Now I think you were correct and I failed to see what you meant at the time.  I should have spent more time considering your input and less time at crafting a smart sounding reply. Thanks!

Just one correction: orthologous and paralogous are not genes that are distinguished by being in the same location or not. They refer to the ultimate event that lead to the separation as two different genes, in the case of orthology refers to a speciation event and in the case of paralogous genes, to a duplication event. The terms are in my oppinion not great and I wish they would mostly disappear from scientific discussions. 

Just a general comment on style: I think this article would be much easier to parse if you included different headings, sections, etc.  Normally when I approach writing here in LessWrong I scroll slowly to the bottom, do some diagonal reading and try to get a quick impression about the contents and whether the article is worth reading. My impression is that most people will ignore articles that are big uninterrupted long chunks of text like this one

Strong upvoted for visibility and because this sort of posts contribute to create a healthy culture of free speech and rational discussion. 

Thank you for the comprehensive answer and for correcting the points where I wasn't clear. Also, thank you for pointing out that the Kolmogorov complexity of a program is the length of the program that writes that program

The complexity of the algorithms was totally arbitrary and for the sake of the example.

I still have some doubts, but everything is more clear now (see my answer to Charlie Steiner also)

I think that re-reading again your answer made something click. So thanks for that

The observed data is not **random**, because random is not a property of the data itself.
The hypotheses that we want to evaluate are not random either, because we are analysing Turing machines that generate those data deterministically.

If the data is HTHTHT,  we do not test a python script that is doing:

random.choices(["H","T"], k=6)

What we test instead is something more like

["H"] +["T"]+["H"]+["T"]+["H"]+["T"]

And

["HT"]*3

In this case, this last script will be simpler and for that reason, will receive a higher prior.

If we apply this is a Bayesian setting, the likelihood of all these hyptohesis is necessarily 1, so the posterior probabilty just becomes the prior (divided by some factor), which is proportional to the length of the program. This makes sense because it is in agreement with Occam's razor.

The thing I still struggle to see is how I connect this framework with probabilistic hypothesis that I want to test, such as the data was generated by a fair coin. One possibility that I see (but I am not sure this is the correct thing) is testing all the possible strings generated by an algorithm like this:

i=0
while True:
   random.seed(i)
   random.choices(["H","T"], k=6)

The likelihood of the strings like HHTHTH is 0 so we remove them and then we are left only with the algorithms that are consistent with the data.

Not totally sure of the last part

The part I understood is, that you weigh the programs based on the length in bits, the longer the program the less weight it has. This makes total sense.

I am not sure that I understand the prefix thing and I think that's relevant. For example, it is not clear to me if once I consider a program that outputs 0101 I will simply ignore other programs that output that same thing plus one bit (e.g. 01010).

I also find still fuzzy (and know at least I can put my finger on it) is the part where Solomonoff induction is extended to deal with randomness.

Let me see if I can make my question more specific:

Let's imagine for a second that we live in a universe where only the next programs could be written:

• A) A program that produces deterministically a given sequence of five digits (there are 2^5 of this programs
• B) A program that produces deterministically a given sequence of 6 digits (there are 2^6 of them)
• C) A program that produces 5 random coin flips with p=0.5

The programs in A have 5 bits of Kolmogorov complexity each. The programs in B have 6 bits. The program C has 4

We observe the sequence O = HTHHT

I measure the likelihood for each possible model. I discard the models with L = 0

A) There is a model here with likelihood 1

B) There are 2 models here, each of them with likelihood 1 too

C) This model has likelihood 2^-5

Then, things get murky:

the priors for each models will be 2^-5 for model A, 2^-6 for model B and 2^-4 for model C, according to their Kolmogorov complexity? 

Yes, this is something I can see easily, but I am not sure how Solomonoff induction accounts for that 

I think this is pointing to what I don't understand: how do you account for hypotheses that explain data generated randomly? How do you compare a hypothesis which is a random number generator with some parameters against a hypothesis which has some deterministic component?

 Is there a way to understand this without reading the original paper (which will probably take me quite long)? 

When you understood this, how was your personal process that took you from knowing about probabilities and likelihood to understanding Solomonoff induction? Did you have to read the original sources or you found some good explanations somewhere?

I also don't get if this is a calculation that you can do in a single step or if this is a continuous thing. In other words, Solomonoff induction would work only if we assume that we keep observing new data? 

Sorry for the stupid questions, as you can see, I am confused.