# [time 326] Re: [time 325] Re: Fisher information and relativity

Tue, 18 May 1999 23:33:55 +0900

Dear Matti,

Your question is meaningful. Indeed it cuts the seemingly continuous
argument of Frieden as I will explain below.

----- Original Message -----
From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
Sent: Tuesday, May 18, 1999 5:48 PM
Subject: [time 325] Re: Fisher information and relativity

[snip]

> > Is Fisher information
> > > still in question when one uses imaginary coordinate x0 =it?
> > >
>
> Coordinates correspond to kind of parameters in Fisher
> information: unfortunately I have not clear picture
> about what kind of parameters are in question.

Parameters are coordinates in the book at least as I read till now. Other
examples may be in the book.

What troubled and
> still troubles me is whether the imaginary
> value of parameter is indeed consistent with this
> interpretation.

You seem to point out the gap in Frieden's development of the theory.

Frieden writes in page 64 in section 3.1.2 entitled "On covariance":

[beginning of quotation]
... By definition of a conditional probability p(x|t)=p(x,t)/p(t)
(Frieden, 1991). This implies that the corresponding amplitudes (cf. the
second Eq. (2.18)) obey q(x|t)= + or - q(x,t)/q(t). The numerator treats x
and t covariantly, but the denominator, in only depending upon t, does not.
Thus, principle (3.1) is not covariant. [HK: (3.1) reads: \delta
I[q(x|t)]=0, q(x|t) = (q_1(x|t), ... , q_N(x|t).]

From a statistical point of view, principle (3.1) is objectionable as
well, because it treats time as a deterministic, or known, coordinate while
treating space as random. Why should time be a priori known any more
accurate than space?

These problems can be remedied if we simply make (3.1) covariant. This
may readily be done, by replacing it with the more general principle

\delta I[q(x)]=0, q(x)=(q_1(x), ... , q_N(x)). (3.2)

Here I is given by Eq. (2.19) and the q_n(x) are to be varied. Coordinates x
are, now, any four-vector of coordinates. In the particular case of
space-time coordinates, x now includes the time.
[end of quotation]

Here Frieden transforms the Euclidean coordinates to the coordinates
possibly covariant wrt Lorentz or any other coordinates transformations.

By this transformation of his theory, he misses the I-theorem, which reads
till he introduces the covariant coordinates:

dI
---- (t) < or = 0 for any t.
dt

This has been assuring that the information I decreases as t increases.
Hence I takes a minimum value as t goes to infinity (since I > or = 0), and
this fact has been ensuring the validness of taking the solution of the
variational problem (3.1) as the physical reality:

\delta I[q(x|t)] = 0. (3.1)

Just when he introduces the covariant coordinates and hence pure imaginary
time, this I-theorem breaks down and he loses the foundation upon which the
validity of variational principle has been relying.

He then instead postulates the variational principle as one of his three
axioms for "the measurement process" in pages 70-72. (In fact there is no
quotation of I-theorem after page 63 till chapter 12 in page 273 entitled
"Summing up" according to the index.)

This means that the introductory part till page 63 is just an illustration
which leads to the introduction of his axioms 1 to 3 in pp. 70-72, not a
justification of the axioms in any sense.

And his axiom 1:

\delta (I - J) =0,

with axiom 2:

I=4 \int dx \sum_n \nabla q_n \cdot \nabla q_n

and

J= \int dx \sum_n j_n(x),

(here n varies from 1 to N, N denoting the number of independent
measurements done.)

is almost the same requirement as the usual variational principle which
gives Lagrangian of the system under consideration.

Thus his contribution is just that the free energy part I is given as above
in his axiom 2. That the form of Fisher information I gives the free energy
part of Euler-Lagrange equation may be a progress of human knowledge. This
is but a small calculation which was described in [time 321], and does not
seem to need a hard covered book.

Frieden's purpose might be in his philosophy. However, he abandons himself
his philosophy (i.e. I-theorem) as you pointed out:

> whether the imaginary
> value of parameter is indeed consistent with this
> interpretation.

The imaginary value of parameters is not consistent with Frieden's own
philosophy, I-theorem. So he just assumes the principle of the least action
as axiom 1 in his derivation of Lagrangian. Here is no new thing except for
an observation that the free energy part follows from the form of the Fisher
information.

Another point which shows the shallowness of his theory is that he does not
give any consideration about time. As in the above quotation, he thinks at
first that time is given. Then he comments that time should be considered an
inaccurate unknown parameter as other space coordinates, and turns to time
as a component of the covariant coordinates. This is a too easy way for one
to construct a unification of physics.

In conclusion, Frieden's theory looks like but a repetition of the principle
of the least action except for the discovery of the relation between Fisher
information and the free energy.

Best wishes,
Hitoshi

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