Saving your life with Bayes Theorem

[ciphergoth]

The report I quoted earlier says that 3-8% of new HIV cases among gay men are due to transmission through fellatio. So, how risky is fellatio? It depends on what proportion of men who have sex with men are engaging in high risk activities. We can analyse this with standard tools from probability theory, along with a bunch of simplifying assumptions.

Our sample population is, say the set of all men in the UK who started the year 2000 HIV-negative and had sex with men that year. Choose a man from that population at random; we'll use letters to designate salient facts that might or might not be true of that man.

L means that man engaged in no activities considered "high-risk" in that year - say, no fucking without a condom, no broken condoms, no IV drug use.

V means that that man contracted HIV that year.

P(V) means the probability he contracted HIV that year. P(~V) means the probability he didn't. Since either he did or he didn't, P(V) + P(~V) = 1, ie P(~V) = 1 - P(V). P(V|L) (reads as P(V given L)) is the probability he contracted the virus, given that he stuck only to low-risk activities.

To simplify things, assume that all the men in our sample performed fellatio that year, and that all of the infections not due to "high-risk" behaviour are due to fellatio.

If we make this assumption (and some other simplifying assumptions I won't go into), then what the survey is telling us is that P(L|V), the probability that someone who contracted the virus that year didn't do anything "high-risk", is 3-8%; call it 5%. What we want to know is how much avoiding high-risk behaviour reduces our risk of getting the virus; we want to know the ratio of the risk-takers risk to the fellatio-only risk:

P(V|~L) / P(V|L)

P(V|~L) means the probability he contracted the virus given that he did *not* stick to low-risk activities that year.

In order to work this out, we need one other salient figure: P(L), the proportion of men who stuck to safe activities that year. That figure plugs into this formula:

P(V|~L)   P(L)P(~L|V)        P(L)
------- = ----------- = 19 --------
P(V|L)    P(~L)P(L|V)      1 - P(L)

So if you think only half the population of men who have sex with men always have safe sex and so forth, you can conclude that safer sex is only 19 times safer than high-risk sex. If on the other hand 95% of that population practice safer sex, then combining that with the statistics tells you that safer sex is 361 times safer.

Unfortunately, I have no idea what P(L) is. Make your own guesses and see what the formula tells you...

Comments

very small number

[hbergeronx.livejournal.com]

what if nearly no-one practices? Doesn't that squash the difference entirely? Or make not practicing safer?

Re: very small number

[ciphergoth.livejournal.com]

I've assumed P(L|V) = 5% or thereabouts based on the research that sparked all this.

If P(L) = 5% = P(L|V), meaning that the proportion of the population practicing safer sex is reflected in the infection statistics, then the formula comes out to 1, meaning that practicing safer sex makes no difference. If P(L) < P(L|V) then the formula comes out to less than one, meaning that not practicing is safer.

So we can be pretty sure that P(L) > P(L|V) - there's lots of research to back *that* conclusion!

does that answer the question?

Re: very small number

[hbergeronx.livejournal.com]

is P(L) the proportion of men who stuck to safe activity, or the proportion of men who stuck to safe activity excepting fellatio? Not really that it matters, mathematically, I guess.

Re: very small number

[ciphergoth.livejournal.com]

Mathematically, as you say, it makes no difference.

But essentially, yeah, P(L) is mean to be the proportion of men for whom fellatio was their most unsafe activity. I made a whole bunch of simplifying assumptions:

To simplify things, assume that all the men in our sample performed fellatio that year, and that all of the infections not due to "high-risk" behaviour are due to fellatio.

just to get a feel for what the figures might be telling us.

Re: very small number

[hbergeronx.livejournal.com]

won't this lump the true safe sex category in with the no safe sex category (i.e. fellatio with a rubber with no rubber at all?)

Re: very small number

[ciphergoth.livejournal.com]

L is meant to cover all behaviour outside the "high risk" category. Yes, it lumps lots of things together, but I only have *one number* to work with, and that's within confidence bounds that vary by 166%!

When the actual report is published, it should be possible to make much better comparisons. At the moment, all I'm trying to do is answer the reaction "fellatio is 8% risky"...

Re: very small number

[hbergeronx.livejournal.com]

i understand. but I was hoping to find a flaw there, such that if a large percentage of people practiced 'true' safe sex, it might explain why "everyone else" might be computed as safer than "safe sex except bj". I agree that P(L) > P (L|V), since it seems almost tautologically obvious.
As you say, it's a difficult number to understand, because it is probably a product of a number of probabilities, such as risk for transmission, likelyhood of having a HIV infected partner, and the likelyhood that one uses a condom for oral sex on a given encouter.
I'm probably as interested in the result as yourself- I just have a different level of access to the concepts and understanding necessary.