October 26, 2021

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Fisher’s fundamental theorem (part 4)

I wrote an article summarizing my work relating natural selection to information theory:

• John Baez, the fundamental theorem of natural selection.

Check it out! If you have any questions or see any errors, let me know.

Just for fun, here’s the summary and introduction.

Summary. Suppose we have it n different types of self-replicating entity, with the population from jotype changing at a rate equal to Pine tree times fitness the end of this type. Suppose fitness the end it is any continuous function of all populations P_1,  dots, P_n. To leave Pine tree be the fraction of replicators that are of the joth type. Then p = (p_1,  dots, p_n) is a time-dependent probability distribution and we show that its velocity measured by Fisher’s information metric is equal to the variance of the physical form. In approximate terms, this says that the speed at which information is updated through natural selection is equal to the variance in fitness. This result can be seen as a modified version of Fisher’s fundamental theorem of natural selection. We compare it to the original result of Fisher played by Price, Ewens and Edwards.

Introduction

In 1930, Fisher stated his “fundamental theorem of natural selection” as follows:

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

Some tried to make this statement accurate as follows:

The temporal derivative of the physical form of a population is equal to the variance of its physical form.

But this is only true under very restrictive conditions, so a controversy erupted.

Price proposed an interesting resolution, which later amplified Ewens and Edwards. We can formalize your idea as follows. Suppose we have it n type of self-replicating entity and idealizes the population of the joth type as a function of real value

For more details on an easy-to-use blog format, read the entire series:

• Part One: The Darkness of Fisher’s Original Document.

• Part 2: An accurate statement of Fisher’s fundamental theorem of natural selection and the conditions under which it is maintained.

• Part 3: A modified version of the fundamental theorem of natural selection, which is much more general.

• Part 4: My work on the fundamental theorem of natural selection.

This entry was posted on Tuesday, July 13th, 2021 at 3:40 pm and is filed under probability biology, information and entropy. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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