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

[John Denker <jsd@av8n.com>]

On 9/22/21 10:13 AM, Paul Nord wrote:

> 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.

In the immortal words of Smith & Dale:
	−− Doctor, Doctor, it hurts when I do "this".
	++ So don't do that.

By that I mean:

We agree that "simply subtracting the background will sometimes
result in a negative number of counts".

However, please consider more closely my previous email.
Add a fifth parameter to the *model*.
This will add something to λ on the model side of the equation,
rather than subtracting something from the opposite (data) side
of the equation.

============

This is one of the eleventeen ways in which your intuition may be
clouded by your experience with Gaussians.  Gaussians are gauge
invariant. That is, you can shift a Gaussian left or right and
it's still a Gaussian. This is not true for a Poisson distribution,
unless the number of counts in every observation is huge.

And I say again, if the number of counts was huge you wouldn't
be asking the question. The subtraction would never go negative.

But that's a minor point. Adding to the λ on the model side of
the equation always works. It is conceptually correct, whether
the number of counts is huge or not.

=====

Related to the above, and possibly important here and elsewhere,
for multiple reasons:

Do not imagine there are error bars attached to the raw data
points. For a discussion of this, and an unforgettable diagram,
see here:
  https://www.av8n.com/physics/uncertainty.htm#sec-raw-cooked

In particular, a bin with zero counts would be interpreted by
the oh-so-typical brain-dead fitting routine as having a mean
of zero and (!) a standard deviation of zero, as if the λ itself
were zero for that bin.

In reality, the λ is not zero. The probability contours come
from the nonzero λ, not from the zero observation.

*** The uncertainties belong to the model, and to the dataset
as a whole ... not to any individual raw data point.