It is well known that the Beta distribution serves as a conjugate prior for the Bernoulli distribution, and that when you observe a Bernouilli random variable, you need only increment the appropriate hyperparameter of the Beta distribution.
However when the Bernoulli distribution is "noisy" in the sense that you do not observe the the Bernoulli random variable directly, but instead observe a random variable that is equal to the Bernoulli random variable with probability 1-p and flipped with probability p, where p is known, and represents an error rate in observing the Bernoulli random variable, the posterior distribution obtained is a linear combination of two Beta distributions.
In the case when p=0, p=.5, and p=1, the posterior distribution is again Beta, but for other values of p, this is not the case.
In my particular application analytic tractability is important. Is there a conjugate prior that would be appropriate for this type of problem?
Failing that, it seems there might be a sensible way to update the hyperparameters of the Beta distribution in an approximate sense. Intuitively, when you observe a Bernoulli random variable with error rate p, the information you have about the parameter theta of the Bernoulli distribution is nothing when p=.5 (observations are completely uninformative) and maximum when p=0 or p=1, and somewhere in between for other values of p. More specifically, in the case when p=1 or p=0, the sum of the hyperparameters of the beta in the posterior distribution is 1 greater than the sum in the prior, and for p=.5, the sum of the hyperparameters remains the same. For other values of p, the change in the sum of the hyperparameters should be intermediate, but I'm not sure how to best choose them.