#beta distribution

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Uncertainty Wednesday: Updating (Beta Distribution 2)

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 the updating formula is super simple. If we observe heads we increment α and if we observe tails we increment β. Here is what the Beta distribution looks like for α = 1 and β = 2, i.e. after we have observed tails:

What does this picture tell us? Going from a uniform prior, the observations of tails has shifted a fair bit of probability towards lower values of θ. Remember that θ is the parameter we are interested in. It is the probability that the coin will come up heads. Since we have seen only one outcome and that outcome was tails, we are updating by assigning a lot more probability to values of θ below 0.5 (on the horizontal axis). This is our updated belief.

Is this the only possible update we could have made? Well, if we use the Beta distribution to model our beliefs and the thing we observe has a binary outcome (such as a coin toss), then this is the precise updating as determined by Bayes’ Theorem. If you are so inclined you can find a very accessible derivation of this result here, which also shows how the simple updating rule results.

So let’s keep tossing our coin. I just did and as it turns out got heads. So our updated values are now α = 2 and β = 2 and our new distribution looks as follows

This is symmetric, which shouldn’t surprise us as we have observed both head and tails. It is also starting to shift probability away from the extremes and towards the middle.

Now if you want to play this game by yourself and see how the beta distribution changes after each toss, you can just head over to this query on WolframAlpha. Just add 1 to the value of α (alpha) each time you toss heads and 1 to the value of β (beta) each time you toss tails. Watch the distribution update!

I have just tossed my coin 30 times and observed 17 heads and 13 tails. Here is what the Beta distribution looks like for α = 18 and β = 14

So the beta distribution is starting to bunch up around 0.5, but we can see that the average is slightly above 0.5, in line with having observed more heads rather than tails (making heads somewhat more likely). Next Wednesday I will talk more about what we have learned at each step and also some of the limitations of this approach.

In the meantime, thanks to Eric Novik from Generable for helping me with my understanding of this!

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Uncertainty Wednesday: Updating (Beta Distribution)

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 prior. That all sounds very technical but the idea is simple: when our prior is a beta distribution and the observations are of the 0 or 1 (head or tails) variety, then our posterior distribution after updating is once again a beta distribution.

Now the beta distribution has two parameters, which are commonly referred to as α and β. These parameters can take on lots of values, but there is straightforward choice for initial parameters. We will pick α = β = 1 because for those values, the beta distribution coincides with the uniform distribution. All the plots in this post are generated using Wolfram Alpha.

As a refresher: the x axis on this chart is the parameter θ for which we are expressing our belief. In the coin toss example θ would be the probability that the coin comes up Heads on any one toss (in which case 1 - θ is the probability that it comes up Tails). We know nothing about the coin right now so that probability θ could be anything between 0 (never see Heads) and 1 (every toss comes up Heads).

At this point you may be thoroughly confused. How do the parameters from the beta distribution relate to the parameter for the coin toss? Sometimes people call the α and β hyper-parameters, but I think a better term would have been meta-parameters or belief-parameters. Put differently α and β determine the shape of our belief about θ.

So now what remains to be done is to figure out how we should update α and β after we have observed some outcomes. We are looking for new values of α and β after we have observed either Heads or Tails. As we will see next Wednesday the updating of these values turns out to be super simple.

[drafthorsemath]

I looked on tumblr and I saw that there were no posts on the Beta Distribution.  I said, "Unacceptable", and decided to right that which was wrong.