[From Bill Powers (2009.08.01.1623 MDTY)]
Fred Nickols (2009.08.01.1342 EDT) -0-
Hmm. I take that to mean that when I flip a coin there is in fact (some of the time) a certain flip and a certain position of a certain coin on my thumb that will result in the coin coming up heads (or tails as the case may be). As a flawed and variable output human being, I can't achieve that particular configuration or exert that particular behavioral output in exactly the same way every time.
You're so nice, Fred, that you wouldn't dream of accusing me of cheating. But I did. Don't overcomplicate reorganization.
When I flip the coin in the air, I do it a little differently each time and the coin spins a different number of times and lands at different angles and so on and so forth, with the net result that there is a 50-50 chance of getting heads (or tails if you intend to produce tails). So you have a 50% chance of getting the result you want on the first toss. Viola!
But what if you don't get the right result the first time? Why, you pick up the coin and toss it again. You have a 75% chance of getting the result you want in two tosses, a 99.6% chance of getting it in 8 tosses. That's how random reorganization works: if you don't get the result you want the first time, you reorganize again immediately. And you keep doing that until you succeed.
Suppose there's a jar holding 100 white marbles and 10 black marbles. The jar is covered with a black cloth so you can't see inside it, and all the marbles feel the same. If you announce your intention of pulling out a black marble by picking blindly, what are your odds of doing it if you start with the same mix of marbles every time (that is a broad hint).
If you use the principle of PCT reorganization, the odds are as high as your patience permits. If you don't get the black marble on the first trial, you throw the white one back in, shake the jar, and reorganize again. According to the principle of reorganization, you keep reorganizing every time the wrong result is obtained, so after 65 reorganizations you'd have a 99.9% chance of getting a black marble (that is, a 99.9% chance of not getting 65 white marbles in a row). If you chose white, it wouldn't take so long. Note that I didn't say your intention was to pull out a black marble THE FIRST TIME. I just said that you intended to pull out a black marble. You may have imagined that I meant "the first time," but I didn't say that. You keep pulling marbles out and throwing white ones back in until you get a black one as you intended. Even though you are selecting them at random, you can control for getting a black one with a high probability of (eventual) success.
Is it sinking in? How do you shake exactly 7 pills out of a bottle of pills so you can fill your 7-day pill dispenser? Well, you shake some pills out and count them. If there are less than 7, you shake some more into your hand. If you have more than 7 you throw the excess back in the bottle. You can't control the number of pills that come out very well, but since you can keep reorganizing as long as the number isn't right, you will end up with exactly 7 pills every time. Random variation of output, selection process to control the result.
What are the chances of throwing a 7 at a dice table, getting a straight flush hand, betting on 7 and winning at the roulette wheel? If you're allowed to reorganize, it's essential 100% in each case. Oh, the casino won't let you keep trying until you get what you want? That would be cheating? Tough for the customers; they ought to stay out of places like that.
Do the E. coli reorganization experiment in Chapter 7 of LCS3. Start it with the Run button, then click on Tumble (bottom center) to change the direction of the moving red dot and (you hope) get it to the target circle in the center. After the first Tumble button click, pressing the space bar will do the same thing. Slow the speed if you have trouble reacting fast enough.
On each Tumble-click or space-bar press, a reorganization takes place: the red dot becomes a spinning bacterium for a moment (a nice touch by Bruce Abbott), then takes off in a new direction. The chances of getting the right direction as a result are pretty small, but they aren't zero. When you get the hang of it, you will find that an average of 10 to 15 tumbles is enough to get to the target. The cumulative number of tumbles per hit is shown at the bottom of the screen. You can click on a radio button to have a model do the clicking; it works faster but takes roughly the same number of tumbles on average to hit the target.
You can click another radio button to see how many random changes are needed to hit the target just by jumping around the two-dimensional plotting area at random. I get numbers like 50 times as many changes, with a far larger ratio if we reorganize in 3 dimensions. Maybe JRK can calculate the ratio for reorganizing in more dimensions than 2 at the same time. I think evolution, to be a feasible process, requires E. coli reorganization.
Best,
Bill P.