Sunday, May 31, 2015

Uncommon Descent in Numbers - 2nd edition

Three years ago, I put up some pictures showing the number of comments and threads at Uncommon Descent. Now seems to be a good occasion to up-date some of this information.

1. Google Trends

Look for yourself: The phrase Uncommon Descent was most searched for in 2008. After that, everybody had bookmarked the site, so further googling became unnecessary. The same holds true for The Panda's Thumb - both sites are equally popular...

2. Threads per Month

The number of new threads per month peaked in 2011, but is still on a high level - though it seems to be decreasing. What makes all the difference is "News" - a.k.a. Denyse O'Leary - adding her news items. While in 2011/2012, those often were left uncommented, since 2013, they attract the attention of her fellow editors (though I got the impression that some commentators use them for their off-topic-remarks, while others just cannot let the copious factual inaccuracies stand uncommented.)

Monday, May 25, 2015

The Natural Probability on M(Ω)

Two weeks ago, Dr. Winston Ewert announced at Uncommon Descent a kind of open mike. He put up a page at Google Moderator and asked for questions. Unfortunately, not many took advantage of this offer, but I added three questions from the top of my head. The experience made me revisit the paper A General Theory of Information Cost Incurred by Successful Search again, and when I tried - as usual - to construct simple examples, I run into further questions - so, here is another one:

In their paper, the authors W. Dembski, W. Ewert, and R. Marks (DEM) talk about something they call the natural probability:

Processes that exhibit stochastic behavior arise from what may be called a natural probability. The natural probability characterizes the ordinary stochastic behavior of the process in question. Often the natural probability is the uniform probability. Thus, for a perfect cube with distinguishable sides composed of a rigid homogenous material (i.e., an ordinary die), the probability of any one of its six sides landing on a given toss is 1/6. Yet, for a loaded die, those probabilities will be skewed, with one side consuming the lion’s share of probability. For the loaded die, the natural probability is not uniform.
This natural probability on the search space translates through their idea of lifting to the space of measures $\mathbf{M}(\Omega)$:
As the natural probability on $\Omega$, $\mu$ is not confined simply to $\Omega$ lifts to $\mathbf{M}(\Omega)$, so that its lifting, namely $\overline{\mu}$, becomes the natural probability on $\mathbf{M}(\Omega)$ (this parallels how the uniform probability $\mathbf{U}$, when it is the natural probability on $\Omega$, lifts to the uniform probability $\overline{\mathbf{U}}$ on $\mathbf{M}(\Omega)$, which then becomes the natural probability for this higher-order search space).
As usual, I look at an easy example: a loaded coin which always shows head. So $\Omega=\{H,T\}$ and $\mu=\delta_H$ is the natural measure on $\Omega$. What happens on $\mathbf{M}(\Omega)= \{h\cdot\delta_H + t\cdot\delta_T|0 \le h,t \le 1; h+t=1 \}$? Luckily, $$(\mathbf{M}(\{H,T\}),\mathbf{U}) \cong ([0,1],\lambda).$$ Let's jump the hoops:
  1. The Radon-Nikodym derivative of $\delta_H$ with respect to $\mathbf{U}$ is $f(H) = \frac{d\delta_H}{d\mathbf{U}}(H) = 2$, $f(T) = \frac{d\delta_H}{d\mathbf{U}}(T) = 0$
  2. Let $\theta \in \mathbf{M}(\{H,T\})$, i.e., $\theta= h\delta_H + t\delta_T$. Then$$\overline{f}{(\theta)} = \int_{\Omega} f(x)d\theta(x)$$ $$=f(H)\cdot\theta(\{H\}) + f(T) \cdot\theta(\{T\})$$ $$=2 \cdot h$$
Here, I have the density of my natural measure on $\mathbf{M}(\Omega)$ with regard to $\overline{\mathbf{U}}$, $$d\overline{\delta_H}(h\cdot\delta_H + t\cdot\delta_T) = 2 \cdot h \cdot d\overline{\mathbf{U}}(h\cdot\delta_H + t\cdot\delta_T).$$ But what is it good for? For the uniform probability, DEM showed the identity $$\mathbf{U}=\int_{\mathbf{M}(\Omega)}\theta d\overline{\mathbf{U}} .$$ Unfortunately, for $\int_{\mathbf{M}(\Omega)}\theta d\overline{\delta_H}$, I get nothing similar: $$\int_{\mathbf{M}(\Omega)}\theta d\overline{\delta_H} = \frac{2}{3}\delta_H + \frac{1}{3}\delta_T$$

So, again, what does this mean? Wouldn't the Dirac delta function be a more natural measure on $\mathbf{M}(\Omega)$?

I hope that Dr. Winston Ewert reacts to all of the questions before Google Moderator shuts down for good on June 30, 2015...

Monday, May 11, 2015

Five Years of "The Search for a Search"

The Journal of Advanced Computational Intelligence and Intelligent Informatics published the paper The Search for a Search: Measuring the Information Cost of Higher Level Search of William A. Dembski and Robert J. Marks II (DM) in its July edition in 2010. With the five year jubilee of the publication coming, it seems to be appropriate to revisit a pet peeve of mine...

(Shell game performed on Karl-Liebknecht-Straße in Berlin, photograph by E.asphys)

Imagine a shell game. You have observed the con artist for a while, and now you know:

  1. The pea ends up under each of the three shells (left, middle, and right) with the same probability, i.e., $$P(Pea=left)=P(Pea=middle)=P(Pea=right)=1/3$$
  2. If the pea ends up under the left or the middle shell, you are able to track its way. So, in these cases, you will find the pea with probability 1 $$P(Finding\,Pea|Pea=left)=P(Finding\,Pea|Pea=middle)=1$$
  3. However, if the pea ends up under the right shell, in 999 times out 1000, you make a mistake during your tracking and be convinced that it is under the left or the middle shell - the probability of finding this pea is 1/1000$$P(Finding\,Pea|Pea=right)=1/1000$$

You are invited to play the game. Should you use your knowledge (method $M_1$), or should you chose a shell at random (method $M_2)$?

