Responding to Dr. Ewert

I’ve added the following to A Question for Winston Ewert at The Skeptical Zone.

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Dr. Ewert has responded nebulously at Uncommon Descent. I’d have worked with him to get his meaning straight. I’m not going to spend my time on deconstruction. However, I will take quick shots at some easy targets, mainly to show appreciation to [Elizabeth Liddle] for featuring this post as long as she has. Here, again, is what I put to Dr. Ewert:

Your search process decides when to stop and produce an outcome in the search space. A model may do this, but biological evolution does not. How do you measure active information on the biological process itself? Do you not reify a model?

Dr. Ewert seemingly forgets that to measure active information on a biological process is to produce a specific quantity, e.g., 109 bits.

One approach is to take the search space not to be the individual organisms, but rather the entire population of organisms currently alive on earth. Or one could go further, and take it to be the history of organisms during the whole of biological evolution. One could also take it to be possible spacetime histories. The target can then be taken to be spacetimes, histories, or populations that contain an individual organism type such as birds.

These search spaces roll off the tongue. But no one knows, or ever will know, what they actually contain. Even if we did know, no one would know the probabilities required for calculation of the active information for a given target. And even if we did know the probability of a given target for a given search, we would not be able to justify designating a particular probability distribution on the search space as the natural baseline. By the way, Dr. Ewert should not be alluding to infinite sets, as his current model of search applies only to finite sets.

Another possibility is to model evolution as a process which halts upon finding the target, but distinguish between the active information derived from the evolutionary process itself and the active information contributed by the stopping behavior. The stopping behavior cannot induce birds to show up in the first place, it can only select them as the output of the search when they arrive. By looking at the number of opportunities for birds to arise, we can determine how much active information was added by the stopping process. It was shown in Conservation of Information in Search: Measuring the Cost of Success that the active information available from such a process is only the logarithm of the number of queries. Any other active information must derive from the evolutionary search itself.

Dembski and Marks define search differently in the cited paper than Dembski, Ewert, and Marks do. The result that Dr. Ewert invokes does not apply to the active information of a search, as presently defined. With the current definition, we can specify a process that goes through elements of the finite search space, one by one, until it recognizes an element of the target. Then the active information of the process is due almost entirely to recognition of the target by the stopping process. I hope this gives you some idea of what’s wrong with Ewert’s claim. Perhaps one of the cognoscenti will supply more of the details in a comment.

Both approaches effectively end up adjusting for the number of trials. Getting a royal flush is improbable, but if you play five million hands of poker it is no longer surprising. Similarly, obtaining a bird is rendered much more probable given the number of chances for it happen in the history of universe. It is a very important point to keep in mind that we cannot simply look at the probability of the individual events but also the number of trials.

Dr. Ewert errs, and has brought to the fore a major weakness of the current definition of search. Here the search space is the set of all five-card poker hands, and the target is the subset containing the royal-flush hands. A search that halts after one step and yields a royal flush with probability 1/2 has exactly the same active information as a search that yields a royal flush with probability 1/2 after five million or fewer steps. In short, a very important point to keep in mind is that the number of trials actually does not enter into the calculation of active information.

For birds to have been produced by an evolutionary process, the universe must have been biased towards producing birds.

Must the universe have been biased against producing flying insects that walk on all fours? (This is not a cheap dig at religion, but instead a substantive response to Robert Saying the Bible is not a book about science is like saying a cookbook is not a book about chemistry Marks. I had forgotten Leviticus 11:20 until I Googled for scientific discussion of why there are no four-legged insects.)

At least a hint, Dr. Ewert?

Posted in The Skeptical Zone.

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I repeat my invitation to Dr. Winston Ewert to join us here for discussion of several questions I raised. It helps immensely to display mathematical formulas, rather than talk about them vaguely. However, he has replied at Uncommon Descent, where that is impossible. I’m genuinely astonished to see:

The sarcastic fringehead

The blog title I’ll always regret having missed.

