Genetic Inheritance - Carrier Probability Question

[mathmannix]

I'm having trouble wrapping my head around this one. It should be a straight-up probability question.

It is about the probabilities of Hemophilia (or Haemophilia, or Hæmophilia) inheritance. For those who don't know, Hemophilia is an X-linked recessive genetic mutation; Females (XX) who have one defective X-chromosome will be carriers (and have a 50% chance of passing the chromosome to their children, sons or daughters), Females (XX) who have two defective X-chromosomes will have hemophilia (and have a 100% chance of passing the chromosome to their children, but I don't know if this has ever been recorded), and Males (XY) who have their one X-chromosome defective will have hemophilia (and will automatically pass the disease to their daughters as carriers, but never to their sons.)

Queen Victoria was a carrier for hemophilia. That means that one of her two X-chromosomes was affected, and presumably her husband Albert's X chromosome was unaffected, because then he would had suffered from the disease. She had 9 children. 1 of her 4 sons was affected (Leopold, who had hemophilia and passed it to his daughter), and at least 2 of her 5 daughters (Alice and Beatrice) were carriers (known because some of their sons and their daughters' sons had hemophilia.)

Let's take the case of one of her other daughters, Helena. Based on her ancestry, we can say that there was a 50% chance she was a carrier, right?

But hang on. Helena married and had children - three sons and two daughters. As far as I can tell, It was never recorded that any of her sons suffered from a bleeding disorder like hemophilia (even though one of them died at one week old), so I will assume they each had a normal X chromosome. Helena's two daughters never had children, so we cannot say if they were carriers or not; they give us no new information.

But based on her three sons, I want to say it was less likely that she was a carrier. But how likely? I mean, if she were a carrier, then there would only be a (.5)^3 = 0.125 chance that her 3 sons would not inherit her faulty X-chromosome. Those three events (sons) can be assumed to be independent, but they're not independent of the probability that Helena herself was a carrier.

The gene can be passed down the female line without a haemophiliac son being born, but as the family line continues and no haemophiliac sons are born, it becomes less likely that a certain ancestor had the gene and passed it on through the female line.

Or is this just faulty logic and I should just keep saying that Helena had a 50% chance of being a carrier, and her daughters had 25% chances?

[doogly]

She had a 50% chance of getting the gene.

But now you want to say, did she actually get it? You can update the prior because you have new information. This is a job for Bayes' Theorem!

p(A|B) = p(B|A) * P(A) / P(B)

P(A|B) = probability that helena has the gene given that we know one son does not
P(A) = probability helena has the gene = .5
P(B|A) = probability that one son does not have the gene given that helena has it = .5
P(B) = probability that her son does not have the gene = 1 (or maybe we're not sure he doesn't?)

But if we use P(B)=1, we would now, after the first son, say that the probability Helena was a carrier is only 25%. If we have two more observations of hemophialess sons, we would get all the way down to a 1/16.

Again, this is not to say her original chances were 1/16; they were 1/2. That's because we are adding evidence to a single chance event.

[mathmannix]

So then can I say that there's a 1/32 chance that her daughters were carriers?
But it doesn't change the sons' probabilities of having gotten it to 1/32 because that would be circular, so they're still at 25%?

[doogly]

The sons' probability of having gotten it is now 0, because we're now in the business of updating based on evidence. They can't be characters, they have it or don't, and we observed that they don't.

If Helena were still with us and churning kiddies out for the sake of England, we'd say well, there's a 1/16 chance you're a carrier, and so your *next* son has a 1/32 shot. Based on the sons. Based on the daughters, we'd have to know whether they were carriers or not to update the probability that their mother is a carrier (and thereby update the probability that her next son is a carrier) - in order to know that about them, we'd have to look at their children.

[mathmannix]

OK, let's say that Helena's daughter Marie Louise had married and had two sons, neither of whom had hemophilia. How could I calculate her probability, because it changes her mother's probability?

It would change all the probabilities, of course. Helena still originally had a 50% chance of inheriting the gene, which gave Marie Louise a 25% chance of inheriting it. Then Marie Louise had two normal sons, making the odds that she actually had the gene 0.125 based on her children only, but her odds should be lower because of her brothers, right? I got confused again.

