In their article

*The Search for a Search*^{1} the authors William Dembski and Robert Marks seem to assume that any search on a space induces a probability measure on the search space, as they write in section 3.1 (

*Active Information*):

"

*Let U denote a uniform distribution on Ω characteristic of an unassisted search and φ the (nonuniform) measure of Ω for an assisted search. Let ***U**(**T**) and φ(**T**) denote the probability over the target set **T** ∈_{[sic]}Ω."

Such an induced measure gives the probability to find a subset

**T** of Ω when performing the corresponding search. But for the assisted searches mentioned in the text (a

*"perfect search"* -

*"indicating an assisted search guaranteed to succeed"* - and another search

*"indicating an assisted search guaranteed to fail"*), such probability measures obviously do not exist! Thus these cases are not covered by their

**Theorem 1** (

*Horizontal No Free Lunch*), as the

*Kullback-Leibler-distance* is defined only for probability measures.

But perhaps the authors intended their theorem only to be valid for those searches which induce a measure on the search space - indeed, they use the term

*uninformed assisted searches* when describing their results. Unfortunately, this term is not defined - neither in the current text, nor in their previous publications

*Conservation of Information in Search - Measuring the Cost of Success*^{2} or

*Efficient Per Query Information Extraction from a Hamming Oracle*^{3}.

Unfortunately, even in this case, the theorem does not work: assuming the easiest nontrivial case of guessing an element of a set with two elements (tossing a coin), Ω = {head, tail}, the two strategies

*naming an element at random* (uniform distribution on

$\backslash Omega\backslash ,$) and

*sticking with one element* (dirac distribution) give the same success rate for a fair coin. The same result occurs when averaging over the probability of showing

*head* (assuming that this probability is uniformly distributed on [0,1]), contradicting the statement of William Dembski and Robert Marks:

*"Because the active entropy is strictly negative, any uninformed assisted search (φ) will on average perform ***worse** than the baseline search". (Highlighting mine)

In my opinion, the fundamental problem stems from identifying the search space Ω

_{(n)} and the space of (deterministic) searches. This is doubtless elegant, but does not seem to work.

### References:

^{1}W. A. Dembski and R. J. Marks II, "The Search for a Search: Measuring the Information Cost of Higher Level Search," Journal of Advanced Computational Intelligence and Intelligent Informatics, Vol. 14, No. 5, pp. 475-486, September 2010

^{2}W. A. Dembski and R. J. Marks II, "Conservation of Information in Search: Measuring the Cost of Success," IEEE Trans. on Systems, Man and Cybernetics A, Systems and Humans, Vol. 39, No. 5, pp. 1051-1061, September 2009

^{3}W. Ewert, G. Montanez, W. A. Dembski, and R. J. Marks II, "Efficient Per Query Information Extraction from a Hamming Oracle," Proc. of the 42nd Meeting of the Southeastern Symposium on System Theory, IEEE, University of Texas at Tyler, pp. 290-297, March 7-9, 2010