The Search for a Search - Measuring the Information Cost of Higher Level Search/Critique of the Draft

The following is a critique of a previous draft of the paper. Some of the problems mentioned here were rectified, others seems to be still there...

Section IV The Search for a Good Search
Just a short summary of a further problem: In this section, a probabilistic hierarchy on top of $$\Omega\,$$ is introduced: $$\mathbf{M}(\Omega)$$ is the collection of all probability measures on $$\Omega\,$$. It is claimed (and hopefully proven) that $$\mathbf{M}(\Omega)$$ is itself a compact metric space with Borel sets, so the process can be reiterated to get $$\mathbf{M}(\mathbf{M}(\Omega))$$ and so on. Define for convenience $$\mathbf{M}^0 := \Omega$$, $$\mathbf{M}^1(\Omega) := \mathbf{M}(\Omega)$$, $$\mathbf{M}^{n+1}(\Omega):= \mathbf{M}(\mathbf{M}^{n}(\Omega))$$, and voilà, you have quite something!

Now, an important integral is introduced to link the different spaces $$\mathbf{M}^{n}:=\mathbf{M}^{n}(\Omega)$$: For an arbitrary probability measure $$\Theta_{k+1}\,$$ on $$\mathbf{M}^{n}$$ (and therefore $$\Theta_{k+1}\in\mathbf{M}^{n+1}\,$$), define

$$\Theta_k = \int_{M^k}\mu d\Theta_{k+1}(\mu)$$.

This is a vector-valued integration and no less than three references are given for it:


 * Nicolae Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces, New York: Wiley, (2000)
 * I. M. Gelfand, Sur un Lemme de la Theorie des Espaces Lineaires, Comm. Inst. Sci. Math. de Kharko. 13(4) (1936): 35–40.
 * B. J. Pettis, On Integration in Vector Spaces, Transactions of the American Mathematical Society 44 (1938): 277–304.

So, we're talking about Bochner integrals or even Pettis-Gelfand integrals. Very interesting stuff! Only one problem: Though Marks and Dembski undertook some pain to get the space right over which they are integrating, they have a problem with the function in this integral: The Bochner (and Pettis-Gelfand integral) $$\int f d\mu$$ is defined for a function $$f: X \rightarrow B$$, where $$X\,$$ is a measure-space and $$B\,$$ is a Banach space, i.e., a complete normed vector space.

And though this works out for $$\mathbf{M}^{n}, n > 1$$, you get a problem for $$\mathbf{M}^{0}$$:

Example: Let $$\Omega\,$$ be a deck of cards. $$\Omega\,$$ can be seen as a discrete metric space. Let's try to calculate $$\Theta_0\,$$ for $$\Theta_1\in\mathbf{M}^1(\Omega)\,$$ being the uniform probability measure. Then

$$\int_{\mathbf{M}^0(\Omega)} \mu d\Theta_1(\mu)$$ $$=\sum_{\mu \in \Omega} \mu \Theta_1(\mu) $$ $$=\frac{1}{52}(\heartsuit 2 + \heartsuit 3 + \ldots \heartsuit K + \heartsuit A $$$$+ \diamondsuit 2+ \diamondsuit 3+ \ldots \diamondsuit K+ \diamondsuit A$$$$+ \spadesuit 2+ \spadesuit 3 + \ldots \spadesuit K+ \spadesuit A$$$$+ \clubsuit 2+ \clubsuit 3+ \ldots \clubsuit K+ \clubsuit A)$$

Conclusion
When W. Dembski and R Marks state their Horizontal No Free Lunch theorem, they talk about an exhaustive partition of &Omega; all of whose partition elements have positive probabilities. You can build a partition of &Omega; of elements of the form $$\Omega^m \backslash (\Omega \backslash T)^m \,$$ only if $$m=1\;\,$$.

In this case, the whole result is trivial: If you are allowed only one shot onto an unknown target, just shoot at random!

But for bigger m, i.e., searches as they are generally understood (and described in their paper as assisted searches), the theorem falls flat. And with it, the rest of the paper.