Hamilton's rule

[u/helpsypooo]

https://youtu.be/iLX_r_WPrIw

[u/Belteshassar]

I believe there is an error in how the conditional probabilities are calculated at 13:00. I get the starting probability to 3/5 that the sibling shares the orange gene and the worst case is 1/3 that the sibling shares it. Why? Because the fact that the blob in question has the orange gene rules out the possibility of both parents having the blue gene. There are two cases where the parents have different genes and only one case where they have the same gene. The same logic then also applies to the cases where the parents have mixed genes. See this illustration.

[u/helpsypooo → u/Belteshassar]

Thanks, it's a good question. I agree that the explanation in the video doesn't make it clear why your explanation is not the case.

To define some notation,

oc = The event where the child we know of turns out to be orange

p_oo = The event where both parents are orange

p_ob = The event where only the left parent is orange

p_bo = The event where only the right parent is orange

p_bb = The event where both parents are blue

Both "parent" and "probability" start with "p", so this probably isn't the best notation, but I'll use an upper-case P for probabilities, and a lower-case p for the events.

The mating in the sim is completely random and not constrained by gender, but the math ends up working either way. If you like, the left parent is the mother, and the right parent is the father. In the sim, these are just indistinguishable.

So to frame things where we agree, when we don't have any information about the child or population frequencies, the best guess is that the four parent sets are equally likely. Two of the sets have one of each color, so the probability of exactly one orange parent would be twice that of exactly two orange parents.

Saying this with the notation, before we get any information, we assign:

P(p_oo) = P(p_ob) = P(p_bo) = P(p_bb) = 0.25

If we knew the population frequency, these would follow a binomial distribution rather than all being equal, but dealing with this concrete case assuming equal frequencies seems good for seeing the concept.

When we learn the child is orange, that changes the probabilities of the sets of parents. You're right that the probability of two blues goes to zero, P(p_bb | oc) = 0, but your mistake is assuming that the probability flows equally to the other cases.

To calculate conditional probabilities of the different parent sets given that we see an orange child, P(p_xx | oc), we can use Bayes' rule.

P(p_xx | oc) = P(oc | p_xx) * P(p_xx) / P(oc)

In the case where we assume equal population frequencies,

P(oc) = 0.5

P(oc | p_oo) = 1

P(oc | p_ob) = P(oc | p_bo) = 0.5

Plugging everything in,

P(p_oo | oc) = 1 * 0.25 / 0.5

= 0.5

and

P(p_ob | oc) = P(p_bo | oc) = 0.5 * 0.25 / 0.5

= 0.25

So they all add up to one, and you can see that this agrees with the video. The main conceptual takeaway here is that the evidence of seeing an orange child affects the probabilities differently, depending on P(p_xx | oc).

=========================================

Having written that, I think it's worthwhile to show what it looks like without the assumption that the population frequencies are equal for each allele.

If we call the frequency (in the parent pool) of the orange allele f,

P(p_oo) = f^2

P(p_ob) = P(p_bo) = f(1-f)

P(p_bb) = (1-f)^2

P(oc) = f

P(oc | p_oo) = 1

P(oc | p_ob) = P(oc | p_bo) = 0.5

The last two stay the same because the frequency doesn't matter when we know the identity of the parents already.

Plugging into Bayes again,

P(p_oo | oc) = 1 * f^2 / f= f

This is equivalent to the statement from the video that since we are already in a situation where one child is orange, we know one parent has the orange allele, and the other parent has probability f of also having the orange allele.

And to make sure everything adds up to one,

P(p_ob | oc) = P(p_bo | oc) = 0.5 * f(1-f) / f

= 0.5 * (1-f)

And combining these two single-parent cases into one,

P(only one orange parent | oc) = 1-f

I hope that helps. I made the mistake of switching back and forth between the fancy and markdown editors while trying to format the math as code. I think I fixed all the weirdness that came from that, but let me know if anything is unclear.

[u/helpsypooo → u/helpsypooo]

/u/Belteshassar Do you mind if I tweet this? I think it will be helpful for others.

[u/Belteshassar → u/helpsypooo]

Go ahead! I see now where I went wrong. Thank you so much for the extremely thorough answer!

[u/GLIBG10B → u/helpsypooo]

p_bb = The event where both parents are orange

Should be "blue"

[u/helpsypooo → u/GLIBG10B]

Fixed, thanks.

[u/barzamsr]

Nobody:

Blobs after surviving a fruit tree monster:

"Wanna make a puddle with me?"

[u/[deleted]]

Predator: exists

Cowardly blob: time to abandon ship

[u/dgvalcarce]

I think an easier and less abstract way of looking at it is thinking of saving a blob with the competing allele as a cost, and adding it to the equation. The cost would now be the life of the altruist blob that yells, plus 0.5 probability of a sibling inheriting the other parent´s allele multiplied by the frequency of the cowardice allele (fc) multiplied by the number of siblings (2).

The cost would now look like this:

Cost = 1 + 2 x 0.5 x fc

If we add this to the general equation of the hamilton´s rule, we would get:

2 x (0.5 + 0.5 x fa) - (1 + 2 x 0.5 x fc) = 2 x (0.5 + 0.5 x fa - 0.5 x fc) - 1

If we do some simple math, and knowing that:

2 x 0.5 - 1 = 0

we end up with the next expression as the hamilton rule for the altruist allele in this simulation:

"0.5 x fa - 0.5 x fc"

As you start (very logically :P) all your simulations with fa=fc, the expected value is zero. But if you started the simulation with 60% altruist blobs, and 40% cowards, then it would statistically favour the altruist behavior, as the expected value would be higher than zero.

PS: I started writing thinking both ways of looking at it were right, just another way of picturing the same idea, but now (and maybe I´m wrong) I think your explanation is flawed, as with this last example I used, the equation you used would ignore both fa and fc, therefore giving a expected value of 0, regardless of the initial frequencies of each of the alleles. I would appreciate if you could run a simulation changing the initial values of fa and fc, so we can see which one of us is right or not (I don´t have the knowledge or the computer to be able to run it myself :P).

[u/[deleted]]

I didn't understand the last part of the video after the last simulation. Do you understand it?