# The LCI and Bertrand’s Box

Tom English has recommended that we read Dembski and Marks’ paper on their Law of Conservation of Information (not to be confused with the Dembski’s previous LCI from his book No Free Lunch). Dembski also has touted the paper several times, and I too recommend it as a stark display of the the authors’ thinking.

Most people won’t take the time to carefully read a 34-page paper, but I submit that the authors’ core concept of “conservation of information” is very easily understood if we avoid equivocal and misleading terms such as information, search, and target. I’ll illustrate it with a setup borrowed from Joseph Bertrand.

The “Bertrand’s box” scenario is as follows: We’re presented with three small outwardly identical boxes, each containing two coins. One has a two silver coins, one has two gold coins, and one has a silver coin and a gold coin. We’ll call the boxes SS, GG, and SG. We are to randomly choose a box, and then randomly pull a coin from the chosen box.

It’s clear that we’re equally likely to end up with a gold coin or a silver coin, so the probability of getting a gold coin (our preferred result) is `1/2`. This unconditional probability of `1/2` gets updated when we choose a box. If we choose GG, then our odds increase from `1/2` to `1` — that is, they change by a factor of `2`, so we’ll say that choosing GG gives us a probability gain (`β`) of `2`. Likewise, SG gives us a `β` of `1`, and SS a `β` of `0`. Note that the probability of choosing a given box (`1/3`) doesn’t exceed `1/β` for that box. This is an example of the following universal fact of probability:

If event `E1` updates the probability of event `E2` by a factor of `β`, then the probability of `E1` is at most `1/β`.

Since `β` is `P(E2|E1)/P(E2)`, the above statement says that `P(E1) ≤ P(E2)/P(E2|E1)`. This is very simple to derive, starting with the following truism:

`P(E1 & E2) ≤ P(E2)`

restating:

`P(E2|E1)*P(E1) ≤ P(E2)`

and dividing both sides by `P(E2|E1)`:

`P(E1) ≤ P(E2)/P(E2|E1)`

QED.

This says that a large gain in probability is obtained at the “cost” of an unlikely event. Dembski and Marks’ LCI is nothing more than an application of this fact to cases in which `P(E1)` and `P(E2)` are based on uniform distributions (which I’ll show in a comment*). So not only is their LCI trivially derivable, but it’s also based on two assumptions of uniform probability, a.k.a. tornado-in-a-junkyard assumptions.

Contrast the simplicity of this concept with the import of the authors’ claims that “the Law of Conservation of Information shows that Darwinian evolution is inherently teleological” and “LCI underwrites the conclusion that Darwinian evolution is teleologically programmed with active information”. These are claims that can be made to appear plausible only through copious amounts of obfuscation and equivocation, which is exactly what you’ll find if you read the paper.

* It appears that subscripts don’t work in comments, so I’ll add this as a footnote to the OP. In case anyone from the Evo Info Lab ever reads this and has some doubts, the following shows that Dembski and Marks’ LCI is an application of

`P(E1) ≤ P(E2)/P(E2|E1)`    (Eq. 1)

Assume:
1) The probability distribution over `Ω1` is uniform.
2) The unconditional (i.e. prior to `E1` or `~E1` being realized) probability distribution over `Ω2` is also uniform.

(I should note that Dembski and Marks never actually mention the second assumption, but it holds for all of their examples, and without it their LCI is false. It was Atom, another member of Evo Info Lab, who pointed out this tacit assumption to me.)

If we define `E1 ⊆ Ω1` as the set of all outcomes that confer a probability of at least `q` on `E2 ⊆ Ω2`, then it follows that ` q ≤ P(E2|E1)`. Furthermore, we define `p1` and `p2` as the probabilities conferred on `E1` and `E2` by uniform distributions over `Ω1` and `Ω2` respectively. Then substitution into Eq. 1 is straightforward, giving:

`p1 ≤ p2/q`

which is precisely Dembski and Marks’ LCI.

## 25 thoughts on “The LCI and Bertrand’s Box”

1. Having read through Life’s Conservation Law I find it is a longer and more popular version of their previous Search For a Search papers. But the same objection applies.

They point out that fitness surfaces are much smoother than they would be if fitness values were assigned randomly to genotypes. Natural selection (with mutation) will be much more successful on the real fitness surfaces than it would be on “white noise” fitness surfaces. Single mutations change fitness by much less than it would change if you mutate every site in the genome.

The calculation of “active information” simply expresses that smoothness of fitness surfaces. But their vision of a quantitative proof of teleological “front loading” omits the possibility that the laws of physics constrains the kinds of fitness surfaces we could have. Not everything in our universe interacts strongly with everything else. My typing these words does not cause the roof of your home to leak. That’s because of inverse-square properties of physical laws.

If the Teleological Designer set up the universe so that these were the laws of physics, that is basically a fine-tuning argument. It argues that the Teleologist could have set up the universe so that my typing damages your roof. It does not at all show that given the physical laws that we have we need assume that a Designer has further intervened.

