Today’s post is going to be a bit more abstract and a bit less applied than usual, because today I want to take Python’s Sympy symbolic algebra library for a spin. As a narrative subject, we’ll be looking at the medieval English Longbow. We’ll start with some physics by modelling the deterministic trajectory of the arrow. Then we’ll add some statistical noise to make it a bit more realistic. Lastly, we’ll do some simple Bayesian inference on the range of the longbow. And we’ll do as much of it as we can via symbolic algebra.
These are English longbowmen. The English (or Welsh) Longbow is around 1.8-2m tall, can fire 10-12 arrows per minute in the hands of a skilled archer, and its steel-tipped arrows can penetrate the armour of a medieval knight. The “draw-weight” of a longbow is considerable, and skeletons of longbow archers often have enlarged left arms. Over a lifetime, the archer’s body was actually deformed by their tool of trade. So, how far could it shoot an arrow? Let’s figure that out with physics. According to Longbow Speed Testing (!), the velocity of an arrow leaving the bow is 172-177 feet per second. Say 53m/s in metric.
The trajectory of a projectile can be computed using classical mechanics:
- is the height of the arrow
- is the horizontal distance that the arrow has travelled
- is the angle at which the arrow is fired
- is the velocity of the arrow as it leaves the bow
- is the acceleration due to gravity (~ on the surface of the earth)
- (We’re ignoring wind resistance)
Let’s start by loading this into Sympy:
At the time of writing, the hunt is still on for Malaysia Airlines flight MH370. Bayesian search theory has become topical (again). Bayesian search has been used to find crashed planes, lost hikers, sunken submarines and even missing hydrogen bombs. Bayes’ theorem is perfectly suited to search because it provides a mathematical framework for deductive reasoning.
Let’s try it out.
Here’s our (semi-fictionalised) search scenario: In 217BC, Rome and Carthage are at war. Dido’s curse still haunts the two civilisations. Carthaginian General Hannibal Barca has just annihilated a Roman army at Lake Tresimene, 180km northwest of Rome. He had already inflicted a series of crushing defeats on the Romans to the point that, after Lake Tresimene, Rome was left virtually without any field army at all. The great fear was that Hannibal would now march his war elephants on the city of Rome itself. In times of dire emergency, the Roman republic allowed for the temporary appointment of a dictator. Five days after Lake Tresimene, the senate appointed Quintus Fabius Maximus as dictator. The first question for him was: where is Hannibal now?
In this post I’ll demonstrate one way to use Bayesian methods to build a ‘dynamic’ or ‘learning’ regression model. When I say ‘learning,’ I mean that it can elegantly adapt to new observations without requiring a refit.
Today we lay our scene in London between the years 1760-1850. The good people of London have bread and they have wheat, which is, of course, what bread is made from. One would expect that the price of bread and the price of wheat are closely correlated and generally they are, though in periods of turmoil things can come unstuck. And Europe, between 1760-1850, was a very eventful place.
This is the first of a series of posts on Bayesian learning. The learning mechanism which is built into Bayes’ theorem can be used in all sorts of clever ways and has obvious uses in machine learning. Today we’ll build our intuition of how the mechanism works. In the next post, we’ll see how we can use that mechanism to build adaptive regression models.
Bayesian statistics work in the same way as deductive reasoning. Bayes’ theorm is a mathematical form of deductive reasoning. It took me a long time to understand this simple but profound idea. Let me say it again: Bayes’ theorm gives us a mathematical way to do deductive reasoning: it allows us to take a piece of evidence, weigh it against our prior suspicions, and arrive at a new set of beliefs. And if we have more evidence, we can do it over and over again.
Let’s do an example. It occured to me that archaeology is an excellent illustration of how this works.
We are archaeologists
We’re excavating the site of a Roman village. Our job is to try and pinpoint what year(s) the village was likely inhabited. In probability terms, we want to work out
p(inhabited) for a range of years. As we uncover relics like pottery and coins, we use that evidence to refine our beliefs about when the village was inhabited.
In the Asterix comics we are never told just what breed Obelix’s little dog Dogmatix is. In fact, he is specifically referred to as being an “undefined breed” in his first appearance in Asterix and the Banquet.
I’ve compiled four hypotheses:
- West Highland White Terrier: In the first two Asterix movies, the part of Dogmatix was played by a West Highland white terrier (a “Westie”). Of all the present-day dog breeds, westies look most like Dogmatix. There is one major hole in the westie hypothesis: it’s extremely unlikely that a Westie would be found in Gaul circa 50BC. That’s because the dogs didn’t exist back then.
- Melitan: A melitan was a breed of lapdog popular with Romans and Greeks in ancient times. They were small companion dogs, typically pampered things who existed solely to amuse their owners, typically noblewomen. “The Melitan was a small, fluffy, spitz-type dog, commonly white in colour… with a very appealing, pointed, fox-like muzzle.” In appearance, they sound nothing like Dogmatix. But the Melitan hypothesis is very strong for another reason: these small dogs actually existed as pets in 50BC. There’s no record of them amongst Gauls, however. We’d have to imagine that rowdy, warring Obelix came into possession of a small Roman lapdog during a raid.
- A Gallic war dog, a.k.a. Irish Wolfhound: What we call an Irish wolfhound is a very ancient breed of dog which Celts were known to use as guard dogs and in combat. Caesar is supposed to have mentioned them in his Commentaries on the Gallic War, though I couldn’t find the reference. This is the only breed of dog which we might realistically find in a Gaulish village in 50BC, which makes it a good hypothesis for Dogmatix. The obvious flaw: wolfhounds are huge. Maybe Dogmatix could be a wolfhound, but he’d have to be an anemic, albino, malformed runt of a wolfhound. He is known to accompany Obelix on raids and bite Romans.
- Schnauzer: The last hypothesis: Dogmatix could be a schnauzer. This is unlikely for so many reasons: (1) schnauzers didn’t exist in 50BC; (2) they aren’t native to Gaul/France; (3) they look nothing like Dogmatix except, except, except that schnauzers have whiskers that perhaps vaguely resemble Dogmatix’s unusual moustache.
Which breed does Dogmatix belong to…?
Our goal is to determine which of these four breeds Dogmatix belongs to. First, we need some data, then we need a method for decision making.