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

[Paul Nord <Paul.Nord@valpo.edu>]

There is a 5th parameter, the background.  We've got heavy shielding, but
cosmic rays and other things do get through.
It's easy enough to take a long measurement of the background and come up
with a count rate.  But then at some point simply subtracting the
background will sometimes result in a negative number of counts.

We're not really hitting the limits of Poisson statistics with most data
points.  But all of the traditional analysis methods that a physics major
would tend to use require some choices in binning the data, what to do with
zeros, etc.  Many early measurements of half-life suffer from these
problems.  It's almost impossible to compare such numbers to modern
measurements.

Perhaps this is more of a graduate exercise.  But I was curious.  I'd like
to find a "here's the probability of the answer being x given the data"
that does not depend on choices made by the experimenter.

Paul




On Wed, Sep 22, 2021 at 11:21 AM John Denker via Phys-l <
phys-l@mail.phys-l.org> wrote:

> On 9/22/21 8:14 AM, Paul Nord wrote:
>
> > 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.
>
> I assume the number of counts in any given interval is not large.
> Otherwise you wouldn't be asking the question. Generic least-squares
> fitting would do the job. Crude but effective.
>
> Core problem #1 is that "least squares" routines assume the statistics
> are Gaussian. The logarithm of a Gaussian is a quadratic. Adding the
> quadratics is isomorphic to multiplying the probabilities to find the
> joint probability. Hence least squares.
>
> That's a problem because when the counts are small, a Poisson distribution
> doesn't resemble a Gaussian. Large counts yes, small counts no.
>
> Core problem #2 is the time dependence. You can find a squillion youtube
> videos on how to find *the* rate of a Poisson process when the rate isn't
> changing.
>
> ===========
>
> As a warmup exercise, let's temporarily switch from analysis of data to
> synthesis of data. That, suppose we wanted to simulate the experiment.
> Given the four parameters N1(initial), N2(initial), τ1, and τ2, you
> can figure out the Poisson rate λ at any given time. Then for any given
> interval near that time, you can synthesize an appropriate number of
> events for that interval.
>
> So that's the model. Now the task is to adjust the parameters of the
> model to fit the real observed data.
>
> Given a set of parameters, then for each observation it is straightforward
> to calculate the probability that the model, using those parameters, would
> have emitted the observed event-count.  Take the log of the probability.
> Sum over all observations. That's the objective function. Search the
> parameter space to optimize the objective function.
>
> That's it. That's a decent way to formalize the problem.
>
> Now it's "just" a four-dimensional search problem. Pain in the neck.
> If necessary you can stabilize it by first doing a 2D fit to estimate
> the fast exponential, then a 2D fit for the slow one, then a joint 4D
> fit.
>
> I realize this does not answer the question, insofar as it is not
> particularly Bayesian. It is still maximum_a_priori (boo!) i.e. the
> conditional probability of the data given the parameters ... when
> what you really want is maximum_a_posteriori i.e. the probability
> of the parameters given the data.
>
> You could fix that given a Bayesian prior on the parameters, but at
> the moment I have no clue what the prior should be.
>
> In the curve fitting business the priors often take the form of
> "regularizers" but it's not obvious what regularizers to use. I
> suppose you could constrain the search to keep the lifetimes within
> (say) a factor of 2 of the book value......
>
> Given a decent amount of /total/ data the priors shouldn't matter
> too much. The total number of events can be large even if the number
> of events in any particular observation is not.
>
> If the total number of events is tiny then a reliable analysis may
> be simply impossible.
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