The Search for a Search  Measuring the Information Cost of Higher Level Search
The divine comedy Creationism 
Running gags 
Jokes aside 
Blooper reel 
“”Intellectual Standards. Are we holding ourselves to high intellectual standards? Are we in the least selfcritical about our work? Are we sober or immodest about our work? Do we demand precision and rigor from our each other? Do we examine each other's work with intense critical scrutiny and speak our minds freely in assessing it? Or do we try to keep all our interactions civil, gentlemanly, and diplomatic (perhaps so as not to give the appearance of dissension in our ranks)? Does the mood of our movement alternate between the smug and the indignant  smug when we hold the upper hand, indignant when we are criticized? Do we react to adverse criticism like firsttime novelists who are dismayed to discover that their masterpiece has been trashed by the critics? Or do we take adverse criticism as an occasion for tightening and improving our work?

—William Dembski, 2002^{[1]} 
"The Search for a Search  Measuring the Information Cost of Higher Level Search" is an article by William Dembski and Robert Marks which appeared in the Journal of Advanced Computational Intelligence and Intelligent Informatics ^{[2]}(abstract, full text) The article is a followup to Conservation of Information in Search: Measuring the Cost of Success which appeared in Ieee Transactions On Systems, Man, and Cybernetics. ^{[3]}
The gist of the article is that Dembski and Marks attempt to demonstrate some difficulties with information searches, building on the existing "no free lunch" theorem, which states (colloquially) that no process for optimizing a search is "better" than any other when applied to all searches. Dembski extends this idea further, and though he does not touch on intelligent design in this paper, he has elsewhere made it clear that he interprets NFL as supporting intelligent design,^{[4]} an idea that has been strongly criticized by no less than David Wolpert, one of the authors of the original NFL paper.^{[5]}
Contents
Original Text  Annotations 

Abstract: Needleinthehaystack problems look for small targets in large spaces. In such cases, blind search stands no hope of success. Conservation of information dictates any search technique will work, on average, as well as blind search. Success requires an assisted search. But whence the assistance required for a search to be successful? To pose the question this way suggests that successful searches do not emerge spontaneously but need themselves to be discovered via a search. The question then naturally arises whether such a higherlevel “search for a search” is any easier than the original search. We prove two results: (1) The Horizontal No Free Lunch Theorem, which shows that average relative performance of searches never exceeds unassisted or blind searches, and (2) The Vertical No Free Lunch Theorem, which shows that the difficulty of searching for a successful search increases exponentially with respect to the minimum allowable active information being sought. 
We will occupy ourselves with the Horizontal No Free Lunch Theorem and show how its assumptions  and its proof  are seriously flawed. 
Introduction[edit]
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[...] Endogenous and active information together allow for a precise characterization of the conservation of information. The average active information, or active entropy, of an unassisted search is zero when no assumption is made concerning the search target. If any assumption is made concerning the target, the active entropy becomes negative. This is the Horizontal NFLT presented in Section 3.2. It states that an arbitrary search space structure will, on average, result in a worse search than assuming nothing and simply performing an unassisted search. [...] 
This is a very surprising claim in the light of the conventional No Free Lunch Theorems (NFL): given the usual assumptions of the NFL, we would expect any search to be as good (or as bad) as random search. And an unassisted search seems to be a random search...
To make our ideas clearer, we will introduce a simple example: a Shell Game, using one, two or even three peas. In the language of Dembski and Marks, our search space is Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega = \{1,2,3\}\, , and the possible targets are the nonempty subsets of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega . Judging from the examples they used in their earlier papers, Dembski and Marks do not mind that the player knows the size of the target  so we can assume that exactly one pea is used in the game. (A random number of peas would provide a new puzzle: how would it to be determined? We could put under each shell a pea with a probability p, and repeat this process if no pea was placed  as the authors don't allow for the empty target. Or we could first decide how many peas to place, and then hide them at random, resulting in another distribution... Though it seems to be farfetched, we have to come back to this little problem later on.) 
Information in Search[edit]
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All but the most trivial searches require information about the search environment (e.g., smooth landscapes) or target location (e.g., fitness measures) if they are to be successful. Conservation of information theorems^{[6]}^{[7]}^{[8]} show that one search algorithm will, on average, perform as well as any other and thus that no search algorithm will, on average, outperform an unassisted, or blind, search. But clearly, many of the searches that arise in practice do outperform blind unassisted search. How, then, do such searches arise and where do they obtain the information that enables them to be successful? 
