Summary. Key ideas about probability models and the objectives of statistical analysis are introduced. The differences between frequentist and Bayesian analyses are illustrated in a very special case. Some slightly more complicated models are introduced as reference points for the following discussion.

Starting point

We typically start with a subject-matter question. Data are or become available to address this question. After preliminary screening, checks of data quality and simple tabulations and graphs, more formal analysis starts with a provisional model. The data are typically split in two parts (y : z), where y is regarded as the observed value of a vector random variable Y and z is treated as fixed. Sometimes the components of y are direct measurements of relevant properties on study individuals and sometimes they are themselves the outcome of some preliminary analysis, such as means, measures of variability, regression coefficients and so on. The set of variables z typically specifies aspects of the system under study that are best treated as purely explanatory and whose observed values are not usefully represented by random variables. That is, we are interested solely in the distribution of outcome or response variables conditionally on the variables z; a particular example is where z represents treatments in a randomized experiment.

We use throughout the notation that observable random variables are represented by capital letters and observations by the corresponding lower case letters.