## Bayes' Rule

Scientific research seeks to make inference about population parameters from data. The way this is done is to gather data, and use it to test and investigate a particular theory. The theory will often be formulated in terms of parameters which are unknown and need to be estimated from the sample data. Of course scientists will have some prior belief about these parameters. The Bayesian view of statistics provides a paradigm that scientists can implement to establish their latest beliefs about the parameters of a model. If we allow theta to represent the parameters of interest then we can imagine that these parameters are the settings on a machine that produces x values (the data). That is x is made under the influence of theta. This is symbolized as x|theta (x given theta). This is theoretically what happens -- the truth is that we don't know the theta values, we only see their effects in x. So what we really need to do is infer from the data the values of theta from x -- that is theta|x (theta given x).