribution that adequately approximates the true distribution. Note that we do not require that contains the true distribution, and in practice that is rarely the case. Indeed, as Burnham & Anderson state, "A model is a simplification or approximation of reality and hence will not reflect all of reality"—hence the saying "all models are wrong". The set is almost always parameterized: . The set defines the parameters of the model. A parameterization is generally required to have distinct parameter values give rise to distinct distributions, i.e. must hold (in other words, it must be injective). A parameterization that meets the requirement is said to be identifiable. An example Suppose that we have a population of children, with the ages of the children distributed uniformly, in the population. The height of a child will be stochastically related to the age: e.g. when we know that a child is of age 7, this influences the chance of the child being 1.5 meters tall. We could formalize that relationship in a linear regression model, like this: heighti = b0 + b1agei + εi, where b0 is the intercept, b1 is a parameter that age is multiplied by to obtain a prediction of height, εi is the error term, and i identifies the child. This implies that height is predicted by age, with some error. An admissible model must be consistent with all the data points. Thus, a straight line (heighti = b0 + b1agei) cannot be the equation for a model of the data—unless it exactly fits all the data points, i.e. all the data points lie perfectly on the line. The error term, εi, must be included in the equation, so that the model is consistent with all the data points. To do statistical inference, we would first need to assume some probability distributions for the εi. For instance, we might assume that the εi distributions are i.i.d. Gaussian, with zero mean. In this instance, the model would have 3 parameters: b0, b1, and the variance of the Gaussian distribution. We can formally specify the