2. I don’t understand this part:

Note that the probability of choosing a given box (1/3) doesn’t exceed 1/β for that box.

1/β for GG=1/2
1/β for GS= 1/1
1/β for SS= 1/0

What am I not seeing?

3. gah!

I’ve always had problems with left-right reversal, and it extends to greater than or less than, even in words! My number line is terrible, and I’m prone to subtract instead of add.

Thanks 🙂

4. Got to page 13. Oh, dear. At least it sounds as though Dembski has finally seen the problem that ID faces.

But, boy.

WEASEL again. And from the fact that his own program MESA doesn’t produce IC critters, and AVIDA does, he draws the conclusion that you can make any conclusion you like from an evolutionary simulation! Um, no you can’t. One Black Swan, Dr Dembski.

And so dismissive about Lenski’s beautiful, and mind-boggling work on E-coli!

OK, I guess I’d better get to grips with the math….

5. Even if the LCI didn’t exist, there still wouldn’t be any evidence that necessity and chance can produce a living organism from non-living matter nor transform a population of E. coli into something other than E. coli.

6. Elizabeth:
But, boy.

WEASEL again.

These folks love “Weasel” because (1) it is Richard Dawkins, and they can give the impression that anyone who doesn’t want to be an atheist must then agree with them, and (2) there are many straw-man mischaracterizations of what Dawkins was trying to do. Which actually was just to provide a dramatic teaching example to counter the repeated creationist mischaracterization of natural selection as purely a theory of “random” change.

7. How different would two strains of E. coli need to become, before you’d be willing to accept that they were different? Would it help to give a new strain a different name?

From the viewpoint of, say, a cat, we might find that evolution has not transformed primates at all. Yeah, there are humans and gorillas and orangutans, but so what? They are still all primates! Yet the difference to us is critical. The difference between two strains of E. coli might be just as critical, at least to the bacteria.

8. Flint:
How different would two strains of E. coli need to become, before you’d be willing to accept that they were different? Would it help to give a new strain a different name?

From the viewpoint of, say, a cat, we might find that evolution has not transformed primates at all. Yeah, there are humans and gorillas and orangutans, but so what? They are still all primates! Yet the difference to us is critical. The difference between two strains of E. coli might be just as critical, at least to the bacteria.

Great reply to the one-trick pony.

9. Joe

there still wouldn’t be any evidence that necessity and chance can produce a living organism from non-living matter nor transform a population of E. coli into something other than E. coli.

“E. coli” is a name WE give to a particular collection of genomes, usually based upon their general phenotype for convenience. We could choose to name each different genome variant by some clumsy label – most completely, its full genotype sequence AATTATATATTACGCGGTATA … As soon as you change to, say, AATTATATATAACGCGGTATA, you have a new label. “Chance” can do that (never seen the need for that “necessity” bit, meself). And it can do it again: AATTATATATAACGCGGTGTA… And again: AATTATACATAACGCGGTGTA … Since every cell on earth is more fundamentally ‘named’ by its gene sequence (which may be unique, or may be shared by several instances), the point at which we humans decide that we have something we’d like to give a different name to is pretty arbitrary. Single changes don’t tend to be noticeable. A few thousand such changes do.

10. Possibly worth noting that cloning organisms don’t actually speciate, in the sense we apply that word to sexually reproducing organisms.

You just get different strains/lineages that differ substantially.

11. Allan and Flint-

Don’t blame me because your position doesn’t have any supporting evidence.

12. LoL! Your position is all tricks and no treats….

13. BTW, guys, Joe G’s heckling is within my rules, but I do suggest that if you want to engage with them, you start a new OP to do so.

14. “My” heckling? That ‘s a tad one-sided…

But anyway perhaps we need a sandbox 2.0

15. How does natural selection take the random variation and turn it into non-random change?

Natural selection is an output with three randomized inputs.

16. Elizabeth: BTW, guys, Joe G’s heckling is within my rules, but I do suggest that if you want to engage with them, you start a new OP to do so.

I don’t think his boilerplate incredulity hurts us. He’s a poster child of why ID is a negative argument.

17. Except ID is not just a negative argument. Eliminating evolutionism is just one mandatory component of the design inference.

That you refuse to understand that says quite a bit about you and your relegation to cheerleading…

18. Better yet, I want an ID proponent to work backwards from the Dog/Wolf lineage and demark the basal form from which no prior ancestor was possible (preferably by identifying those traits with de novo CSI): Was it Canis lepophagus, Eucyon davisi, Hespercyon gregarius, Miacis cognitus, or perhaps some other species?

19. I think this is asking for too much – namely, some reading and understanding. Hard enough to get across that the descendents of primates will ALWAYS be primates, forever. Even that concept is pretty advanced, you know.

20. Will descendents of prokaryotes always be prokaryotes, forever?

Will descendents of fish always be fish, forever?