A better reason for Dr Ewert to enter The Skeptical Zone

As I explained in my last post, Winston Ewert has solicited questions on his research with William Dembski and Robert Marks, and I have raised several at The Skeptical Zone. I avoided upstaging DiEb, who followed Ewert’s procedure, and submitted questions through a Google Moderator page. As you can see from the following note I left at DiEblog, an immoderate moderator at Uncommon Descent has haplessly given Ewert a better reason to answer at The Skeptical Zone than I have.

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I hope you don’t mind my observation that your post relates to one of three questions you posed at Ask Dr Ewert (link expires June 30, 2015). Ewert, who collaborates with Dembski and Marks, evidently intends to answer selected questions at Uncommon Descent. You’ve been banned there since raising the questions, have you not? Correlation does not imply causation. But if you cannot comment on his answers to your questions, then he will in fact have ensconced them in a sham forum.

At The Skeptical Zone: A question for Winston Ewert

I’ve invited Winston Ewert to join a technical discussion at The Skeptical Zone. I do solemnly vow to keep it perfectly civil. It would be better to comment there than here. But suit yourself.

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I actually have three technical questions for Winston, but plan on one post apiece. He should respond first to questions he receives through Google Moderator, including those from DiEb, who has added a relevant post to his blog. Hopefully he will join us here when he’s done with that.

Let’s be clear from the outset that off-topic remarks go straight to Guano. (If you attack Winston personally while I am trying to draw him into a discussion of theory, then I will take it personally.) You shouldn’t make claims unless you have read, and believe that you mostly understand, the material in all three sources in note 3, apart from the proofs of theorems. Genuine requests for explanation are, of course, welcome. They’re especially welcome if you’ve made a genuine effort to get what you can from the sources.

The overall thrust of my questions should be clear enough to Winston, though it won’t be to most readers. I’m definitely not laying a trap for him. The first two questions have answers that are provably right or wrong. The third is more a matter of scientific modeling than of math. I’m starting with it because TSZ isn’t yet configured to handle embedded LaTeX (mathematical expressions).

Questions

1. What is the formal relationship between active information and specified complexity?

2. What is the formal relationship between active information and average active information per query? Does the conservation-of-information theorem apply to the latter?

3. Your search process decides when to stop and produce an outcome in the search space. A model may do this, but biological evolution does not. How do you measure active information on the biological process itself? Do you not reify a model?

Notes

1. There’s an answer that covers both Dembski's 2005 version (the probabilistic complexity minus the descriptive complexity of the target) and the algorithmic version of specified complexity. For the latter, it’s apparently necessary to restrict the target (no longer called a target) to a single-element set.

2. The conservation-of-information theorem applies to active information. Winston and his colleagues have measured only average active information per query (several closely related forms, actually), which seems unrelated to active information, in their analyses of computational evolution and metabiology. Yet they refer to conservation of information in exposition of those analyses.

3. The search process of Dembski, Ewert, and Marks terminates, and generates an outcome. The terminator and the discrim­inator of the search in fact contribute to its active information — bias, relative to a baseline distribution on outcomes, in favor of a target event. However, biological evolu­tion has not come to a grinding halt, and has not announced, for instance, Here it is — birds! It seems that Winston, in his ENV response to a Panda’s Thumb post by Joe Felsenstein and me, tacitly assumes that a biologist has provided a model that he can analyze as a search, and imputes to nature itself the bias that he would measure on the model of nature. If so, then he erroneously treats an abstraction as though it were something real. Famously, The map is not the territory. Perhaps Winston can provide a good argument that he hasn’t lapsed into reification.

Deobfuscating a theorem of Ewert, Marks, and Dembski

Back in July, I found that I couldn’t make heads or tails of the theorem in a paper by Winston Ewert, Robert J. Marks II, and William A. Dembski, On the Improbability of Algorithmic Specified Complexity. As I explained,

The formalism and the argument are atrocious. I eventually decided that it would be easier to reformulate what I thought the authors were trying to say, and to see if I could generate my own proof, than to penetrate the slop. It took me about 20 minutes, working directly in LaTeX.

I posted a much improved proof, but realized the next day that I’d missed something very simple. With all due haste, here is that something. The theorem is best understood as a corollary of an underwhelming result in probability.

