Dembski's and Mark's "The Search for a Search": Trying to work out one of their examples

Three years ago, on April 1st, 2010, W. Dembski's and R. Marks's paper The Search for a Search^[1] was accepted for publication. Recently, W. Dembski referred to it as a seminal paper in his article Before They've Even Seen Stephen Meyer's New Book, Darwinists Waste No Time in Criticizing Darwin's Doubt at the Discovery Institute's Evolution News and Views.. The first result of their paper is the Horizontal No Free Lunch Theorem which shows that average relative performance of [assisted] searches never exceeds unassisted or blind searches. Hopefully this paper is covered at the 2013 Summer Seminar on Intelligent Design In the natural sciences from July 12-20, 2013 in Seattle, WA - at least in the past, Marks and Dembski have been speakers at this seminar. Wouldn't it be a nice warm-up exercise for all these bright students gathered in Seattle to work out an example of an assisted search and see for themselves how the Horizontal No Free Lunch theorem works? I just ask because I haven't seen any such example calculated by Marks, Dembski or one of their students. They have mentioned a few kinds of assisted searches - like Dawkins's weasel, the Easter Egg hunt, etc., but they have never shown how those fit their model.

So I tried to work out a kind of simplified Easter Egg hunt in the language of Dembski and Marks - and I invite everyone to try to elaborate similar examples. My claim: any search which uses a kind of feedback (an oracle, a fitness function, etc.) cannot be described meaningfully in the terms of Marks and Dembski - which cast a doubt on their far-reaching conclusions about assisted searches in general. But see for yourself:

Again a proud number as a headline at Uncommon Descent: 10,000!

Again a proud number as a headline at Uncommon Descent: 10,000! The whole thing reads:

This is the 10,000th post at UD. We would like to thank all of our loyal readers, lurkers, commenters, writers, webmaster, contributors and all the others who have made this a wonderful run so far!

So congratulations! But I just have to pour some water in Barry Arrington's wine:

1) It comes actually a little bit late...

The last announcement of a milestone was the thread on Uncommon Descent's 9,000 post on May 14, 2012 - I commented on this event here. At this time, more than 200 threads were created per month, so the 10,000-ceiling should have been shattered in August 2012 (or September 2012) at the latest. What happened? Denise O'Leary - who put on thread after thread under the witty name "news" - left and took here energy over to The Best Schools' Blog, where she now baffles a non-existing audience (which isn't allowed to respond). I don't know why she did so, but it cut the number of threads per month by 75%.

2) The loyal commenters aren't that loyal any longer...

More than 3,300 editors have commented on Uncommon Descent. But over the last months, the number of unique editors per month is decreasing: in October, November and December 2012 it fell even below 100 - the smallest numbers since mid-2005!

3) But at least the number of comments is up - somewhat...

After a period of drought in the first half of the year 2012, the numbers have risen again. Most comments are made by the regulars, therefore the number of deleted comments became very small in November and December 2012. Not small enough though: a couple of my remarks never appeared...

Some Annotations to the Previous Post

1. Joe, at this point I'd advice students to draw a decision tree. Some would draw one with six nodes in the first layer, representing the machines $M_1, M_2, ... , M_6$ and then 36 nodes in the second layer, representing each of the possible outcomes from $1,2,...,6$ for each of the machines. At each of the branches, they put the possibility to get from one node to the next, and at the end of the the diagram they write down the 36 probabilities for the outcomes which they get by multiplying the probabilities on the branches which lead to the outcome. However, others would opt for a much easier design, summarizing the machines $M_2, M_3, ..., M_6$ as $\overline{M_1}$, and the non-desirable outcomes $\{1,2, ...,5\}$ as $\overline{6}$, which leads to the following graph:

Could you please correct your miscalculation, Dr. Dembski?

William A. Dembski wrote "a long article […] on conservation of information. " at Evolution News and Views (ENV), an outlet of the Discovery Institute. Others have commented on more sophisticated problems, either at Uncommon Descent or at The Skeptical Zone. Here I want just to correct some simple math which occurs in a toy example used in the article:

To see how this works, let's consider a toy problem. Imagine that your search space consists of only six items, labeled 1 through 6. Let's say your target is item 6 and that you're going to search this space by rolling a fair die once. If it lands on 6, your search is successful; otherwise, it's unsuccessful. So your probability of success is 1/6. Now let's say you want to increase the probability of success to 1/2. You therefore find a machine that flips a fair coin and delivers item 6 to you if it lands heads and delivers some other item in the search space if it land tails. What a great machine, you think. It significantly boosts the probability of obtaining item 6 (from 1/6 to 1/2).

