Let’s say you have a small n of randomly sampled testing from a population with x% positive and a larger N of non random testing of people who come into hospitals with a larger X% positive. Can you combine both samples to inform your estimate of the true population prevalence?
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Yes, but with the following qualifiers: 1. If n is sufficiently big, you don't need N. A surprisingly small n is enough to give you a sense of prevalence amongst a population within a certain confidence level. This is the basic premise behind sampling in quality control.
2. N is relevant insofar as you know what % of infected go to the hospital (in real life that % is dynamic, but anyways...). So you could say, between 10% and 50% of infections go to the hospital. If you have 100 patients, then you know there are 200 to 1000 infections.
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What I’m wondering is if under different testing regimes we can make different inferences of how much ex ante we should expect X>x (ie, do we think someone who gets tested is one hundred or twice as likely to be positive as the general pop?)
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Depends on your x% no? if x% is 100%, then equally likely. If x% is 1%, then they are much much much much more likely to be infected than the random person.
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3. The bigger n and N get relative to the total population, the more informative they are, with n being more powerful. 4. As the other guy said, you can use Bayesian probability, i.e. what's the probability that we got this result given our hospital or random testing result.
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