The denominator of Bayes Rule is arguably the most important part of the formula. Motivation: Why? Let’s go through the formula piece by piece for a second:

What do we mean by sensitivity? Sensitivity refers to a measure of how much influence the prior of choice affects the associated posterior distribution. A general rule of thumb is, when looking at Bayes Rule, the component (either likelihood or prior) with the most extreme LOWER value will affect the posterior the most. Now, It is also true that the MORE data we have, the LESS of an…

Two more types of priors are uninformative and informative priors. Uninformative Priors Like we mentioned in the previous post on priors, all priors contain some information. Although Uninformative priors are often used in attempt to produce an “objective analysis”, there is no such thing! …

Let’s use a Bayes Box to illustrate how priors and likelihoods affect the shape of the corresponding posterior distributions! A Bayes Box is a table that walks you through the calculation of the posterior. There is a column for each part of the equation: Let’s look back at our apple…

In the fashion where a mathematical proof introduces a base case as the first example, we will do this for priors as well. A flat prior essentially demands that p(theta) is held at a constant value (usually a natural number). So what does having a constant value prior mean to…

In this post, we will be introducing the likelihood’s neighbor: Priors. They are exactly what it sounds like. They represent our most current, basic understanding and interpretation of a phenomena. Here, we present a few ways of understanding what a prior is. We will go through each one.

You may recall that likelihoods are constructed from the product of many individual likelihoods. In order for us to claim this, we must ensure our sample we are basing our likelihoods from is independent and identically distributed or random. However, this can be a tricky condition to meet. Thanks to…

There are many equivalence relations you might come across in your own mathematical journeys, but there is one particularly useful one acknowledged in Bayesian Statistics. Before we take a look at it, let’s do a little thought exercise: Suppose we have a probability of flipping heads for a coin …

For those of you who would prefer a more mapped out relationship between probability distributions and likelihoods, please refer to the map below. Here is an example using integrals to show that Likelihoods do not sum to 1 for all values of theta: