Note: this takes some great work by many others including Kevin Systrom, Frank Dellaert and Adam Lerer.

This is Bayes' Theorem as we'll use it:

This says that, having seen new cases, we believe the distribution of is equal to:

• The likelihood of seeing new cases given times ...
• The prior beliefs of the value of without the data ...
• divided by the probability of seeing this many cases in general.

A likelihood function function says how likely we are to see new cases, given a value of .

Any time you need to model 'arrivals' over some time period of time, statisticians like to use the Poisson Distribution. Given an average arrival rate of $\lambda$ new cases per day, the probability of seeing new cases is distributed according to the Poisson distribution: