A while back I wrote a post titled “In Praise of Consistency,” which argues that a powerful way to examine one’s beliefs is to check their internal consistency. An example that’s currently on my mind is the right of private companies to turn away unvaccinated customers. On the right there are a lot of people who seem to think that shouldn’t be allowed, but at the same time belief that a baker should be able to refuse making a wedding cake for a gay couple. This strikes me as highly inconsiste...

Now that we have spent the last few Uncertainty Wednesdays on modeling beliefs as probability distributions, we can now get to the topic of updating. Updating is what we are supposed to do with our beliefs when we have new observations. We first encountered a similar idea in the extensive example of a cancer test which we used to derive Bayes’ theorem. In that post I wrote that “[Bayes’ theorem] relates the probability of the world being in state B *before* we have observed a signal to the pr...

Today’s Uncertainty Wednesday is a quick wrap up to the formalization of beliefs as probability distributions. Let’s first start with a simple question. What probability distribution reflects the least knowledge about a coin? How do you express “I don’t know, the coin could be anything from only heads to only tails to anything in between”? The answer is a uniform distribution as shown here:The horizontal axis is the probability of the coin coming up heads – ranging from 0 to 1 and the vertica...

Last Uncertainty Wednesday, I introduced the idea of beliefs. Today we will make this idea more precise. We started with an extreme belief, the one that a coin is so biased that we will only observe “heads” (H). More realistically one might belief that a coin is fair, but has some possibility of being slightly biased in either direction (e.g., more likely to observer H or more likely to observe T). So how do we formalize this? A belief is simply a probability distribution. For instance the on...

The last few weeks in Uncertainty Wednesday we took a detailed look at the various problems with p-values. Now we will start to work towards an alternative approach based on Bayesian inference. Along the way though we have a chance to think about more foundational ideas, including the subjective versus objective and beliefs versus truths. Today we will start by looking at beliefs and how observations may influence those. As before with p-values we will use coin flips as they are easy to follo...