I'm curious how I should think about the risk of hanging out with a friend (who is 2x pfizer vaccinated). It seems like a good opportunity for bayesian thinking in the real world, but I'm really unclear how to think about it.

Info: he tested positive on 4 UK lateral flow tests (LFTs), all from the same box (on 2 different days). After this, his roommates took two tests from the same box & both were negative.

He has subsequently taken 3 PCR tests + a LFT each day, which have been negative.

However, false positives seem to be very rare even for LFTs. They're ~1/1000 (number range from .9968 in original studies to .9997 more recently)

https://www.gov.uk/government/news/new-analysis-of-lateral-flow-tests-shows-specificity-of-at-least-999

https://www.bmj.com/content/372/bmj.n706

But false negatives seem common for everything, including PCRs. It seems there's a 20-67% false negatives (20% being best it ever gets, on day 8 of infection)

https://www.acpjournals.org/doi/10.7326/m20-1495

https://pubmed.ncbi.nlm.nih.gov/33301459/

Given this, what are the chances he had covid (from maybe 10-20x lower-than-average-risk prior, but in oxford/England)?

There are a lot of moving parts and uncertainties in this question.

I'd say the biggest contributor toward "false positive" probability would not be failure of the test kit itself, but error due to the user and/or environment in which the test is done. There are lots of instructions about avoiding eating recently before the test, avoiding contact with various other surfaces and so on, and it may be accidentally possible to pick up some contamination that invalidates the test.

What is the chance of this? Controlled testing is always going to provide lower bounds here, not upper bounds. However, if it was negligible then they wouldn't bother to be so careful with the instructions, and if it was large then millions of people would be getting false positives every day, but all we can do here is guess. Four false-positive tests over two days is less likely than one, but they're not going to be independent of one another so we can't just multiply the numbers that we don't have anyway. All I could really say is that I would be surprised if it was above 2% (50:1 odds ratio), but it could easily be as low as 0.1% (1000:1 odds ratio).

The subsequent negative tests (PCR and LFT) do reduce the probability, but not by much. Asymptomatic cases are common, especially so in vaccinated people, and usually feature a short window of detection. The differential probability between "was infected" and "was not infected" is almost entirely conditional probability of this window ending between the first LFT and the start of the subsequent tests, which (depending upon timing) could be 50% or higher.

Given the number of weekly cases in the region, even among people who are vaccinated, I'd say this puts the balance of probability toward your friend having been infected.

Probably of more interest to you would be the probability of your friend being

currently infectious, which seems quite low. Having received results from 3 PCR tests suggests that quite a number of days has passed since the last positive test. That alone in an asymptomatic person is a strong indicator that that they are not infectious now, and clearance times are generally reduced in vaccinated people. The odds ratio would depend upon just how many days it has been.The fact that the all the subsequent tests were negative suggests even more strongly that they are no longer infectious, even if some of those may have been false negatives. Unfortunately there aren't really any credible studies on distributions of how infectious individuals are at various stages of their recovery.

We do have observational results of number of infection chains started by people who leave isolation some days after testing negative. That suggests that the chance of infecting any other person 7 days after a negative test that is subsequently confirmed with another negative test, is less than 1 in 1000.