Continuations

The last few Uncertainty Wednesdays had us look at how to model beliefs using probability distributions and then update those with a specific example of using the beta distribution. You may have noticed something odd about the way we updated the parameters of the beta distribution: add 1 to α when we observe heads and add 1 to β when we observe tails. This wipes out any and all ordering information. So let’s say you have a total of 100 observations. With this update rule the only thing that m...

Last Uncertainty Wednesday, I introduced the beta distribution to model our prior belief about the probability of Heads in a coin toss. We saw that for the parameters α = β = 1 the beta distribution gives us a uniform prior, which expresses the highest degree of uncertainty (you may want to revisit the earlier post on entropy for that). Now I will toss an actual coin while writing this post. Wait. It came up tails (i.e. not heads). What should our new values be for α and β? As it turns out th...

It’s been six weeks since the last Uncertainty Wednesday, so I strongly suggest you go back first and read that post which provides an introduction to the idea of updating. Take your time, this new post won’t go away! The distribution that we will use for modeling our coin is the Beta distribution. Why choose that one? Because the beta distribution when combined with likelihood function for a coin toss gives us another beta distribution. This is known technically as a so-called conjugate prio...

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...

Before covering p-values and why they are so problematic, I thought it would be a good idea to provide a bit of a recap of Uncertainty Wednesday. I have been writing this series for almost two years now, beginning summer of 2016. Uncertainty is everywhere in the world and yet we are generally poorly equipped to reason about it. That happens to be true even for people, like myself, who have studied some probability and statistics. My approach in Uncertainty Wednesday has been to consider that ...