In the last recipe, we used a frequentist method to test a hypothesis on incomplete data. Here, we will see an alternative approach based on Bayesian theory. The main idea is to consider that unknown parameters are random variables, just like the variables describing the experiment. Prior knowledge about the parameters is integrated into the model. This knowledge is updated as more and more data is observed.
Frequentists and Bayesians interpret probabilities differently. Frequentists interpret a probability as a limit of frequencies when the number of samples tends to infinity. Bayesians interpret it as a belief; this belief is updated as more and more data is observed.
Here, we revisit the previous coin flipping example with a Bayesian approach. This example is sufficiently simple to permit an analytical treatment. In general, as we will see later in this chapter, analytical results cannot be obtained and numerical methods become essential.