From the p-value debate at Andrew Gelman's blog:

For every measurement of the macroscopic universe that will ever be performed by humans, that measurement will have some resolving power. Let’s pretend that the highest resolving power will come from an electronic A/D converter of some measurement instrument yet to be devised, and that it has 256 bits of resolving power. Today, very high quality A/D converters have maybe 31 bits

So under this hypothesis that the best ever measurement instrument will have 256 bits of resolving power, any scientific hypothesis involving sample spaces larger than 2^256 different possible finite outcomes is not a testable scientific hypothesis. PERIOD.

Now, let’s examine some physical reality of the universe: according to Wikipedia current approximate calculations give the number of protons/electrons in the universe as around 10^80

This means to have my hypothetical 256 bit A/D converter we would have to accurately count all the electrons in approximately 1/1000 of the entire universe. I assert that this will never occur, so every probability sample space on scientific measurements has less than 2^256 distinct discrete possible outcomes, each distinct outcome has a perfectly ordinary probability associated to it.

Continuous probability distributions are purely convenience for not having to work with an exactly known quantity of discrete outcomes, and not having to carry around sums that contain 2^15 terms and soforth.

Perhaps I'm wrong, perhaps some day we'll have a measurement which is accurate to 1 part in 2^257, the argument still survives. The exact number doesn't matter, its some big finite number. This is exactly why I love the nonstandard analysis of Edward Nelson (IST) because it's all about assigning a predicate to "bigger than we'll ever actually define, but still finite". Integrals are big finite sums, very big, extremely big, bigger than you'll ever actually define if you write towers of $10^{10^{10^{...}}}$ for the rest of the existence of the human race. It's big enough eventually.

3 Responses leave one →
1. March 16, 2013

Daniel,

I read your stuff all the time, but seemingly never get around to commenting. We have somewhat similar backgrounds, which differ from normal stat folks (me: separate master's degrees in Physics, pure Math, and Statistics).

You're exactly right about the inherently discrete nature of statistics. Although I had a long and intimate acquaintance with measure theory before statistics, I think E. T. Jaynes assessment was right:

(1) Measure theory isn't a more rigorous approach to statistics, it's just more general.
(2) The increased generality is completely irrelevant for applications.

Like many coming to statistics from a pure math background, I thought measure theory was the royal road to statistical insight. The exact opposite was true. I have yet to encounter a single instance in which measure theory even came close to solving any foundational problems with the philosophy of statistics.

-joseph

P.S. I spent some time back in the day on Non Standard Analysis. I'll get around to adding my \$.02 worth at some point. It's a really obscure topic to most people.

P.P.S. In your Continuum Mechanics related work, do you ever run across the name Clifford Truesdell? I'm curious to know if his influence, such as it was, is completely lost.

• March 16, 2013

Hi Joseph,