Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

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

Fitting multiple exponential decays for the case with unknown decay rates and unknown initial number of atoms is a notoriously tricky numerical problem because it’s easy to get good fits with very different parameter values. (Acton’s classic book “Numerical Methods that Usually Work” discusses the issue.)

A Bayesian approach can help in this case if you happen to have additional information on top of the counts. If you do, you can build this information into the prior and maybe rule out lots of the posterior parameter space.

My Matlab example was for the case of a single exponential decay. If the experimental data includes two decaying isotopes, my code can be extended to produce the posterior probability for 4 parameters. The numerical integrations over a 4d mesh of points will get quite a bit more costly in terms of computer time, but it should still take less than a few minutes on a modern laptop. I’m pretty sure though that the posterior probability will be quite broad and produce large error bars for the estimated decay constants unless extra info is built into the posterior. But that’s not be a problem with the estimation method. It’s a feature of the problem.

Sent from my iPhone

On Sep 22, 2021, at 8:29 AM, Brian Whatcott <betwys1@sbcglobal.net> wrote:

 Francois' Take 2 is not playing happily with the other children at present. I may have erred in the transcription.
On Wednesday, September 22, 2021, 10:00:15 AM CDT, Brian Whatcott <betwys1@sbcglobal.net> wrote:

Like Francois, I played with a MATLAB code for exponential decay. Galloping off rapidly in all directions, I generated two exponential decay series with noise, to address the issue of extracting data for two decay species, like this.....[spoiler: extracting multiple decay parameters from observations is an on going topic in the literature]>> >> x = (0:0.2:5)';y = 2*exp(-0.2*x) + 0.01*randn(size(x));y2 = 3*exp(-0.3*x) + 0.01*randn(size(x));
f = fit(x,y,'exp1')plot(f,x,y)
f =
General model Exp1: f(x) = a*exp(b*x) Coefficients (with 95% confidence bounds): a = 2.06 (1.941, 2.179) b = -0.1894 (-0.2159, -0.1629)

f2 = fit(x,y2,'exp2')plot(f2,x,y2)
f2 =
General model Exp2: f2(x) = a*exp(b*x) + c*exp(d*x) Coefficients (with 95% confidence bounds): a = -0.197 (-7.766e+04, 7.766e+04) b = -0.2926 (-1567, 1566) c = 3.203 (-7.766e+04, 7.767e+04) d = -0.3006 (-98.32, 97.72)>>
[Conclusion: stop wasting time!]
Next I plotted Francois' 'nice-shaped posterior' (forgive the mental image) whichI show here:https://imgur.com/gallery/TpP4ZdY

...to confirm his code for a noiseless error-free data set.



On Tuesday, September 21, 2021, 04:24:44 PM CDT, 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.
_______________________________________________
Forum for Physics Educators
Phys-l@mail.phys-l.org
https://www.phys-l.org/mailman/listinfo/phys-l

_______________________________________________
Forum for Physics Educators
Phys-l@mail.phys-l.org
https://www.phys-l.org/mailman/listinfo/phys-l

_______________________________________________
Forum for Physics Educators
Phys-l@mail.phys-l.org
https://www.phys-l.org/mailman/listinfo/phys-l