Active information defined… for the fourth time in nine years?

Yesterday, a couple of folks let me know of a paper that crypto-creationist [ETA: perhaps under reform] George Montañez had just posted at arXiv, “The Famine of Forte: Few Search Problems Greatly Favor Your Algorithm.” Below you’ll find my response to one of them. I should explain a few things, by way of introduction.

Montañez is a former advisee of the “Charles Darwin of intelligent design,” Baylor University professor Robert J. Marks II. Last I heard, he was pursuing doctoral studies in machine learning at Carnegie Mellon University. He worked not only with Marks, but also with William A. Dembski, the “Isaac Newton of information theory,” and Winston Ewert, the “Pooh Bear of evolutionary informatics,” on applications of measures of active information. He is still affiliated with them at the Evolutionary Informatics Lab. I refer to the core of affiliates who actually contribute to the output of the Lab — Marks, Dembski, Ewert, and Montañez — as Team EIL. The first three of them have a book scheduled for release by World Scientific on January 30, 2017. The title is Introduction to Evolutionary Informatics. I am trying to pull together a series of posts with the same title.

My email note follows.

[ETA: George Montañez has kindly responded here at TSZ. Contrary to what I guess below, he is not presently collaborating with the authors of the book.]


There’s an important development in the paper. In their studies of ev, Avida, etc., Team EIL did not apply their measures of active information, for which they had “conservation of information” (CoI) theorems, but instead their egregiously misnamed measures of average active information per query (AAIPQ). In it’s simplest form, AAIPQ is the endogenous (not active) information  $I_\Omega$ divided by the expected number of queries for the sampling process to first hit the target,

     $$I_\oplus = \frac{I_\Omega}{E[Q]}.$$

Active information was defined as the difference of endogenous information and exogenous information,

     $$I_+ = I_\Omega - I_S,$$

and endogenous information was thus the sum of active information and exogenous information,

     $$I_\Omega = I_+ + I_S.$$

You don’t even need to know the definitions of endogenous information and exogenous information to see that AAIPQ is active information per expected number of queries plus exogenous information per expected number of queries,

     \begin{align*} I_\oplus     &= \frac{I_\Omega}{E[Q]} \\    &= \frac{I_+ + I_S}{ E[Q]} \\     &= \frac{I_+}{ E[Q]} + \frac{I_S}{ E[Q]}. \end{align*}

The upshot is that Team EIL indicated falsely in papers where they applied AAIPQ to evolutionary models that they had mathematical justification for their “conservation of information” rhetoric. Their CoI theorems were in fact irrelevant. I brought this up in one of my “Ask Dr Ewert” questions [posted here in The Skeptical Zone], and Ewert responded with unadulterated bullshit at Uncommon Descent.

Montañez has just provided a new definition of active information — the fourth in nine years, and quite different from the previous ones — along with conservation theorems. There is indeed averaging in the new measure. It serves as a replacement for AAIPQ. The new theorems say nothing relevant to AAIPQ. The release of Introduction to Evolutionary Informatics has been pushed back to January 30, and I’ll hazard a guess that Montañez and Ewert have applied the new measure to the data from the old studies, and that the new measurements will appear in the book. As for the arXiv publication, you know how uncharacteristic that is of stealth creationists. But it makes perfectly good sense if Marks et al. are using the work in the book, and need to cite a source. They’re perhaps hoping to change the reference to a peer-reviewed, forthcoming paper prior to publication of the book.

Montañez says nothing to indicate that his work is revisionary, as is the norm for Team EIL. Nor does he indicate that his Famine of Forte theorem (with conservation of active information redux as a corollary) is analogous to the “Algorithmic Specified Complexity Is Rare” theorem of Ewert et al. (By the way, I proved a closely analogous “Active Information Is Rare” theorem a couple years ago, using the then-current definition of active information, but decided not to share it with Team EIL.) In all sincerity, I had hoped that the outstanding faculty in machine learning at Carnegie Mellon would set Montañez straight. No such luck.

I’ll have to adjust the part of my introduction to evolutionary informatics dealing with average active information per query. The story I have to tell now is more complicated, but also more juicy. Marks et al. are damned if they use the new measure, because it appeared in an unreviewed, self-published paper after World Scientific began taking orders for the book (advertising it as based on peer-reviewed publications). Also, their new calculations will not have passed peer review, even if the paper by Montañez has. They’re damned if they don’t use the new measure, because they obviously were wrong, as I showed above, in suggesting that their CoI theorems of the time were relevant to AAIPQ, and because Montañez has now acknowledged tacitly that they were wrong.

