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Re: [Phys-L] Bayesian Inference in Half-Life measurement



John,

The experiment:
Two copper isotopes occur naturally 63Cu and 65Cu. We irradiate a sample
of copper with thermal neutrons producing some number or 64Cu and 66Cu
atoms. These have half lives of 12.7 hours and 5.12 minutes. After
removing the copper from the neutron bath we measure the activity of the
sample with a simple geiger counter. Within the undergraduate lab period
we can take data for most of the short half life decays. Students have to
come back every day to take another data set each of the next few days to
get data to find the longer half life.

Traditionally we have them fit a double exponential function to this data.
But this is fraught with problems and subtle choices in the analysis which
will change the result. And it's not how these half lives are really
measured. Though, obviously in a precision measurement you'd want to have
continuous data collection.

Francois suggested an interesting monte carlo method. I'll have to take a
look at that.

Paul

On Tue, Sep 21, 2021 at 4:24 PM John Denker via Phys-l <
phys-l@mail.phys-l.org> wrote:

On 9/21/21 10:28 AM, Paul Nord wrote:

That title doesn't google well.

Agreed.

1) I reckon it's likely that you are generally on the right
track. This is something a lot of textbooks get wrong.

2) It would help to spell out in plain English the background
of what you are doing, and the objective.

3a) If I had to guess, I'd say there's an experiment to measure
the half life by counting decays, and it's tricky because:
— The number of decays in any reasonable interval is small.
— The observed numbers are subject large-percentage fluctuations
due to Poisson statistics, even though the underlying physics
is not changing.
— Ordinary "textbook style" least-squares curve fitting to
extract the rate fails miserably. That's because "least
squares" usually means maximum likelihood, which is jargon
for maximum a priori, but any sane person would want maximum
a posterori.

If so, you could try googling this:

https://www.google.com/search?q=%22maximum+a+posteriori%22+%22fitting%22+%22poisson%22+data

3b) But I'd rather not guess. Please clarify.
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