1.3 Bayesian modeling
Models are simplified descriptions of a given system or process that, for some reason, we are interested in. Those descriptions are deliberately designed to capture only the most relevant aspects of the system and not to explain every minor detail. This is one reason a more complex model is not always a better one. There are many different kinds of models; in this book, we will restrict ourselves to Bayesian models. We can summarize the Bayesian modeling process using three steps:
Given some data and some assumptions on how this data could have been generated, we design a model by combining building blocks known as probability distributions. Most of the time these models are crude approximations, but most of the time that’s all we need.
We use Bayes’ theorem to add data to our models and derive the logical consequences of combining the data and our assumptions. We say we are conditioning the model on our data.
We evaluate the model, and its predictions, under different criteria, including the data, our expertise on the subject, and sometimes by comparing it to other models.
In general, we will find ourselves performing these three steps in an iterative non-linear fashion. We will retrace our steps at any given point: maybe we made a silly coding mistake, or we found a way to change the model and improve it, or we realized that we need to add more data or collect a different kind of data.
Bayesian models are also known as probabilistic models because they are built using probabilities. Why probabilities? Because probabilities are a very useful tool to model uncertainty; we even have good arguments to state they are the correct mathematical concept. So let’s take a walk through the garden of forking paths [Borges, 1944].