134 thoughts on “Active information defined… for the fourth time in nine years?

  1. Before we try to understand whether Montañez’s Expected Active Information Per Search measure is well-defined, is useful, or is new, let me make a couple of points about what Montañez’s version of the No Free Lunch theorem implies for evolution:

    Basically, nothing. The original NFL was not aimed at evolution, but William Dembski argued in his 2002 book No Free Lunch: Why Specified Complexity Cannot Be Purchased without Intelligence that it showed that evolutionary processes could on average do no better than ones that made changes completely at random.

    This was immediately refuted by critics (starting with Richard Wein and Jason Rosenhouse). Here is my attempt to explain what they discovered. Basically, the NFL asks about average behavior over all ways that fitnesses could be associated with genotypes. Most of those are “white noise” fitness surfaces. Real fitness surfaces are much smoother than that, because physics.

    Tom and I have also addressed here whether the Conservation of Information theorems of Dembski, Ewert, and Marks imply that on average “evolutionary searches” perform no better than random. They prove this by including in evolutionary searches all sorts of crazy ones, including ones that deliberately try to do badly. Once we eliminate all of the crazy ones, and confine ourselves to ones that have genotypes that have fitnesses, their result collapses.

    As far as I can see Montañez’s results don’t change any of this. Am I right about that, Tom?

  2. Thanks for this, Tom. can I ask an elementary (not to say stupid) question? You write,

    In it’s simplest form, AAIPQ is the endogenous (not active) information divided by the expected number of queries for the sampling process to first hit the target,

    Active information was defined as the difference of endogenous information and exogenous information,   

    and endogenous information was thus the sum of active information and exogenous information,

    Is that what you meant to write? I mean, endogenous info is first referred to as ‘not active’ and then defined as the sum of active and not active information.

    Anyhow, that confused me.
       

  3. phoodoo:
    I find it rather ridiculous, but not surprising given Tom Englishes prior record of cheesy criticisms, that Tom is trying to argue technical points here, that the authors are clearly not going to respond to here, so it is really just a pointless exercise in Tom claiming his information is correct.

    Is the math right, are the presumptions right, are the conclusions right-who the hell knows, because any objection English or Joe might have clearly needs to be addressed precisely by the authors.Such criticisms are meaningless, unless equal responses are heard from the other side.

    Joe’s line “This was immediately refuted by critics (starting with Richard Wein and Jason Rosenhouse).” is a typical example of such nonsense.This is exactly how wikipedia tries to slant the discourse of any subject they have their particular pet viewpoint (materialism).So and so has criticized the work as being inaccurate…

    Anyone can make such a claim. That is supposedly to be clearly against the rules at Wikipedia, but that never stops them.

    What you seem to be saying here is that you are not competent to assess the accuracy of this post. I’m with you there! The difference is that you think Tom is wrong, based on other writings you are also incompetent to assess, because you WANT a certain view to be correct. I leave science to the scientists. If there’s a consensus, I go along, if not, I’ll keep an open mind. You should try that.

  4. Perhaps it’s not surprising that I agree with phoodoo here.

    This OP contains both bombastic inflammatory sounding attacks and obscure difficult math.

    It’s a bit like seeing the WWE on CSPAN and it inclines me to dismiss the content before going to the effort to understand it.

    peace

  5. The original No Free Lunch argument of Dembski was very “mathy” and designed to persuade people that on average, evolution doesn’t work. The many careful critiques of NFL had to work hard to explain all this to a wider audience, and explain why Dembski’s use of the NFL Theorem did not refute evolution. I congratulate all those authors for their outstanding work.

    Now here come phoodoo and FMM saying that these critiques are too mathy for them and that therefore the arguments are meaningless. Neither of them gives a single reason why the critics’ arguments are wrong.

  6. Moved a comment to Guano. Address the post, not the poster. Any Guano’d comment may be cleaned up to remove the personal insults and resubmitted.

  7. Joe Felsenstein: Now here come phoodoo and FMM saying that these critiques are too mathy for them and that therefore the arguments are meaningless.

