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On Jul 13, 2013, at 5:36 PM, Ian Tickle <[log in to unmask]> wrote:

> On 8 July 2013 18:29, Douglas Theobald <[log in to unmask]> wrote:
> 
> > > Photons only have a Poisson distribution when you can count them:
> > > QM says it meaningless to talk about something you can't observe.  
> >
> > Aw, come on --- QM is a theory, it says no such thing.  The claim
> > that "it's meaningless to talk about something you can't observe" is
> > a philosophical principle, not science.  There are many
> > interpretations of QM, some involving hidden variables, which are
> > precisely things that exist that you can't observe.  Heck, I'd say
> > all of science is *exactly* about the existence of things that we
> > only infer and cannot observe directly.  Remember, when you get the
> > readout from a detector, you are not directly observing photons even
> > then --- you are formally inferring things that you can't observe.
> > There's a whole chain of theory and interpretation that gets you
> > from the electronic readout to the claim that the readout actually
> > represents some number of photons.
> 
> In science we make a clear separation of what is 'data' from what is
> 'model' (or 'inference').  

Often we do, but it is not all that clear cut.  Especially in Bayesian
analyses, "data" and inference are often interchangeable. One man's
inference is another man's data.  As a very pertinent example, the true
intensities are inference, but in terms of the Wilson distribution they
are data (p(J|sdw)).  

> Of course one can be pedantic and argue that everything is really
> inference since our brains interpret everything that goes on 'out
> there' by means of inference from its sensory inputs. Obviously I
> don't take seriously the premise of the 'Matrix' movies (excellent
> though they may be!) that such inputs are just a simulation!  

You can accept that everything is inference without slipping down the
slope to the invalid conclusion that the Matrix is true.  

> At some point you have to believe what your eyes are telling you, as
> long as there is a clear chain of believable cause-and-effect between
> the observation and the inference of that observation.  However, we
> are rightly suspicious of any model that is not supported by data (in
> fact inference in science requires data). Note that I always mean
> 'observation' to be synonymous with 'measurement' (as in 'Fobs'), not
> 'observation' in the weaker sense of 'seeing'.

So you agree that all scientific measurements/observations, aside from
trivially "seeing" something, are indirect.  My argument is that you can
in fact measure (or estimate) the background under the intensity using
the model:

Is = Ib' + Ij

Ib

where we experimentally measure Is (the spot) and Ib (the background
around the spot), and we assume that both Ib' and Ib come from the same
(Poisson) distribution with mean B, and that Ij (a sample from the true
spot intensity J) comes from a Poisson as well.

Given that model, you can actually measure Ib' and Ij.  It's an indirect
measurement, but all measurements/observations are.  Here there's really 
no conceptual difference from how we measure photons with a detector:
we assume some model for how photons interact with our detector and we
behave as if we are measuring photons with the output from the detector
(in your words, the model is a "clear chain of believable
cause-and-effect"). All (non-seeing) measurements are ultimately
indirect and model-based, a form of inference.  Your claim that we can't
measure Ib' or Ij is based on the your rejection of the above model ---
but it is a circular argument to reject that model by saying we can't
measure Ib' or Ij.

Bringing in QM is of no help, since QM is consistent with (non-local)
hidden variable interpretations where photons are in fact
distinguishable and exist as real things with definite physical
properties before we measure them.  

> > Again, this is your own personal philosophical interpretation of QM
> > --- QM itself says nothing of the sort.  For instance, Bohm's pilot
> > wave interpretation of QM, which is completely consistent with
> > observation and QM theory and calculation, states that individual
> > photons *do* go through one slit or the other.  But this is really
> > off point here, I think --- as I said, I don't want to get into a QM
> > debate.
>
> Neither do I, I would just observe that under no QM interpretation can
> you determine which slit any individual photon went through and I
> would argue that it's therefore not even a meaningful question to ask
> which slit it went through.

Again, that is one of many possible QM philosophies.  I think, however,
that if we are to ever find a better theory than QM, then we must ask
questions like "which slit did the photon go through".  Otherwise we
have a science-stopper. Just because QM, in its present form, can't tell
us which slit a photon went through does not logically imply that the
photon did not go through a slit.  

