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 probability *after* we have observed signal H.” Now in that quote and in the example we used probabilities and not probability distributions. We had found the following formula, which is known as Bayes’ rule:
P(B | H) = [P(H | B) / P(H)] * P(B)
As a reminder P(B) is the probability of event B before we have observed anything, so in the case of cancer this would be the rate at which this cancer occurs in the relevant population. P(B | H) is the updated probability conditional on having received a positive cancer test (H as in high). We saw that the updating occurs through the factor P(H | B) / P(H) which consists of the sensitivity of the test P(H | B) divided by the total probability of seeing a positive test P(H).
What we are looking for now is to come up with a similar version of Bayes’ rule for beliefs expressed as probability distributions. We want something that looks roughly like:
posterior belief = update factor * prior belief
where again the update factor captures the likelihood of the observations, keeping in mind here that we are now dealing with distributions. The beliefs and the update factor are both functions which makes the formula for this quite daunting looking. We will write it – with some abuse of notation – as follows to keep things simple
p(θ | x) = [p(x | θ) / p(x) ] * p(θ)
where θ is the parameter we are interested in, such as the probability of Heads for our coin, and x denotes our observations. We see the numerator of the update factor now is p(x | θ) – this is a function, the so-called likelihood function, which maps θ into p(x | θ). The denominator is p(x) which is the probability of the observations. That in turn is quite complicated if we unpack it, since it is an integral over all the possible values of θ and their probabilities (what comes out though is a scalar, meaning just a number, not a function).
So what we are really doing is multiplying two functions: the likelihood function and the probability density function of our prior belief which gives us a new function that represents our updated or posterior belief. This is complicated for the general case and along with calculating p(x) by evaluating the integral will require numerical approximations.
Thankfully though it turns out that there are elegant and simple solutions for some types of probability distributions, such as the Beta distribution which I had introduced as an example of a possible belief. If you have a Beta distribution as the prior belief for the probability parameter in a coin toss then the posterior belief also takes the form of a Beta distribution! Next Wednesday we will use this fact to show how we can easily update our beliefs on a coin toss (provided we buy into using the Beta as the shape of our prior).