Is the average active information a suitable measure of search performance?

(Though the following doesn't include any maths, the reader is expected to be familiar with William Dembski's and Robert Marks's paper The Search for a Search and should have glanced at On a Remark by Robert J. Marks and William A. Dembski)

One of my problems with the modeling of searches by William Dembski and Robert Marks is that I don't see how every assisted search can be described as a probability measure on the space of the feasible searches. But nevertheless, Winston Ewert insisted that

All assisted search, irrespective of the manner in which they are assisted, can be modeled as a probably distribution biased towards selecting elements in the target.

Marks and Dembski claim that the average active information is a measure of search performance - at least they write in their remark:

If no information about a search exists, so that the underlying measure is uniform, then, on average, any other assumed measure will result in negative active information, thereby rendering the search performance worse than random search.

Their erratum seems indeed to proof the remark in a slightly modified way:

Given a uniform distribution over targets of cardinality k, and baseline uniform distribution, the average active information will be non-positive

(The proof of this statement in the erratum is correct - at least as far as I can see...)
So, lets play a game: From a deck of cards one is chosen at random. If you want to play, you have to pay 1\$, and you get 10\$ if you are able to guess the card correctly. But you are not alone, there are three other people A, B and (surprisingly) C who'll announce their guesses first. They use the following search strategies:

• A: he will announce a card according to the uniform distribution
• B: he will always announce ♦2
• C: He has access to a very powerful oracle, which gives him the right card. Unfortunately - due to an old superstition - he is unable to say ♦2, so every time this card appears he will announce another one at random

Which strategy will you follow? Siding with player A or B gives you a chance of 1/52 for a correct guess, so you will loose on average ca. 81¢ per game. However, if you pose your bet with C, you will win 8.81$ a game in the long run! That's because the probability of a correct guess is 1/52 for both our players A and B, while C's chance for success is 51/52.

But what would Marks, Dembski or Ewert do? They calculate the average active information according to the formula given in the erratum. This E[I₊] is 0 for player A, but -∞ for B and C. As negative active information on average renders the search performance worse than random search, they have to stick with player A.

So, either average active information is not a good measure of search performance, or not every assisted search, irrespective of the manner in which they are assisted can be modeled as a probably distribution. Or it is a bit of both...

A new erratum for Dembski's and Marks's The Search for a Search

Last month, I made a post On a Remark by Robert J. Marks and William A. Dembski where I addressed errors in a section of Dembski's and Marks's paper The Search for a Search. I exchanged emails over this with Winston Ewert, one of the people at The Evolutionary Informatics Lab ( a former?/grad? student of Marks). He informed me:

You also make brief mention that the HNFLT proof assumes a partition of the search space and thus cannot handle the overlapping targets. This is problematic because any target on the original search space will become an overlapping target on the multi-query search space. I think you are correct on this point. Thanks for bringing it to our attention. We've added a description of the problem as well as an alternate proof for the uniform case on overlapping targets to the pdf: http://evoinfo.org/papers/2010_TheSearchForASearch.pdf.

Here are some thoughts:

9,000!

9,000! is the proud headline of a post by Barry Arrington at Uncommon Descent. The whole thing reads:

The post before this one was UD’s 9,000th. Thank you to all of our readers for your support as we celebrate this milestone.

So congratulations! But a comment by SCheesman pours a little water into the celebratory wine:

I wish I could celebrate, but I fear 9000 is a reflection of a vast inflation in the number rate of postings in the last year or two, with a corresponding decline in comments.
I owe a good deal of what I know today about ID from UD, both from a scientific and theological perspective, and used to enjoy the long threads and back-and-forth between proponents and opponents.
But now, many, if not most posts get nary a comment, and the ones engendering some debate often are lost in the crowd. Since the recent purge of participants who failed to pass what amounted to a purity test, it’s been pretty quiet here. The most lively recent discussion featured a debate between OEC’s and YEC’s. Now I enjoy that sort of thing (like on Sal Cordova’s old “Young Cosmos” blog), but it’s hardly what UD used to be known for.
Maybe the new format gets more visitors than it used to, but I’d be interested in seeing the stats, including comments per post, posts per month, unique visitors etc. over the last few years.
I miss the old days. I expect a lot of us do.

I'll try to satisfy the curiosity as good as I can.

Posts per month