> > I disagree.  Following that logic, we could not talk about the error
> > in our estimate of the gravitational force on the Earth from the
> > Moon (because, if our theories of gravity are correct, the force
> > exerted on the Earth is the sum of the gravitational pull of all
> > massive objects in the universe, and it is physically impossible for
> > us to, say, remove the Sun and then independently measure the force
> > exerted by the Moon).
> 
> The estimate of the gravitational force on the Earth from the Moon is
> an inference, not an observation: it relies on Newton's theory of
> gravitation being correct. Of course I'm not denying that you can talk
> about the error estimate for an inference, but I was talking about
> errors in observations (= measurements).  

As I explained above, *all* observations/measurements rely on some model
or theory being correct, and all observations/measurements are indirect
in this sense.  There really is no practical difference between
inference and observation.  All non-trivial observations/measurements
rely on inference.  

> It's physically impossible to move the Sun, but not logically
> impossible (i.e. we can imagine it happening).  In contrast separating
> Ispot into its component Itrue & Iback is a logical impossibility; we
> can't even imagine it without allowing a logical contradiction: since
> photons are indistinguishable the same photons in Ispot have to be
> blocked and not blocked at the same time.  

There is no logical contradiction in imagining that photons are
distinguishable --- the idea that photons are (irreducibly)
indistinguishable is based on a particular interpretation of one
particular physical theory, not logic. If history is any guide, all of
our present theories are wrong.  There very well may be a better theory
than QM in which we can label individual photons.  How is that illogical?

In any case, Bohm's interpretation of QM is observationally equivalent
with QM theory, and in Bohm's interpretation, photons have identity
(i.e., they are identical in the classical sense, but not irreducibly
indistinguishable).  Since Bohm's interpretation is physically possible,
then it is also logically possible.  

As an aside, I note that physicists (and the rest of us) don't really
act like photons are indistinguishable.  We all talk very comfortably
about the photons hitting my eyes from my LCD here in Waltham and those
other photons hitting the ADSC detector at the SIBYLS beamline in
Berkeley.  

> I contend that if a 'measurement' is a logical impossibility (but not
> necessarily a physical one) then it's meaningless to talk about an
> error distribution for such a 'measurement'.
> 
> > My point is, however, that F&W implicitly assumes that the
> > background we measure (in Ib) comes from the same process as the
> > background under the spot (Ib').  In other words, the underlying
> > model is:
> >
> > Is = Ib' + Ij
> >
> > Ib
> >
> > where we experimentally measure Is (the spot) and Ib (the background
> > around the spot), and we assume that both Ib' and Ib come from the
> > same (Poisson) distribution, and that Ij (a sample from the true
> > spot intensity J) comes from a Poisson as well.

> >
> > I agree that Is-Ib is approximately Gaussian for reasonably large Is
> > and Ib. The problem is P(Is-Ib | J,sdJ).  Why is the mean of this
> > Gaussian J?  Why not J^2, or J/4?  How do you derive the Gaussian
> > P(Is-Ib |J,sdJ)?  The claim that the mean of Is-Ib is J requires
> > some sort of justification --- otherwise I might as well claim that
> > the mean of Is-Ib is the number of quarters in my pocket.
> > 
> > We both agreed that Is and Ib are dependent, and so that necessarily
> > means that P(Is-Ib | J,sdJ) cannot be the whole story.  Dependence
> > means that the distribution of Is-Ib will also be a function of B
> > and sdB, unless you've integrated them out somehow (and even then
> > the form of the marginalized distribution depends on B and sdB).
> > 
> > F&W assumes that P(Ij|J,sdJ) is Poisson.  If you think F&W doesn't,
> > then explain how you get the Gaussian P(Is-Ib | J,sdJ).  Again, why
> > is the mean of Is-Ib equal to J?  I gave one derivation, based on
> > the model that P(Ij|J), P(Ib|B), and P(Ib'|B) are all Poisson.  If
> > you disagree, then what is your derivation of the Gaussian P(Is-Ib |
> > J,sdJ)?
> 
> OK let's go through it step-by-step and you tell me where I went
> wrong. Note that I do not assume a prior P(Ij|J,sdJ), in particular I
> do not assume that it's Poisson, or even that Ij or P(Ij|J,sdJ) exist:
> I contend that Ij is certainly not observable.
> 
> 1. The true values of the Bragg & background components of a spot are
> J and B, so the true total coherent scattering is S = J + B.
> 
> [Note: true values are unknown constants, which obviously do not have
> uncertainties, i.e. if we were doing a simulation we might assume say
> J = 10, B = 20; there's no assumption of P(Ij) here.  Of course
> estimates of true values do have uncertainties, but the Wilson prior
> P(J), the posterior P(J|Is,Ib) and hence <J> and var(<J>) only come in
> later after we invoke Bayes' theorem in F & W.]