These are good questions, but unfortunately, Dembski and Marks will not answer them. But let us at first formulate our search problem in the language of the No Free Lunch Theorems: these are generally about minimizing a (somewhat) unknown function f over a space Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\,
, that is, you know that a function f belongs to a set of functions Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Gamma\,
and you are allowed to evaluate the function at any element of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\,
. As any evaluation is costly in the world of computer science, you try to keep the number of evaluations low. A search is a special form of minimization: imagine the set Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Gamma\,
consisting from three functions Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): f_1, f_2, f_3\,
from Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega = \{1,2,3\}\,
into the set Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal{Y}=\{0,1\}
, such that
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): f_i(x) = \left\{\begin{matrix} \,0 \quad &: &\quad x = i \\ \,1 \quad &: &\quad x \neq i \end{matrix}\right. This is the set of functions matching our problem of looking for a single pea under three shells! Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Gamma\, has a special property: it is closed under permutation (c.u.p.), that is for each permutation Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \sigma \in \mathcal{S}_{\Omega} and each function Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): f \in \Gamma\, we find that , too. This is the kind of sets for which the No Free Lunch theorems hold: you cannot find a strategy beating random search without repetition when playing this game: if you are allowed to guess once, your chance will be 1/3, if you allowed to guess twice, your chance will be 2/3, and if you are allowed to guess three times (this may happen in the world of mathematics!), your chance is 1 to find the pea  under any strategy you can imagine! Now look at Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Gamma'\, , consisting form the three functions Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): g_i: \Omega \rightarrow \{0,1,2,3\}\, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): g_i(x) = \left\{\begin{matrix} \,0 \quad &: &\quad x = i \\ \,i \quad &: &\quad x \neq i \end{matrix}\right. This set isn't c.u.p.: Take the permutation Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \sigma = (1 \rightarrow 2, 2 \rightarrow 3, 3 \rightarrow 1)\, and look at Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): g_1 \circ \sigma\, : Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (g_1 \circ \sigma)(x) = \left\{\begin{matrix} \,1 \quad &: &\quad x = 1 \\ \,1 \quad &: &\quad x = 2 \\ \,0 \quad &: &\quad x=3 \\ \end{matrix}\right. Clearly not a member of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Gamma'\, ! Indeed we can find a strategy faring better than random search, at least if we are allowed to guess twice:
Of course, this set of functions is very artificial, but all the interesting minimization problems we find in nature are of the nonc.u.p. kind. 
Blind and Assisted Queries and Searches[edit]
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Let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_1 \in \Omega_1\, denote a target in a search space, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_1\, , so that for uniformity the probability of success, p, is ratio of the cardinality of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_1\, to that of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_1\, . The probability, p, is the chance of obtaining an element in the target with a single query assuming a uniform distribution. If nothing is known about the search space or target location, the uniformity of the distribution follows from Bernoulli’s Principle of Insufficient Reason (PrOIR) ^{[9]}^{[10]}^{[11]} Bernoulli’s PrOIR states that in the absence of any prior information, “we must assume that the events ... have equal probability”^{[10]}^{[11]}. Uniformity is equivalent to the assumption that the search space is at maximum informational entropy. 
Interestingly, at the NFL theorems, such a uniformity of probability is assumed  though it is certainly possible to formulate the theorems for a family of distributions  as long as this family is c.u.p. again. But keep in mind that assuming that the events have equal probability doesn't make it true. What does this mean for our shell game? If we assume that the pea is hidden under each shell with probability 1/3, it is okay for us to pick a shell at random  or stick with picking a certain shell. But if the man who hides the pea has a preference for one shell, a random guess will still result in a successrate of 1/3  while other strategies may have other rates of success: greater than 1/3 if it shares the proclivity of the master of the game, less than 1/3 if it doesn't! 