    I have no problem with the math. I love math

    I have a problem with mixing math with bombastic attacks

    peace

  8. Winston Ewert and Robert Marks and Bill Dembski are my friends. But as I’ve said before, I don’t feel going down the road of information theory and more complex math is the most effective way to frame design claims to biologists and even the general public. So apologies to my old friends.

    Even if we accept for the sake of argument the math is right, it doesn’t lend itself to straight forward analysis of biochemical systems.

    I talk to IDists who are students and faculty of biology and biochemistry, and the mathy stuff doesn’t resonate with them. I have a modest background in math myself ( including grad study of Shannon’s theorems, non-Euclidean genometry, etc.) and I have a hard time applying the ideas to biological systems. In fact, I gave up trying!

    I’ve been formulating alternative ways of expressing the design argument that minimizes information theory and unnecessary mathiness.

  9. walto, to Tom:

    Is that what you meant to write? I mean, endogenous info is first referred to as ‘not active’ and then defined as the sum of active and not active information.

    Anyhow, that confused me.

    walto,

    When Tom says this…

    In it’s simplest form, AAIPQ is the endogenous (not active) information divided by the expected number of queries for the sampling process to first hit the target…

    …he’s saying that when you do the division, you need to put the endogenous information (which includes both active and not active information) in the numerator, and that putting just the active information in the numerator won’t give the correct answer.

  10. keiths:
    walto, to Tom:

    walto,

    When Tom says this…

    …he’s saying that when you do the division, you need to put the endogenous information (which includes both active and not active information) in the numerator, and that putting just the active information in the numerator won’t give the correct answer.

    Oh, OK, so by “(not active)” in that first sentence, he means “(not only active)” or (“not just the active”). That makes sense. Thanks.

  11. Joe Felsenstein: Now here come phoodoo and FMM saying that these critiques are too mathy for them

    Apparently you also can’t understand simple English either Joe. I didn’t say it was too mathy, I said it was meaningless to only hear one side’s version of what is correct.

    That you can’t understand this distinction, just goes to show how incapable you are of making subtle conclusions even if you did understand the math, which we have no evidence for.

  12. phoodoo: Apparently you also can’t understand simple English either Joe. I didn’t say it was too mathy, I said it was meaningless to only hear one side’s version of what is correct.

    I agree. You’re quite right, Evolution News and Views should allow comments, and Uncommon Descent should not so often ban people when they do an effective job of refuting their posts.

  13. Joe Felsenstein,

    You have a website which allows comments Joe? Can I go there and insult you? Please let me know the address.

    Whose fault is it you can’t follow the rules at UD?

    You can say anything stupid you want at this site, and if Patrick feels like it he can allow it or not. Does that make this site better?

  14. Alan Fox,

    But I was talking about Tom’s posts Alan. Besides even Joe’s wants people to be able to say anything they want here without interference. So surely he supports my right to criticize.

  15. stcordova,

    Winston Ewert and Robert Marks and Bill Dembski are my friends. But as I’ve said before, I don’t feel going down the road of information theory and more complex math is the most effective way to frame design claims to biologists and even the general public. So apologies to my old friends.

    Even if we accept for the sake of argument the math is right, it doesn’t lend itself to straight forward analysis of biochemical systems.

    I talk to IDists who are students and faculty of biology and biochemistry, and the mathy stuff doesn’t resonate with them. I have a modest background in math myself ( including grad study of Shannon’s theorems, non-Euclidean genometry, etc.) and I have a hard time applying the ideas to biological systems. In fact, I gave up trying!

    I’ve been formulating alternative ways of expressing the design argument that minimizes information theory and unnecessary mathiness.

    I think the most telling issue is that we cannot create a straight forward mathematical model of current proposed evolutionary mechanisms that support major transitions like prokaryotic to eukaryotic .

    I agree that trying to create a argument through mathematics that natural selection cannot solve the problem is silly.

    If the mechanism is discovered that shows how DNA of specie A becomes DNA of specie B, the mathematical model will be straight forward.

  16. fifth,

    This OP contains both bombastic inflammatory sounding attacks and obscure difficult math.

    Algebra is “obscure difficult math”?

  17. fifth,

    I have no problem with the math. I love math

    You sound like Donald Trump talking about “the Hispanics”.

    Not to mention hat you completely contradicted yourself, as Trump so often does. The math is obscure and difficult, and you have no problem with the math.

