Section 1: The two types of Fisher Information I,J
There is a decisive difference between the experiences of
(a) passively observing a lamp voltage of 120.0 v on a meter and
(b) becoming an active part of the electrical phenomenon by sticking your finger in the lamp socket
Ouch! Sorry, this follows the old textbook advice to better understand physics by “participating” in it, as in the usual cited example of kicking a wall to better understand the reactive force of mechanics. Actually, there are much better uses of the “participatory” idea in physics, as discussed below.
Curiously, science in general may be derived quite literally out of the above difference. This amounts to a distinction between what is observed in a message (the meter reading), and what is (the electrical current flow). This basic distinction was apparently first noted by Plato (see below).
The difference may be expressed on the level of information by the incredibly simple form
I – J = extremum. (1)
This is called the EPI principle (see below). The numbers I and J are the outputs of integrals that define values of the “Fisher information,” a concept invented by the statistician and geneticist R.A. Fisher in the early 1920’s. (In fact, this was the original “information”, predating by some 20 years C.E. Shannon’s currently more well-known information form.) Why do two Fisher informations arise in eq. (1), and why is their simple difference extremized?
The information that is acquired in a message does not generally arise out of nothing.
Any acquired information is usually about something. That something is generally called an information source, and it has information level J. That is, the information level J is required to completely describe it. The source is an effect like lamp voltage in the above. By comparison, its measured value in a message, observed as the above meter reading, conveys information level I to the observer. In general this cannot exceed level J since measurements are generally imperfect (see below). The meter constitutes an information sink. Together, the source and sink define an information measurement channel.
An effective flow of information
J → I (2)
from source to sink takes place during the process of measurement. The flow proceeds as in (2) not only because (usually) an effect precedes its measurement, but also because the flow physically consists of carriers of information that illuminate the source and proceed to the sink. This is illustrated by the following information channel.
Example: an optical microscope
Information source effect. Here the information source is the effect governing the position of the subject on a microscope stage. This source effect is also a probabilistic effect. That is, the subject’s position is a random number (called a random “sample”) from some probability law. The probability law defines the source effect under study, and the source effect contains information amount J (i.e., requires an amount J of information for its complete description). The probability law (the source effect) is unknown, and is to be found.
The flow in eq. (1) is provided by illuminating particles that carry information.
The subject is illuminated by photons. The photons bounce off the subject and proceed to the observer. Therefore, these are the carriers of information in the example, and create the information flow (2). The photons convey a message. This is an approximate value of the position of the subject. Since this datum or measurement is imperfect, it contains but a finite amount I of information about the true position value. Therefore, since J is the total information in the source effect, I must be some fraction of J (as in eq. (16) below).
The datum is also, as described above, a random sample from the probability law that describes the unknown source effect. In this way the photons provide data of information level I about a source effect of information level J. All well and good, but can the source effect actually be found from such considerations?
The information carriers also act as perturbers of the source effect.
That is, the photons also fill a second role of the communication channel. Since they illuminate and probe the source, they necessarily perturb it to some degree. This perturbation ultimately gives rise to the extremum principle (1) preceding, as shown below. In turn, the extremum principle, when solved, gives the physical effect at the source, i.e. the probability law that was sought. The exact manner by which this calculation is carried through is the subject of the books [1a-c] and other references cited below.
Hence, the probe particles of the channel fill a dual role. They both provide the observations and enact an extremum principle that defines the physical law that generates the observations. This describes a self perpetuating or autopoietic process. It also describes an Escher-like cycle that intimately connects the human observer with the source effect he/she wants knowledge of. In this respect it is a good example of Wheeler’s “participatory universe” as described below.
In summary, all uses of the EPI principle eq. (1) require, as in the preceding, the identifications of the (i) source effect, (ii) information flow route, (iii) information carriers, (iv) message, and (v) measurements of the channel.
An instructive second example arises in the analysis of in situ breast cancer (see prediction (8) below). Here
(i) the source effect is the cancer mass growth law expressed as an unknown probability; it holds information J about the true age t of the cancer; information J is “bound to” the effect (see (f)-(i) below);
(ii) the information flow J→ I is from a growing cancer mass to a neighboring healthy cell. The healthy cell receives information level I about the age t of the cancer.
(iii) the information carriers are here positive ions of lactic acid that are secreted by the cancer mass to the neighboring healthy cell. (Also, the clinician acts as a remote observer of the flow J→ I via the photons of a microscope.)
(iv) the message conveyed to the cell is “die” (since the ions act to poison it); the lactic acid ions perturb both the cancer mass and the neighboring cell, i.e. both the source and the receiver. (The clinician does not receive this message since he is decoupled from it due to system decoherence [1a,b] ); finally,
(v) the measurement is the time at which the healthy cell is observed by the clinician to receive the “die” message. This is a generally imperfect version of the true onset time of the cancer, and for this reason the acquired information I in the measurement is finite.
Information levels I and J are, as we saw, those of the data in the message (the “information sink”) and of the source. These are sums or integrals involving the unknown probability law defining the source effect. First consider information I.