Consider Q queries (samples) from Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_1\, without replacement. Such searches can be construed as a single query when the search space is appropriately defined. For a fixed Q, the augmented search space, , consists of all sequences of length Q chosen from Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_1\, where no element appears more than once. Note that, appropriately, the original search space, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_1\, , is for Q = 1. In general Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (2)\qquad\Omega_Q\,= \frac{\Omega_1!}{(\Omega_1  Q)!} The requirement of sampling without replacement requires Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): Q \leq \Omega_1 . If nothing is known about the location or the target, then, from Bernoulli’s PrOIR, any element of is as likely to contain the target as any other element. Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (3)\qquad p_Q =\frac{T_Q }{\Omega_Q} where Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_Q \in_{[sic]} \Omega_Q consists of all the elements containing the original target, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_1 . When, for example, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_1\, consists of a single element in Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, (i.e., Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_1 = 1\, ) and there is no knowledge about the search space structure or target location, then Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): p_1 = 1/\Omega_1 \, and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (4)\qquad p_Q = \frac{Q}{\Omega_1} = Q\;p_1 From Eqs. (2) and (3), we have Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): =\frac{Q}{\Omega_1}\Omega_Q Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): =\frac{Q(\Omega_11)!}{(\Omega_1 Q)!}.\qquad(5) 
These innocuous notations are the main reason of all the trouble to come. Let us first have a look how a deterministic minimization algorithm is described in the language of the No Free Lunch theorems: at the nth step, you have evaluated the function f (n1) times, thus you know a sequence Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): <\omega_1,f(\omega_1)>, <\omega_2,f(\omega_2)>, \dots, <\omega_{n1},f(\omega_{n1})>
. Considering this knowledge, you will chose a new element Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \omega_n\,
where you will evaluate the function. Formally, a strategy for solving a minimization problem is a funcion Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): D\,
which will map sequence of length 0 up to Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_1 1\,
to elements of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_1\,
(with Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \mathcal{Y} = \cup_{f \in \Gamma} f(\Omega)\,
):
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): D: (\Omega \times \mathcal{Y})^{\ast} \rightarrow \Omega With , we get a new sequence and can go on. If our functions Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): f \in \Gamma only take to values, 0 if the element is the minimum, i.e., it is found, and 1 otherwise, we can drop the second element in each pair as it doesn't have any influence on our considerations: if the element is the minimum, fine, we are done and carry on the same way we would have done without finding it  we have already a winning sequence. Such search strategies are completely determined by the elements  and not the values of the functions. And so, the clumsy writing of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): D(\omega_1,\omega_2,...,\omega_{n1})=\omega_n can be abbreviated to Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (\omega_1,\omega_2,...,\omega_{n1},\omega_{n}) . We will call this a 01 strategy and summarize:
These deterministic 01 strategy of length Q can be easily randomized by choosing one of these strategies according to a probability measure on the set of the strategies. Such a measure will be called randomized 01 strategy of length Q  or just 01 strategy of length Q. So, what Dembski and Marks see as augmented searchspace, we regard as the set of deterministic search strategies... 
There appears, at first flush, to be knowledge about the search space for Q > 1 since a sequence in has the same chance of containing a target as an element in with a perturbation of the same elements. This, however, is knowledge we cannot exploit with a single query to . Also, if the space is exhaustively contracted to eliminate all elements allowing a perturbation rearrangement to another element, the chance of success in a single query remains the same. 
Nothing wrong about this! 
Keeping track of Q subscripts and exhaustively contracted perturbation search spaces is distracting. As such, let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, denote any search space, like , wherein Bernoulli’s PrOIR is applicable for a single query. We adopt similar notation for the target subspace, T , and the probability, p, of finding the target using a single query. A single query to a multiquery space can then be considered a search. If a search is chosen randomly from such a multiquery space, we are still preforming a blind (or unassisted) search. 
Well, we will keep track with the subscripts and the different spaces! For instance, guessing twice in our shell game results in Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_2 = \{(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)\} . A search is now guessing an element of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_2 , and a search is successful if our chosen element lies in the target  a subset Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T \subset \Omega_2\, . But be careful: the target the pea is under shell 1 is represented now by . 
Active Information and No Free Lunch[edit]
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Define an assisted query as any choice from that provides more information about the search environment or candidate solutions than a blind search. Gauging the effectiveness of assisted search in relation to blind search is our next task. Random sampling with respect to the uniform probability U sets a baseline for the effectiveness of blind search. For a finite search space with Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, elements, the probability of locating the target T has uniform probability given by Eq. (3) without the subscripts: Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): p=\frac{T}{\Omega}.\qquad(6) 
Here, Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Gamma'\, springs to mind: on a nonc.u.p. set of functions there are searches which do better than blind search. But how to do so on ? In our definition, such a query does not exist: each 01strategy is as good as any other... 
