Note that Dembski has uploaded a revised manuscript which now correctly attributes the measure to Renyi and thanks the many critics for their contributions
I am not a mathematician but let me give it a try and others can amend and revise my comments.
The Kantorovich/Wasserstein distance metric is also known under such names as the Dudley, Fortet Mourier, Mallows and is defined as follows.
d_p(F,G) = \overset{\inf}{\tau_{x,y}} \lbrace E |x-y|^{\frac{1}{p}} \rbrace
where E(x) refers to the expectation of the random variable x and \inf means that the minimum is sought on all random variables X which take a distribution F and random variables Y which take a distribution G.
where \tau_{x,y} is the set of all joint distributions of random variables X and Y whose marginal distributions are F and G.
These metrics define a ‘distance’ between two stochastic distributions and are one of many such metrics that have been mathematically defined. There is a good paper on many of these metrics On Choosing and Bounding Probability Metrics. Different circumstances ask for different distance metrics.
These metrics have found applicability in non-linear equations, variational approaches to entropy dissipation, Phase transitions and symmetry breaking in singular diffusion, random walks, Markov processes and many more. Needless to say these metrics are quite commonly applied in a variety of applications. Applications of this metric to Markov processes may be of interest to evolutionary theory.
Adapted from Central Limit Theorem and convergence to stable laws in Mallows distance
Another way of looking at this is by assuming one has two samples X and Y of the same size X=\lbrace x_1,…,x_n \rbrace and Y=\lbrace y_1,…,y_n \rbrace. The Mallows distance between empirical distributions is
d_p(X,Y)= ( \frac{1}{n} \overset{min}{(j_1,…,j_n)} \sum_{i=1}^{n} \lvert x_i - y_i \rvert )^\frac{1}{p}
where the minimum is taken over all possible permutations of \lbrace 1, …, n \rbrace
Rachev, S. T. (1984), The Monge-Kantorovich problem on mass transfer and
its applications in stochastics, Theor. Probab. Appl., 29, 647-676.
As far as some interesting applications are concerned
6 Comments
Michael Buratovich · 13 August 2004
PvM,
Thanks for that explanation. I quit math after linear algebra and differential equations. which means that I am not a mathematician either. I am still trying to figure out why Dembski brought this up in his reply. I am also completely unsure why this method has more probabalistic power over the Renyi equation that was discussed. I know I'm a math idiot, but can one of you demi-gods out there help on this one.
MB
Pim van Meurs · 13 August 2004
As I am starting to 'understand' these issues, the Wasserstein metric presents a weak topology onto space. While the original Renyi measure provides what Demsbki calls "variational information" it needs to be tied in with the nature of the actual path(s) taken.
Thus he attempts to 'coordinate the variational information with the topology of the underlying probability space".
Where is Dembski going with this? I see this as working towards a measure that may be helpful in suggesting if there exist "probability paths". Probably to show that there exist 'irreducibly complex' systems to which the probability paths may be lacking. But so far I have failed to see how any of these measures may be helpful here.
steve · 13 August 2004
Pim, I think the broad outline of the argument is:
1 The dead-ends so vastly outnumber the workable arrangements evolution can find, if it searches randomly, it can't find them quickly enough
2 Evolution randomly searches all possibilities
3 Therefore, it couldn't have found all these workable arrangements
His critics seem to have said that 1 isn't certain, but 2 is just plain wrong. Since lots of smart people like Wolpert say his arguments are failures, I haven't bothered to study them in depth myself. Life is short.
hmmm · 16 August 2004
that paper on minimum entropy probability paths between genome families is pretty funny. (I figure you probably googled it and didn't have the chance to read it).
those guys have a total lack of understanding of biology. their basic idea is to take a DNA sequence, compute an AGTC frequency vector, and then talk about "minimizing the entropy path integral" during a sequence of base pair substitutions en route to a destination DNA sequence.
"Minimizing the entropy path integral"?!?
where did that come from? it seems these guys really think that the overall composition is influenced *not* by horizontal transfer conditions, or, say, the rest of the genome's base-pair content (reasonable speculations)....but by the idea that base pair frequencies stay away from an entropy maximizing (.25,.25.,25.25) during the move from one sequence to another. DUBIOUS, to say the least. it's more likely that base pair frequencies will track the genome content as a whole, or that local area of the chromosome.
But the best part of the whole paper is when they claim that the *advantage* of their approach is that "this allows us to compare sequences based on their composition as a whole, rather than by sequence alignment". jeeez...as if throwing away all the order information is going to *improve* your understanding of evolutionary relationships between sequences! In these guys' world, AGTC and ACTG are equivalent. Forget sequencing the genome, guys...let's just go back to chargaff's rules!
Pim van Meurs · 16 August 2004
Pim van Meurs · 16 August 2004
See also "A new distance measure for comparing sequence profiles based on path lengths along an entropy surface" by Gary Benson which explains the motivations for this new measure