[doogly]

aight I see what's confusing. the thing is we don't want to double count things.

if we're trying to ask, "what is the probability ML actually got the gene", we roll the evidence we have about her brothers up into the probability H had it, use that to determine her base probability, and then we can update it with the information that ML had two sons who also didn't have it.

if we're going back to ask, "what is the probability H actually got gene," we still use all that information, but we don't *double* use it to lower ML's starting probability.

so you still wind up using all available evidence, just where you account for it depends on how you are posing the problem

[ThirdParty]

The original probability that Helena would inherit the gene but none of her 3 sons would inherit it from her was 50%^4 = 6.25%.

The original probability that Helena would not inherit the gene was 50%.

So, given the information that Helena's 3 sons did not inherit the disease, the probability that Helena was a carrier is 6.25%/56.25%, which comes to 11.11%.

[gmalivuk]

She had a 50% chance of getting the gene.

But now you want to say, did she actually get it? You can update the prior because you have new information. This is a job for Bayes' Theorem!

p(A|B) = p(B|A) * P(A) / P(B)

P(A|B) = probability that helena has the gene given that we know one son does not
P(A) = probability helena has the gene = .5
P(B|A) = probability that one son does not have the gene given that helena has it = .5
P(B) = probability that her son does not have the gene = 1 (or maybe we're not sure he doesn't?)

But if we use P(B)=1, we would now, after the first son, say that the probability Helena was a carrier is only 25%. If we have two more observations of hemophialess sons, we would get all the way down to a 1/16.

Again, this is not to say her original chances were 1/16; they were 1/2. That's because we are adding evidence to a single chance event.

You wouldn't use P(B)=1, though, because on the right side you want your prior probabilities.

The prior odds of Helena having the gene are 1:1. The likelihood of no sons having it if she's a carrier is 1/8 as much as the likelihood of no sons having it if she's not a carrier, so now the posterior odds of Helena carrying the gene are 1:8, for a probability of 1/9

[doogly]

Aight so I worked it out, the better way to get P(B) with more details is rather than

p(A|B) = p(B|A) * P(A) / P(B)

set it up with

P(A|B) = p(B|A) * P(A) / [ P(B|A)P(A) + P(B|not A)P(not A)]

P(A|B) = probability that helena has the gene given that one son does not
P(A) = prior probability helena has the gene = 1/2 (as before)
P(B|A) = probability that one son does not have the gene given that helena has it = 1/2 (still)
P(not A) = 1/2
P(B | not A) = 1

so P(she has it | one son doesn't) = 1/3
and then
P(she has it | three sons don't) = 1/9

so we cool

fractions, man

[gmalivuk]

I'm a fan of the odds ratio formulation of Bayes, precisely because I can never quite remember which probabilities to use in the conditional probability formulation.

Whereas you just multiply the prior odds by the likelihood ratio to get the posterior odds, and a bit of intuition about the direction the new information should shift things should be enough to make sure you do it the right way around.

[ThirdParty]

I find it most intuitive to just use the basic definition of conditional probability:

P(H|E) = P(E&H)/P(E) = P(E&H)/[P(E&H)+P(E&~H)]

Identify the set of possible worlds consistent with the evidence. Figure out what proportion of those worlds (weighted by their prior probabilities, if they're not all equal) are ones in which the hypothesis is true. That proportion is the probability of the hypothesis, given the evidence.

I'm not sure what's gained by memorizing P(E|H)*P(H) in place of P(E&H). It's usually pretty obvious how to compute P(E&H).

[gmalivuk]

People might find that numerator easier because those numbers are usually the ones given directly, but you're right that the general definition is probably the most widely useful one to remember.

[Xanthir]

Yeah, the ratio formulation is super easy to remember - P(A)/P(A) (one of them conditional) = P(B)/P(B) (the same one conditional).

Then remembering that P(A|B) is done by "canceling out" the B in P(A&B)/P(B) is also fairly easy, and you're good with a bit of moving things around.