### Bayesian updating beta prior

Bernoulli Distribution Our example is that of a sequence of coin flips. We can actually use a simple calculation to prove why the choice of the beta distribution for the prior, with a Bernoulli likelihood, gives a beta distribution for the posterior. That is, we wish to quantify our uncertainty in how biased the coin is. What beta distribution is produced as a result? However, perhaps the most important reason for choosing a beta distribution is because it is a conjugate prior for the Bernoulli distribution. I like to think of priors in terms of what kind of information they represent. We couple our prior beliefs with the data we have observed and update our beliefs accordingly. All three posteriors are getting narrower as more data is added. Estimating the proportion of patients who make a full recovery after taking an experimental drug to cure a disease Corporate Finance: The hypothesis with the most relative support is. Starting at different points yields different flows over time. In fact, if you multiply the two together as in Bayes' rule , you get: All of these things are reasonable to consider and I am going to ignore them all. Note in particular that we can use a posterior beta distribution as a prior distribution in a new Bayesian updating procedure. The fairness of the coin will also be stationary, that is it won't alter over time. How does this change our belief on the fairness of the coin? Likelihoods are relatively straightforward to understand because they are based on tangible data. There are a couple of reasons: Recalling Bayes' Rule In the previous article we outlined Bayes' rule. Consider the following scenarios: In this article we are going to expand on the coin-flip example that we studied in the previous article by discussing the notion of Bernoulli trials , the beta distribution and conjugate priors. Updating the prior via the likelihood Now for the easiest part. This will motivate the following rather mathematically heavy sections and give you a "bird's eye view" of what a Bayesian approach is all about. This property of likelihoods is carried over to posterior updating. But all three posteriors are still quite broad, and the breadth of the curves can be thought to represent the uncertainty in my estimates.