The local uniformity principle

This is topic xxx on our philosophy topics list. It doesn't seem to have attracted much discussion, and indeed doesn't have a standard name. Quantitative scientists tend to regard it as obvious, whereas philosophers seem not to have noticed it as an issue. The 4 illustrations below illustrate how the principle is used; here's my attempt to articulate the principle itself. It refers to numerical data.

(i) Probability distributions on continuous variables (and on integers, if the spread is large) may be assumed to be "smooth", unless there is some specific reason why they may not be.
(ii) In the setting of (i), quantitative predictions for data based on the assumption that the data arises from independent random picks from some unknown "smooth" distribution are usually approximately correct, even when the ``independent random picks" part of the assumption is unrealistic.

As usual, let me distinguish the philosophical issue -- why do we think the assumptions in (i) or the predictions in (ii) are reasonable? -- from the empirical issue of when they work out to be give correct predictions.

Note that the usual math setup "builds in" assumption (i) for continuous variables as a default by working with density functions -- unless one is explicitly modeling e.g. a fractal -- but that doesn't address either issue.

4 illustrations of uses

Hitting a bull's eye in darts

A stereotypical textbook exercise is

A darts player aims at the center of the bull's eye (which has diameter 0.50in); the distributions of horizontal and of vertical deviations have mean 0 and s.d. 1.50in. Assuming these deviation have independent Normal distributions, what is the chance of hitting the bull's eye ?

The intended answer is "about 1.4%". The model is not very realistic (one expects smaller horizontal than vertical s.d., and the Normal distribution is doubtful) but setting aside the specific distributional model, the conceptual point is that the "local uniformity principle" says we may assume there is a smooth density function f, and then we estimate the desired chance in terms of f, to a first approximation as area(bull's eye) x f(0) where f(0) is the density at the bull's eye.

xxx coin tossing

seems intuitively reasonable but why?

Asteroid near-misses

99942 Apophis is a 350-meter long asteroid which is confidently predicted to pass Earth just below the altitude (35,000 km) of geosynchronous satellites (which provide your satellite TV) on Friday, April 13, 2029.

There are several ways of estimating the chances of an asteroid collision with Earth. The actual frequency of different sized asteroids at different points in the solar system is of course an empirical issue. But because the (quarter million mile) distance to the Moon is small relative to the solar system, the "local uniformity principle" says we may assume that the chance of an asteroid being at one point near the Earth's orbit is not substantially different from the chance of it being at a different point a quarter million miles away. Then mathematics, and the empirical fact that the ratio (radius of Moon orbit)/(radius of Earth) is approximately 60, implies
Amongst asteroids which pass closer to earth than the Moon's orbit, about one in 3,600 will hit Earth.
(the 3600 arises as 60^2). Hypothetically, if astronomers could and did detect all asteroids of diameter greater than 50 meters passing within the Moon's orbit for a period of years, and found there were on average 3.6 per year, then one could infer that such an asteroid would hit the Earth about once every 1,000 years on average. (Note: this conclusion is a typical estimate in the literature but was not obtained in this way.)

xxx exam scores

xxx last digit of travel expenses 100-1000 dollars

xxx Benford's law

xxx empirical: diff sources mix -- not The Secret Life of Numbers