Let q denote the probability of success of an assisted search. We assume that we can always do at least as well as uniform random sampling. The question is, how much better can we do? Given a small target T in Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, and probability measures U and characterizing blind and assisted search respectively, assisted search will be more effective than blind search when p < q, as effective if p = q, and less effective if p > q. In other words, an assisted search can be more likely to locate T than blind search, equally likely, or less likely. If less likely, the assisted search is counterproductive, actually doing worse than blind search. For instance, we might imagine an Easter egg hunt where one is told “warmer” if one is moving away from an egg and “colder” if one is moving toward it. If one acts on this guidance, one will be less likely to find the egg than if one simply did a blind search. Because the assisted search is in this instance misleading, it actually undermines the success of the search. 
So, Dembski's and Marks's uniform random sampling takes an element of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, at random. In our example  two guesses at a shell game  any element of is chosen with probability 1/6. How could a look like? Well, in the spirit of the Easter egg hunt, someone could say bigger or smaller after your first choice. But how does this induce a probability measure on Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, ? Well, it doesn't! Our decision is not longer dependent only on our first choice, but on additional input  and this input depends on the target. So, there is no describing an Easter egg (shell) hunt. 
An instructive way to characterize the effectiveness of assisted search in relation to blind search is with the likelihood ratio Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \frac{q}{p} . This ratio achieves a maximum of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \frac{1}{p} when q = 1, in which case the assisted search is guaranteed to succeed. The assisted search is better at locating T than blind search provided the ratio is bigger than 1, equivalent to blind search when it is equal to 1, and worse than blind search when it is less than 1. Finally, this ratio achieves a minimum value of 0 when q = 0, indicating that the assisted search is guaranteed to fail. 
Active Information[edit]
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Let U denote a uniform distribution on Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, characteristic of an unassisted search and the (nonuniform) measure on Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, for an assisted search. Let U(T) and denote the probability over the target set Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T \in_{[sic]} \Omega\, . Define the active information of the assisted search as Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): I_+(\varphi\mathbf{U}) := \log_2 \frac{\varphi(T)}{\mathbf{U}(T)} Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): =\log_2 \frac{q}{p}\qquad(7) 
Of course, we can define such a quantity. But does it make sense? Does it even exist? We will see... 
Active information measures the effectiveness of assisted search in relation to blind search using a conventional information measure. It^{[12]} characterizes the amount of information^{[13]} that (representing the assisted search) adds with respect to U (representing the blind search) in the search for T . Active information therefore quantifies the effectiveness of assisted search against a blindsearch baseline. The NFLT dictates that any search without active information will, on average, perform no better than blind search. 
As the NFL theorems are stating that a search which outperforms blind search only exists if there is a nonc.u.p. set of functions, should be defined on a much more interesting (i.e., targetdepending) space than Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\,  as shown even by our simple example above. (Perhaps in this case Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \varphi(\cdot  \Omega) will just be the uniform probability...) 
The maximum value that active information can attain is Failed to parse (lexing error): (I_+)_{max} = − \log_2 p = I_{\Omega} indicating an assisted search guaranteed to succeed (i.e., a perfect search)^{[12]}); and the minimum it can attain is , indicating an assisted search guaranteed to fail. 
So, here we find to examples for assisted searches, the perfect search and its evil twin, the search which is doomed to fail. So, what do the (nonuniform) measures on Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, (as mentioned above)for this assisted searches look like? Obviously, there are no such measures! 
[...] 

Active information can also be measured with respect to other reference points. We therefore define the active information more generally as: Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): I_+(\varphi\psi) Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): =\log_2\varphi(T)log_2\psi(T)\qquad(10) for and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \psi arbitrary probability measures over the compact metric space Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, (with metric D), and T an arbitrary Borel set of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega\, such that Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \psi(T)>0\, . 
Again, this can be defined: but what is it meaning? 
3.2. Horizontal No Free Lunch[edit]
Original Text  Annotations  

The following No Free Lunch theorem (NFLT) underscores the parity of average search performance. 

Theorem 1: Horizontal No Free Lunch. Let and Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \psi\, be arbitrary probability measures and let Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \tilde{T} = \{T_i\}_{i=1}^N be an exhaustive partition of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega all of whose partition elements have positive probability with respect to Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \psi\, . Define active entropy as the average active information that contributes to Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \psi\, with respect to the partition Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \tilde{T} as Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): H_+^{\tilde{T}}(\varphi\psi) Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): :=\sum_{i=1}^N \psi(T_i) I_+^{T_i}(\varphipsi) Then Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): H_{\tilde{T}}(\varphi \psi) \leq 0 with equality if and only if Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \varphi(T_i) = \psi(T_i) for Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 1 \leq i \leq N . 