So I find this a bit odd, as it seems to me we are saying the same
thing, but you evidently see a difference.  You accept that S=J+B is
a valid model, which is very similar to the model I've been proposing
(in fact to me it appears identical, except that you deny a distribution
for Ij).  But you also claim that we can never measure J or B (where B is
by definition the background under the true intensity that combines to
form S). Yet you say that "it's meaningless to talk about something you
can't observe".  What gives?  This seems contradictory to me --- where
am I misunderstanding you?

A few words about Bayesian vs frequentist interpretations of
probability.  From a Bayesian POV, true values are in general not
constants and they have uncertainties.  (A constant is simply a random
variable that has such small uncertainty that its distribution is a
delta function with mean equal to the true value.) The Bayesian view of
probability holds that uncertainties are only in our minds;
uncertainties (or probabilities) are not properties of physical objects.
Thus, from a Bayesian POV, it is completely natural (and necessary) to
assign a probability distribution to any unknown value, like B and J. In
frequentism it might not make any sense, but here we are discussing a
Bayesian method (F&W), not a frequentist one.  Perhaps this is actually
the root of our disagreement. 

> 2. We make a measurement of S and get Is.  If we made multiple
> measurements we would expect to see (say) a Poisson distribution, in
> which case for a single measurement we can estimate var(S) = S ~ Is.

OK.

> 3. We also estimate B from the adjacent background Ib and similarly we
> estimate var(B) = B ~ Ib.

OK.

> 4. So now we have estimates of S (~ Is) and B (~ Ib), so therefore we
> have an estimate of J = S - B (~ Is - Ib).

Now, you agree that Is is Poisson with mean S, and that we can measure
Is.  You also agree that Ib is Poisson with mean B, and that we can
measure Ib.  You also assume that S = J + B.  Yet, it is a well-known
statistical fact that a Poisson with mean S=J+B is equivalent to the sum
of two Poissons, with means J and B respectively.  You can easily prove
this using moment generating functions.  So your model here and my model
are mathematically equivalent.  The existence of P(Ij|J) and Ij is 
mathematically implied by your assumptions, and Ij is necessarily
Poisson.  

> 5. We have established that the difference of 2 Poissons with means S
> and B is well-approximated by a Gaussian with mean (S-B) = J and
> variance (S+B).

Not so fast.  The difference of two Poissons is a Skellam distribution.
If the means of the two Poissons are S and B, then the Skellam has two
parameters S and B.  When S and B are equal, the Skellam distribution
very quickly approaches a Gaussian as S and B get larger.  This is
basically equivalent to the result that Randy got with his Mathematica
plots. However, when S and B are unequal, the Skellam is always skewed
(if S>B, right skewed, if S<B, left skewed).  If S and B are very
different (say one is 5 times the other), then the approach to a
Gaussian is much slower.  So now that I've talked this out, I think 
I've pinpointed where the Gaussian approximation comes in with F&W 
--- it's not with the Poissons, as I thought, but with the Skellam.  

And now I've gone back and reread F&W 1978 more carefully; they 
apparently make this very argument and cite Skellam:

"Firstly, whilst we accept the data are certainly not exactly normally
distributed, we do contend that the normal distribution is an adequate
approximation for our purposes. From a theoretical point of view we are
encouraged in this belief, since the merged intensity I is made up of
sums of differences of theoretically Poisson-distributed counts. Such
operations on Poisson variables reduce them to normality quickly (Irwin,
1937; Skellam, 1946)." p. 518.

(Note that what they call the "merged intensity I" is what we are
calling Is-Ib.)

> 6. So P(Is-Ib,J) is Gaussian with mean J, and var(J) = (S+B) ~ (Is+Ib).
> 
> I don't see what's wrong with that.

So I don't really see anything wrong with that, except that you missed
the fact that your derivation is mathematically identical to mine, since
your assumptions necessarily imply Poisson P(Ij|J).  

Cheers,

Douglas