Sunday, September 28, 2014

Conservation of Information in Evolutionary Search - Talk by William Dembski - part 5

For an introduction to this post, take a look here. As I ended part 4 quite abruptly, this section starts in the middle of things....

Part 4: 45' 00" - 52' 50"

Topics: What is Conservation of Information? Example continued.

William Dembski: These tickets have probability 1/2, 1/2, 1/2, 1/2, and this one ticket has probability 1. If I happen to get this ticket, I have probability 1/2 of choosing curtain 1, but it is also probability 1/9 of getting that ticket. When you run the numbers, at the end of the day, by using these tickets, I'm not better of than I was originally. It is still only a probability of 1/3 of finding curtain 1, of finding the prize there. Once one factors in how did I limit myself to these tickets in the first place. Going from this whole space to this, that is information intensive. I have ruled out certain possibilities, that incurs an information cost. As I said, the cost is 5/9. It is really just an accounting thing. That is what conservation of information is. Once you factor in the information that it takes to get the search, get a search which has improved the probability for finding your original target, we haven't gained anything. It is called Conservation of Information, as the problem can even get worse. At this case, we have broken even, we are back to 1/3 for the probability of getting the prize, but let's say, you really want to improve the probability, you want to guarantee that you get that prize with this tickets. Well, then you have got only one ticket that will work for you.

Conservation of Information in Evolutionary Search - Talk by William Dembski - part 4

For an introduction to this post, take a look here.

Part 4: 31' 25" - 45' 00"

( I had to pause at 45', there is such an elementary mistake in Dembski's math, it was just to funny...)

Topics: What is Conservation of Information?

William Dembski: Now let us get to the heart of things "Conservation of Information". What is that conservation? Let me put on the next slide.

William Dembski: This is probably the most gem-packed slide in this talk. I want to make a distinction between -what I call - probable and improbable events, and probable and improbable searches. An improbable event is just something that is high in improbability: flip a coin a thousand times, get a thousand heads in a row. Highly improbable. It happens: if you believe in a multi-universe, then there is a universe where this is happening, where someone like me is speaking, my double-ganger flips a coin over the next hour and sees 1000 heads in a row. Probable and improbable search, that is where what is the probability that a search is successful. It is not so much asking whether it actually succeeds, it is not concerned with the result. It is concerned with the probability distribution associated with the search. This is an important distinction because so many intelligent design arguments look for a discontinuity in the evolutionary process. We look for highly improbable events. Such as the intelligent design people: you get for instance Thomas Nagel's "Mind and Cosmos". He is basically looking at probabilistic miracles. Think how the origin of life undercuts a materialistic understanding of biology. So he is looking into improbable events. That is what we do when we try to find evidence for a discontinuity. What I'm doing in this talk is saying, look, I'm going to give you evolution, give you common ancestry, all of that. That is no problem. What I'm interested though is the probability of success for a search.

member of the audience: What are we searching for?

William Dembski: It is whatever the target happens to be.

Saturday, September 27, 2014

Conservation of Information in Evolutionary Search - Talk by William Dembski - part 3

For an introduction to this post, take a look here. There is some interaction with the audience (15'30" - 18'00") which I wasn't able to understand fully. Any help is appreciated!

Part 3: 12' 45" - 31' 25"

Topics: What is an evolutionary search?

William Dembski: Now let's add this next term evolutionary. What does evolutionary - when we put it in front of search - add to the discussion? I think it changes one key aspect here. Whereas we were looking at some query feedback, now this query feedback takes the form of fitness: how good is it? Query feedback can be quite general. Maybe the query feedback is nothing, when we examine it. Or maybe the query feedback may just say "I'm in the target" or "I'm not in the target". That would be very simple. Fitness is going to give some sort of range of values that ideally identify how close am I to the target.

William Dembski: There are examples of evolutionary search. There is the Dawkins' weasel example from his book "The Blind Watchmaker", that is the one I'm going to focus on here. Then there are various - what I would regard as - embellishments of that, because I don't think that there is anything fundamentally new about them. There is MSU's Avida program, Tom Ray's Tierra, Schneider's ev. What is at the heart of these programs that these are computer programs which mimic - try to mimic - Darwinian evolutionary processes. What are they supposed to show? That is interesting. Look at the history of this field of evolutionary computing and there is a reason why people wanted to do evolution in the computer. That is because the computer would allow evolution to be done in real time, because we cannot really see it in real time in the wild.

Friday, September 26, 2014

Conservation of Information in Evolutionary Search - Talk by William Dembski - part 2

For an introduction to this post, take a look here. This is quite a short section, with some annotations from me.

Part 2: 09' 40" - 12' 45''

Topics: What is a search?

William Dembski: We talked about information. Let's now look at that second key term "Search". What is a search. There are seven key components in a search.

William Dembski: You have a search space, you have a target - we are looking for something in the search space. There is initialization - where do we start off? There is a query limit - how many things in the search space can we check out? There is query feedback - when we have checked out, when we have located some item - what is it telling us about itself in terms of how it relates to the target? There is an update rule - once we have queried something, what do we query next? And then finally a stop criterion - when do we stop? How do we know that we have done enough? This is very general.