This is as true as it is meaningless: Generally, for a search of two or more guesses, there is no meaningful way to get the partition Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \tilde{T} = \{T_i\}_{i=1}^N . Again, let us have a look at the shell game with two guesses: As mentioned before, our target (the pea is under the first shell) in Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_2 is now Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \{(1,2),(1,3),(2,1),(3,1)\} . Let our strategy be
This strategy works well if the pea is under the first shell  or the second one (target Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \{(1,2),(3,2),(2,1),(2,3)\} ), and fails for the third one (target ). But these three targets aren't looked at by the theorem: they don't form a partition of Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_2  though it was a partition of which created them. What does a partition on Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \Omega_2 look like? Assume, it contains Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_1 =\{(1,2),(1,3),(2,1),(3,1)\} . Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_2=\{(2,3),(3,2)\}\, are the elements left. But what does it mean to search successfully for Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_2 : it means that we have to find two peas  one under the second and another under the third shell! On Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): T_2 , our search fails for sure, and so, this partition is not allowed to be used to evaluate our strategy. In fact, any nontrivial partition feasible for the strategy does not include a set which represents an original problem! 

In our opinion, the problem arises from the elegant  but unfortunately invalid  identification of the search space and the space of the searches: For our set of deterministic 01 strategies, each measure introduces a randomized strategy. For each such strategy, we can calculate the probability to find a target in the original space. Let's go back to our twoguesses shell game. The deterministic 01 strategies are:
Dembski's and Marks's uniform distribution U leads to choosing one of these searches with probability 1/6. If the pea is hidden without any preference for any of the shells, this strategy gets a success rate of 2/3. As does any other randomized 01 strategy  because it is the success rate of the deterministic strategies! And that is no surprise at all, as the formulation of the searches introduced a set of c.u.p. functions. If there is a preference for  for instance  the first shell, the picture changes: Suddenly, there are preferable strategies: sticking with Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): D_1\, , for instance. In this case, the strategy will outperform Dembski's and Marks's random search, another problem for the horizontal no free lunch theorem. 
See also[edit]
 Conservation of Information in Search  Measuring the Cost of Success
 Efficient Per Query Information Extraction from a Hamming Oracle
 Uncommon Descent
 Barry Arrington
References[edit]
 ↑ Becoming a Disciplined Science: Prospects, Pitfalls, and a Reality Check for ID, William Dembski, Keynote address delivered at RAPID Conference (Research and Progress in Intelligent Design), Biola University, La Mirada, California, 25 October 2002
 ↑ Journal of Advanced Computational Intelligence and Intelligent Informatics, Vol.14, No.5, 2010, pp. 475486
 ↑ Ieee Transactions On Systems, Man, and Cybernetics—Part A: Systems And Humans, Vol. 39, No. 5, September 2009.
 ↑ Dembski, W. A. (2002) No Free Lunch, Rowman & Littlefield
 ↑ Wolpert, D. (2003) "William Dembski's treatment of the No Free Lunch theorems is written in jello".
 ↑ C. Schaffer, “A conservation law for generalization performance,” Proc. Eleventh Int. Conf. on Machine Learning, H. Willian and W. Cohen, San Francisco: Morgan Kaufmann, pp. 295265, 1994.
 ↑ T. M. English, “Some information theoretic results on evolutionary optimization,” Proc. of the 1999 Congress on Evolutionary Computation, 1999, CEC 99, Vol.1, pp. 69, July 1999.
 ↑ D. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evolutionary Computation, Vol.1, No.1, pp. 6782, 1997.
 ↑ W. A. Dembski and R. J. Marks II, “Bernoulli’s Principle of Insufficient Reason and Conservation of Information in Computer Search,” Proceedings of the 2009 IEEE Int. Conf. on Systems, Man, and Cybernetics, San Antonio, TX, USA, pp. 26472652, October 2009.
 ↑ ^{10.0} ^{10.1} J. Bernoulli, “Ars Conjectandi (The Art of Conjecturing),” Tractatus De Seriebus Infinitis, 1713.
 ↑ ^{11.0} ^{11.1} A. Papoulis, “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGrawHill, 1991.
 ↑ ^{12.0} ^{12.1} 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. 10511061, September 2009.
 ↑ T. M. Cover and J. A. Thomas, “Elements of Information Theory,” 2nd Edition, WileyInterscience, 2006.