Bayesian Estimation - From Fork Handles to Radar Tracking

Most people living in the UK over a certain age will be familiar with the Four Candles sketch from the 1976 Two Ronnies comedy show. For others, a quick education is available on the Wikipedia page for the sketch or simply watch it on You Tube. The essence of the sketch that we are interested in here is the ambiguity in the request made by the customer and the assumptions made by the shop keeper. What makes the sketch funny is that reasonable assumptions about what the customer was asking for turned out to be wrong. We will consider this in the context of a mathematical theorem, originally proposed in the 18th century and which over its history has been triumphed, discredited, wiped from maths teaching, rediscovered and used to crack German Enigma codes. Welcome to Bayes' theorem and its application to radar tracking. Back to the Four Candles sketch. The customer asked the shopkeeper for "Four Candles" - or that's what the shopkeeper thought. There was ambiguity in the request, so it could have been heard as "Fork Handles", which is what the customer really wanted. Although the shopkeeper did not go through a conscious mathematical process to interpret what he heard, his brain...