  18. colewd:
    stcordova,

    I think the most telling issue is that we cannot create a straight forward mathematical model of current proposed evolutionary mechanisms that support major transitions like prokaryotic to eukaryotic .

    I agree that trying to create a argument through mathematics that natural selection cannot solve the problem is silly.

    If the mechanism is discovered that shows how DNA of specie A becomes DNA of specie B, the mathematical model will be straight forward.

    Yeah, like the mathematical model of the development of the gold deposits of Witwatersrand. The mechanism has been argued for years, whether it’s placer deposits, hydrothermal, or more recently, sulfate reduction during a time of particularly high volcanism.

    But maybe it was all just by design. Why not, if you don’t really care about mechanisms or evidence when it comes to that?

    Glen davidson

  19. colewd: If the mechanism is discovered that shows how DNA of specie A becomes DNA of specie B, the mathematical model will be straight forward.

    Still yammering on about this idiotic nonsense I see?

    Why are you asking for solution to a nonexistent problem? It’s mutation. DNA of species A changes into DNA of species B by accumulating mutations.

    That’s insertions, deletions, substitutions, inversions, fusions and so on. That’s it. Are you going to show signs of finally fathoming this some time in the future?

  20. walto: Is that what you meant to write? I mean, endogenous info is first referred to as ‘not active’ and then defined as the sum of active and not active information.

    Anyhow, that confused me.

    Sorry. I am saying that Dembski and Marks gave a quantity a misleading name, to make it fit into the story they wanted to tell about conservation of information. The definition says that AAIPQ is endogenous information, I_\Omega, per expected number of “queries”, E[Q].

  21. Btw, this I off-topic, but do you know the late Fred Dretske’ book, Knowledge and the Flow of Information? I’m a big fan of his (and still mourning philosophy’s loss).

  22. Hi Tom and everyone else,

    First time here! First off, thanks for the interest in my work. Having people discuss your work (even critically) is better than having a stack of papers nobody reads.

    A few things regarding the paper and the original post:

    1.This paper is an independent work, and has nothing to do with the upcoming book. It is on arXiv because I have been advised to begin putting my papers on arXiv.

    2.The Famine of Forte results are not strictly NFL results, since we’re not looking at the mean performance uniformly averaged over all problems, but instead are considering the proportion of problems for which a fixed algorithm can perform well. This is stated clearly several times in the paper. The two are definitely related, but are nevertheless distinct.

    3. If you read the manuscript, you’ll see the definition of active information (given in section 5.2) remains unchanged. (I treat the quantity q in that case as a random variable, and I+ is defined as the negative log transform of p/q, as usual). My other results are in terms of p/q(T,F), which is a different quantity (since q(T,F) is the expected per query probability of success, where expectation is taken over histories and sequences of distributions). Nowhere in the paper (that I’m aware of) do I call this quantity active information. I’m not even sure I name it…it is simply the bound that arises during the derivations.

    4. I have no beef with AAIPQ or the previous work of the EIL. This is just a different way of doing things, which is better suited for the questions I personally care about (in machine learning).

    5. The Famine of Forte theorem is different from Ewert et al.’s ASC bound, but both can be derived using the same mathematical tools (expected forthcoming publication) and are both conservation theorems, loosely speaking. The latter is a distribution-free bound on the probability of different levels of ASC (bits), while the Forte result is a bound on the proportion of search problems providing a given expected per-query probability of success for a fixed algorithm. So I’m not quite seeing the strong analogy, but may be missing an unexpected connection.

    6. “Montañez has now acknowledged tacitly that they were wrong.” My independent work in machine learning, using a different and independently derived framework, says nothing about the work of the EIL. It only speaks to what my advisor and I found useful for exploring specific questions related to machine learning.

    Thanks again for reading my paper. I can probably answer a question or two on it, if anyone has some. I don’t have tons of time to check this blog, but peer feedback is always appreciated.

  23. I agree that trying to create a argument through mathematics that natural selection cannot solve the problem is silly.

    Nice to hear form you colewd.

    Yes, I tend to like the way Behe approached the issues vs. Dembski.

    In the case of Behe, I would have focused on other well known biochemical systems like ribosomes. Natural selection won’t work to evolve these system (especially ribosomes) because it would not be alive in the first place — the chicken and egg paradox. He would have avoided all the complaints of co-option if he used these as examples.

