How Bayesian analysis works

I assume that we are interested in the value of a parameter, such as the mean or proportion. We start by giving this parameter a prior distribution quantifying our beliefs about where the parameter is located, based on what we believe about it before collecting data. There are lots of ways to pick the prior;  for example, we could pick an uninformative prior that says little about a parameter's value. Alternatively, we could use a prior that gives beliefs based on, say, previous studies, therefore biasing the value of the parameter to these values.

Then, we collect data and use it to compute the posterior distribution of the parameter, which is our updated belief about its location after seeing new evidence. This posterior distribution is then used to answer all our questions about the parameter's location. Note that the posterior distribution will answer all questions with probabilities. This means that we don't say whether the parameter is in a particular region or not, but the probability that it is located in that region instead. In general, the posterior distribution is difficult to compute. Often, we need to rely on computationally intensive methods such as Monte Carlo simulation to estimate posterior quantities. So, let's examine a very simple example of Bayesian analysis.