Gpuccio: Functional Information Methodology

[swamidass (S. Joshua Swamidass)]

@gpuccio, we will continue this for a bit, will likely split it into a new thread.

@gpuccio

10 objects with 50 bits of FI each are not, in any way, 500 bits of FI.

False.

@gpuccio

10 objects with 50 bits of FI each are 500 bits of FI only if those 10 exact objects are needed to give some new defined function.

Exactly, such as the combination of their 10 independent functions. Therefore, by your own definitions, 10 objects with 50 bits are exactly 500 bits of FI (defining function as the sum total of their function).

@gpuccio

Let’s see the difference.

You go on to explain something entirely unrelated.

@gpuccio

These functions are the small safes in my example.

The probability of getting a success in one attempt is, of course, 1:2^50.

How many attempts are necessary to get at least one success? This can be computed using the binomial distribution.

The result is that with 2^49 attempts we have a more than decent probability (0.3934693) of getting at least one success.

Your math is all wrong here. In your example above, that of the 100 safes, the success probability for each trial is 50%, and if you have, say, 20 trials, you expect to get about ten successes.

@gpuccio

How many attempts are necessary to have a decent probability of gettin at least 10 successes, each of them with that probability of success, each of them with 50 bits of FI?

Again, we use the binomial distribution.

The result is that with 2^53 attempts (about 16 times, 4 bits, the number of attempts used before) we get more or less the same probability: 0.2833757

That means that the probability of getting 10 successes is about 4 bits lower than the probability of getting one success. The FI of the combined events is therefore about 54 bits.

This also is all wrong. You need to use a modified version of an extreme value distribution.

@gpuccio

Why is that? Why do probabilities not multiply, as you expect?

Because your math is all wrong. Very easy. These are basic homework problems in an intermediate probability course. In your education to become a physician, you probably never had a chance to learn this. Just turns out you are doing the math wrong.

@gpuccio

It’s because the 10 events, while having 50 bits of FI each, are not generating a more complex function. They are individual successes, and there is no relationship between them.

The conceptual disconnect is so large here that I am tempted too call it confabulation.

@gpuccio

Those ten objects have 500 bits of FI only if, together, they, and only they, can generate a new function.

Of course they can, in all cases, if we define the new function as the sum total of all the functions.

@gpuccio

Please, give me feedback on this point, before I start examining the example of affinity maturation in the immune system.

This will be handled, also, in a new thread.

@gpuccio

By the way, I was forgetting: using the binomial distribution, we can easily compute that the number of attempts needed to get at least one success when the probability of success if 1:2^500 (500 bits of FI) is 2^499, with a global probabilty of 0.3934693.

Nope. As I just explained here: Information is Additive but Evolutionary Wait Time is Not. It appears you had some basic misunderstandings here of FI. I encourage you to catch up. This is an important area of science, and it is good to learn about.