  24. phoodoo: I was discussing the quality of Tom’s posts, they are jerkish.That has nothing at all to do with moderation.

    And yours are as if written by a syphilitic imbecile who’s tiny mind is controlled by a gerbil residing in his rectum. Not a very clever rhetorical, rule-skirting trick either way.

  25. walto: Btw, this I off-topic, but do you know the late Fred Dretske’ book, Knowledge and the Flow of Information? I’m a big fan of his (and still mourning philosophy’s loss).

    I spent some time looking at what SEP has to say about Dretske, a year or two ago. All that stuck with me was “possible worlds.” (By the way, the reason I rarely engage in philosophical debates these days is that I have spent quite a bit of time perusing SEP, and have come to know how little I know about philosophy.)

    The book is screaming “read me” with its title. What comes up over and again in evolutionary informatics (the crypto-creationist kind) is the assertion: The system you studied wouldn’t have done what it did unless you had informed it to do what it did. And with that comes the presumption that you must have known how to do what you informed the system to do. There’s a lot to address here that I don’t feel terribly competent to address. But I will assert that knowing the mechanism of a system is not the same as knowing what occurs when the mechanism operates. I cannot imagine how anything we would call a mechanism would be anything other than constraints on the temporal evolution of a system. I very well may know constraints without knowing the consequences of the constraints. Thus I may gain information by observing a mechanism in operation, even when it is I who specified the mechanism. It would be a grave error to say, when the system does something remarkable, that I must have informed the system to do that specific something.

    There’s a connection of the foregoing to the crypto-creationist idea of “conservation of information,” but I’m not up to making it clear at the moment.

  26. stcordova: Natural selection won’t work to evolve these system (especially ribosomes) because it would not be alive in the first place

    So sayeth stcordova, all knowing, infallible.

  27. Tom English: But I will assert that knowing the mechanism of a system is not the same as knowing what occurs when the mechanism operates.

    If Wolfram accomplished nothing else with his automata, he demonstrated that a very simple algorithm can produce unpredictable output.

    That’s my reading.

    I see genes — some of them — producing effects 20 years after reproduction, that are critical to continuing the species. Design that.

  28. stcordova,

    In the case of Behe, I would have focused on other well known biochemical systems like ribosomes. Natural selection won’t work to evolve these system (especially ribosomes) because it would not be alive in the first place — the chicken and egg paradox. He would have avoided all the complaints of co-option if he used these as examples.

    Very good point. I think ATP Synthase also fits your criteria.

  29. Tom English: There’s a connection of the foregoing to the crypto-creationist idea of “conservation of information,”

    Since I believe that whatever is not explicitly pro-Christian is by definition anti-Christian is it OK if I call you a crypto-Satanist (typo corrected) while discussing your thoughts on this topic?

    peace

  30. GeorgeMontanez: If you read the manuscript, you’ll see the definition of active information (given in section 5.2) remains unchanged.

    This sounds like something specific that Tom should respond to.

    peace

  31. GeorgeMontanez: First time here! First off, thanks for the interest in my work. Having people discuss your work (even critically) is better than having a stack of papers nobody reads.

    Hey, George. I have not yet studied your paper as I would if I were reviewing it as a submission to a journal or a conference. I’m presently trying to focus (something I don’t do very well) on your past measurements of AAIPQ on evolutionary models. If I find later, when I give your paper the attention it deserves, that I’ve gotten something wrong, I will post again, and own up to my error.

    GeorgeMontanez: 1.This paper is an independent work, and has nothing to do with the upcoming book. It is on arXiv because I have been advised to begin putting my papers on arXiv.

    There’s now a note in the OP, above the fold, telling new readers that my guess is wrong, before they read it.

    GeorgeMontanez: 2.The Famine of Forte results are not strictly NFL results, since we’re not looking at the mean performance uniformly averaged over all problems, but instead are considering the proportion of problems for which a fixed algorithm can perform well. This is stated clearly several times in the paper. The two are definitely related, but are nevertheless distinct.

    Then most of what Wolpert and Macready referred to as NFL theorems are not NFL theorems. As I recall, there is a theorem “No Free Lunch Theorems for Optimization” that is somewhat similar to yours. I’m not going to look for it at the moment. I strongly recommend that you review the paper. (Transforming probabilities logarithmically, and making much of the fact that you’re talking about “information,” is about as empty as can be.)

