The Horsham projectOccam's razor in bits: minimum description length
A curve is only worth sending if it saves more than it costs.
The data is one column from the 2021 census: the number of people in Horsham district at each single year of age, from babies under one to people aged 99. That is 100 numbers. They rise to 2,368 at age 55, dip sharply to 980 at age 19, and drop from 1,955 at age 74 to 1,437 at age 75. The whole table counts 146,776 people, a figure two away from the district total above because the census protects privacy by adjusting each table slightly. Sent raw, each number needs 12 bits, so the message is 1,200 bits long.
Now the learner tries to do better by sending a smooth curve and then the corrections. The curve is a polynomial; its degree is how many bends it is allowed. We charge 3.32 bits for every number needed to describe the curve, which is half of log2(100), a standard price when there are 100 data points. The corrections are charged by how large they are: small corrections are cheap to send, large ones are dear. Python fits each curve by least squares and adds up the bill.
Polynomial curves fitted to the Horsham district age profile, 100 ages, our Python run on Census 2021 data| Degree of curve | Typical miss (people) | Whole message (bits) |
|---|
| 0 (a flat line) | 581.4 | 1,126.4 |
| 2 | 286.1 | 1,030.7 |
| 6 | 148.3 | 949.2 |
| 12 | 137.7 | 958.5 |
| 20 | 78.8 | 904.5 |
| 30 | 67.9 | 916.3 |
| 40 | 55.0 | 919.0 |
Read the middle column alone and the advice is simple and wrong: the miss keeps shrinking, so use the biggest curve you can. The right-hand column tells the fuller story. Going from degree 6 to degree 12 cut the miss from 148.3 to 137.7, but the message grew from 949.2 to 958.5 bits. Those six extra numbers did not pay for themselves. Pushing on to degree 20 did pay: the curve finally had enough bends to follow the sharp features, and the total fell to 904.5 bits, the shortest of any degree from 0 to 40. After that, every added bend cost more than it saved.
The learner then checks the verdict a second way, with data the curve has not seen. Fit each curve using only the even ages and test it on the odd ages. Degree 20 again gives the smallest miss, 99.0. Degree 30 gives 32,689.1: between the points it was shown, the curve swings wildly. Two honest limits finish the write-up. Even the winning curve saves only about a quarter of the raw 1,200 bits, and it still misses age 19 by 295 people, because smooth curves are a poor language for a profile with sudden steps. And everything here describes the district as a whole, not the town alone.
Ages 8 to 11
Describe a dot picture to a friend in as few words as you can, then count the words and the mistakes.
Ages 11 to 15
Fit straight lines and gentle curves in Python and plot how the miss shrinks as bends are added.
Ages 15 and up
Code the two-part bill in bits, find the degree with the shortest message, and test it on held-back ages.
Source and method
Age counts are Census 2021 table TS007 for Horsham district from the Office for National Statistics via Nomis, read 30 September 2026, Open Government Licence. The people aged 100 and over are published as one open-ended group and were left out of the fit. The pricing of bits is a common textbook scheme; other fair schemes would give somewhat different totals.
