Am I choosing correct likelihood? I am using Bayes theorem to solve links upvoting problem for reddit/hacker new style website. 
Every link has a probability $P(Q)$, that the link is a high quality one. And every user has probability $P(R)$ that her upvoted link is a good quality one (reliability, thus R). The problem is that I want to update the probability $P(Q)$ after every upvote, which is $P(Q|R)$.
I know how to calculate $P(Q|R)$ for this problem, I am just not sure if I am using a correct likelihood $P(R|Q)$.
My though process is that $P(Q|R)$ means the probability that the link is high quality, given the user has upvoted it. Based on the definition of the problem ($P(R)$ is the probability that the upvoted link was good), I think $P(R|Q) = P(R)$.
Am I correct, or I got it wrong somewhere?
Thank you!
 A: You have to be a bit more careful with your notation. What is the event $R$? A user upvoting a quality link? You also need to specify the probability that a user fails to upvote a quality link, etc.
I'd recommend starting with a naives Bayes model. $P(Q)$ is your prior, say $P(Q)=0.1$ if 10% of the submitted linked are good. Now each user can either be a good upvoter or a spammer. Let $G_i$ be the event that user $i$ is a good upvoter.
Now suppose that the link is upvoted by users $i_1, i_2, \ldots i_p$ (events $U_{i_1}, \ldots U_{i_p}$)
$$P(Q|U_{i_1}, \ldots, U_{i_p}) = \frac{P(U_{i_1}, \ldots, U_{i_p}|Q)P(Q)}{P(U_{i_1}, \ldots, U_{i_p})}$$
I naive Bayes, the (naive) assumption is that upvotes are independent. In this case 
$$P(U_{i_1}, \ldots, U_{i_p}|Q) = \prod_{j=1}^p P(U_{i_j}|Q)$$ and
$$P(U_{i_1}, \ldots, U_{i_p}|\neg Q) = \prod_{j=1}^p P(U_{i_j}|\neg Q)$$
We write $$P(U_{i_j}|Q) = P(U_{i_j}|Q, G_{i_j})P(G_{i_j}) + P(U_{i_j}|Q, \neg G_{i_j})(1-P(G_{i_j}))$$
$P(G_{i_j})$ is your estimate of the probability that user $i_j$ is a good guy. You can estimate that from its history of upvoting. We assume that $P(U_{i_j}|Q,G_{i_j}) = P(U|Q,G)$ is a constant that represents the rate at which good users upvote quality links.
Likewise $P(U_{i_j}|Q,\neg G_{i_j}) = P(U|Q,\neg G)$ is the rate at which bad users upvote good links, etc. In total you have to specify four rates: the rates at which (good/bad) users upvote (quality/not quality) links.
Going back to the main equation, we have
$$P(Q|U_{i_1}, \ldots, U_{i_p}) = \frac{1}{1 + \frac{1-P(Q)}{P(Q)}\frac{\prod_{j=1}^p P(U|\neg Q, G) P(G_{i_j}) + P(U|\neg Q, \neg G) (1-P(G_{i_j}))}{\prod_{j=1}^p P(U|Q,G) P(G_{i_j}) + P(U|Q,\neg G) (1-P(G_{i_j}))}}$$
More generally, you might want to infer the rates of upvotes and the probabilities that users are good from the data. At this point, the easiest approach is to write a generative model and sample from it using MCMC.