    GeorgeMontanez: 3. If you read the manuscript, you’ll see the definition of active information (given in section 5.2) remains unchanged. (I treat the quantity q in that case as a random variable, and I+ is defined as the negative log transform of p/q, as usual). My other results are in terms of p/q(T,F), which is a different quantity (since q(T,F) is the expected per query probability of success, where expectation is taken over histories and sequences of distributions). Nowhere in the paper (that I’m aware of) do I call this quantity active information. I’m not even sure I name it…it is simply the bound that arises during the derivations.

    The definition (perhaps a definition) of active information is in Corollary 1, “Conservation of Expected Active Information” [emphasis added]. You do not define q clearly in the corollary, and it is possible that I erred in linking it to the “expected per-query probability of success q” in Equation 1. Here you’re saying that q is a random quantity, not an expected value. Either way, the meaning of q in your expression is different from the meaning of q in the three previous definitions of active information. In the previous definitions, q was a probability. It was not a random probability. It was not the expected value of a random probability. Having assigned a new meaning to one of the symbols in your expression of active information, you have redefined active information.

    GeorgeMontanez: 4. I have no beef with AAIPQ or the previous work of the EIL. This is just a different way of doing things, which is better suited for the questions I personally care about (in machine learning).

    Totally unacceptable response. It is reasonable to regard

        \[\frac{Q}{I_\Omega} = \frac{Q}{-\!\log_2 p} = \frac{Q}{-\!\log_2 \frac{|T|}{|\Omega|}}\]

    as a normalized measure of performance in hitting the target. Here the random quantity Q is the size of the sample when the sample first includes an element of the nonempty target set T \subset \Omega, and \Omega is the finite sample space. Referring as you do to I_\Omega as endogenous information, you would sensibly refer to the expected performance

        \[E\left[ \frac{Q}{I_\Omega} \right] = \frac{E[Q]}{I_\Omega}\]

    as the average number of queries per bit of endogenous information. You and your coauthors have never provided a rationale for taking the reciprocal of expected performance,

        \[I_\oplus = \frac{I_\Omega}{E[Q]} \not\equiv E\left[ \frac{I_\Omega}{Q} \right],\]

    which is the number of bits of endogenous information divided by the average number of queries, not the average number of bits of endogenous information per query. I have not managed to come up with a rationale, and it is not for want of trying. You are totally wrong — no ifs, ands, or buts — to convert endogenous information into active information with the name you give to I_\oplus, average active information per query. And you cannot tell me that it does not matter what you call the quantity, because you use the name, and nothing but the name, to invoke “conservation of [active] information” in papers where you measure average endogenous information per query (and take the reciprocal of the measurement).

    You have no mathematical justification for saying that the quantity I_\oplus is conserved. That is a huge and glaring defect in your studies of evolutionary models, where you measure only I_\oplus, not the active information. You are obviously too smart not to see the defect when it is laid out for you, and that is why your “I have no beef with AAIPQ” line is totally unacceptable.

    GeorgeMontanez: 5. The Famine of Forte theorem is different from Ewert et al.’s ASC bound, but both can be derived using the same mathematical tools (expected forthcoming publication) and are both conservation theorems, loosely speaking. The latter is a distribution-free bound on the probability of different levels of ASC (bits), while the Forte result is a bound on the proportion of search problems providing a given expected per-query probability of success for a fixed algorithm. So I’m not quite seeing the strong analogy, but may be missing an unexpected connection.

    I applied the term analogous to your theorem, and the term closely analogous to my theorem. You are saying that targets for which a ratio of measures is high are rare, as did Ewert et al.

    GeorgeMontanez: 6. “Montañez has now acknowledged tacitly that they were wrong.” My independent work in machine learning, using a different and independently derived framework, says nothing about the work of the EIL. It only speaks to what my advisor and I found useful for exploring specific questions related to machine learning.

    You are still addressing black-box search. You have improved upon the work that you previously did in black-box search with Marks, Dembski, and Ewert. I believe that your new measure of active information can be applied where you previously applied the misleadingly named measure of AAIPQ. Were your new measure applied where AAIPQ was previously applied, there would be mathematical justification for saying that information is in a sense conserved. Have I misunderstood?

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