The simple

Suppose that $\mu$ and $\nu$ are probability measures on a countable sample space $\Omega$, and that $c$ is a positive real number. What is the probability that $\nu(x) \geq c \cdot \mu(x)$? That’s a silly question. We have two probabilities of the event \[ E = \{x \in \Omega \mid \nu(x) \geq c \cdot \mu(x) \}. \] It’s easy to see that $\nu(E) \geq c \cdot \mu(E)$ when $\nu(x) \geq c \cdot \mu(x)$ for all $x$ in $E$. The corresponding upper bound on $\mu(E)$ can be loosened, i.e., \begin{equation*} \mu(E) \leq \frac{\nu(E)}{c} \leq \frac{1}{c}. \end{equation*} Ewert et al. derive $\mu(E) \leq c^{-1}$ obscurely. [Added 30/12/2018: George Montañez has referred to this post in his BIO-Complexity article “A Unified Model of Complex Specified Information.” I should make explicit something that is implied by the identification, below, of $\nu$ with an algorithmic probability measure: there is no requirement that the probabilities of atomic outcomes sum to unity, i.e., that $\mu(\Omega) = 1$ and $\nu(\Omega) = 1.$ The loosening of the upper bound on $\mu(E)$ assumes that $\nu(\mathcal{E}) \leq 1$ holds for all events $\mathcal{E} \subseteq \Omega.$]

The information-ish

To make the definition of $E$ information-ish, assume that $\mu(x) > 0$ for all $x$ in $\Omega$, and rewrite \begin{align} \nu(x) &\geq c \cdot \mu(x) \nonumber \\ \nu(x) / \mu(x) &\geq c \nonumber \\ \log_2 \nu(x) - \log_2 \mu(x) &\geq \alpha, \end{align} where $\alpha = \log_2 c$. This lays the groundwork for über-silliness: The probability of $\alpha$ or more bits of some special kind of information is at most $2^{-\alpha}$. This means only that $\mu(E) \leq c^{-1} = 2^{-\alpha}.$

The ugly

Now suppose that $\Omega$ is the set of binary strings $\{0, 1\}^*$. Let $y$ be in $\Omega$, and define an algorithmic probability measure $\nu(x) = 2^{-K(x|y)}$ for all $x$ in $\Omega$. (I explained conditional Kolmogorov complexity $K(x|y)$ in my previous post.) Rewriting the left-hand side of Equation (1), we obtain \begin{align*} \log_2 2^{-K(x|y)} - \log_2 \mu(x) &= -\!\log_2 \mu(x) - K(x|y) \\ &= ASC(x, \mu, y), \end{align*} the algorithmic specified complexity of $x$. Ewert et al. express an über-silly question, along with an answer, as \[ \Pr[ASC(x, \mu, y) \geq \alpha] \leq 2^{-\alpha}. \] This is ill-defined, because $ASC(x, \mu, y)$ is not a random quantity. But we can see what they should have said. The set of all $x$ such that $ASC(x, \mu, y) \geq \alpha$ is the event $E$, and $2^{-\alpha} = c^{-1}$ is a loose upper bound on $\mu(E)$.

Tolstoy on the studious deceit of children by the church

Nothing captures my experience with the church better than does this passage from Leo Tolstoy’s The Kingdom of God is Within You (1894).

The chief and most pernicious work of the Church is that which is directed to the deception of children — these very children of whom Christ said: Woe to him that offendeth one of these little ones. From the very first awakening of the consciousness of the child they begin to deceive him, to instill into him with the utmost solemnity what they do not themselves believe in, and they continue to instill it into him till the deception has by habit grown into the child's nature. They studiously deceive the child on the most important subject in life, and when the deception has so grown into his life that it would be difficult to uproot it, then they reveal to him the whole world of science and reality, which cannot by any means be reconciled with the beliefs that have been instilled into him, leaving it to him to find his way as best he can out of these contradictions.

If one set oneself the task of trying to confuse a man so that he could not think clearly nor free himself from the perplexity of two opposing theories of life which had been instilled into him from childhood, one could not invent any means more effectual than the treatment of every young man educated in our so-called Christian society.

(I provide context here.) It’s not exactly surprising that people who refer to indoctrination as Christian education should regard education as indoctrination when it happens to conflict with their beliefs.