5
$\begingroup$

I am trying to read Prior Probabilities (1968), by Edwin T. Jaynes.

In two sections he discusses mollusk-like qualities [of parameter spaces]:

The real problem, therefore, must be stated rather differently [than that of Jeffreys and Bayes]; we suggest that the proper question to ask is: "For which choice of parameters does a given form such as that of Bayes or Jeffreys apply?" Our parameter spaces seems[sic] to have a mollusk-like quality that prevents us from answering this, unless we can find a new principle that gives them a property of "rigidity".

Stated in this way, we recognize that problems of this type have already appeared and have been solved in other branches of mathematics. In Riemannian geometry and general relativity theory, we allow arbitrary continuous coordinate transformations; yet the property of rigidity is maintained by the concept of the invariant line element, which enables us to make statements of the definite geometrical and physical meaning independently of the choice of coordinates.

What does mollusk-like quality actually mean in this context?

$\endgroup$
4
$\begingroup$

The reference is to Einstein's foundational work on relativity theory. He refers to a "mollusc" (non-rigid) space in this work.

For this reason non-rigid reference-bodies are used which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion. Clocks, for which the law of motion is any kind, however irregular, serve for the definition of time. We have to imagine each of these clocks fixed at a point on the non-rigid reference-body. These clocks satisfy only the one condition, that the “readings” which are observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount. This non-rigid reference-body, which might appropriately be termed a “reference-mollusk,” is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. That which gives the “mollusk” a certain comprehensibleness as compared with the Gauss co-ordinate system is the (really unqualified) formal retention of the separate existence of the space co-ordinate. Every point on the mollusk is treated as a space-point, and every material point which is at rest relatively to it as at rest, so long as the mollusk is considered as reference-body. The general principle of relativity requires that all these mollusks can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature; the laws themselves must be quite independent of the choice of mollusk.

The statement concerns transformation groups of continuous parameter(s) to obtain compactness and invariance. It is a topic of consideration in differential geometry. Jaynes is trying to formulate a hybrid between the considerations of a Bayesian statistician accommodating multiple disparate prior constraints (e.g. we have 90% believe $\theta > 0$ or $\theta$ has a unimodal distribution) and the Bayes' / Jeffreys' approach of having a "naive" / uniform possibly improper prior. His hybrid may be envisioned as being "mollusk" like in shape by having a complex space (impact of prior specifications) but a well-defined measurable compact property (preferences of Bayes and Jeffreys).

$\endgroup$
2
  • 3
    $\begingroup$ Although Jaynes' use of "mollusk-like" is undefined and opaque, I have a hard time applying this interpretation in the two places where the term is used. He first uses it in sharp contradistinction to "rigidity," suggesting he is thinking of a mollusk as a relatively shapeless and flabby animal rather than in terms of the shape of its shell. The next use is in a context about "definite geometrical and physical meaning," where "mollusk-like" once again appears to mean the opposite of such definiteness. $\endgroup$
    – whuber
    Dec 28 '15 at 21:31
  • 1
    $\begingroup$ @whuber I added a quote from a cited reference that I stumbled upon whence googling "mollusk" and Einstein geometry. I don't know how illuminating it will be. Both Einstein and Jaynes appeal to the "mollusc" as being a highly general space with some properties (differentiability) that permit some mathematical results to be obtainable (like linearization and local optimization, perhaps). So it's used in the sense of a tradeoff, and the two opposite sentiments are intended to convey as much to reader--I think. $\endgroup$
    – AdamO
    Dec 28 '15 at 21:49
1
$\begingroup$

I think Einstein's use of the mollusc was simply a counterpoint to the concept of a "rigid body". If you were to rapidly accelerate a titanium anvil (from behind, as in a push) it would physically shorten an infinitesimal amount in the direction of travel. A jellyfish, on the other hand, undergoing the same process would be considerably more distorted... .

$\endgroup$
2
  • 1
    $\begingroup$ A different example might be in order, because jellyfish are in a different phylum than molluscs: that is, the two are about as similar as we are to an insect. Regardless, could you explain how this might apply to the parameter spaces in the original question? $\endgroup$
    – whuber
    Jan 16 '20 at 20:04
  • 1
    $\begingroup$ @whuber My thoughts exactly. A jellyfish fits better than most molluscs though since many of the latter have shells, cuttlebones, pens - i think the only "molluscs" i can think of that are truly all soft bodied are the octopus and certain gastropods like seaslugs and nudibranchs. Einstein i think was simply leaning on the literal meaning of the word, which sits in German as a synonym for the native German word "Weichtier", simply soft animal, which is literally the idea he wants to get across. I don't think he was thinking hard biology! $\endgroup$ Sep 1 at 8:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.