I have a problem in which I'm trying to maintain a binomial distribution likelihood function. The wrinkle is that I have uncertainty about the results of an individual trial.
For example, each trial's success may come as a probability value (say, 0.78) rather than as a binary value (1 or 0).
Is there a standard or recommended way to handle this kind of nondeterministic evidence for updating a likelihood? I don't require that it be exactly a binomial likelihood.
Section 5.2 in Sokal and Rohlf's Biometry 3rd edition breaks down expansion of binomial frequencies and may help to see how the binomial proportion probabilities distribute. If 0.78 is the proportion of successes out of N trials, one way to model it is with the frequencies in, for example, a glm in R. An R Companion to Applied Regression has several glm models for binomial or binary outcomes, one of which you may feel is suitable for your research goal and data.
As discussed in the comments, the data you have are probabilities, not binary events. Why would you build a model of data you don’t have (binary) instead of building a model for the data you have (probabilities)?
If your data are probabilities, a more appropriate model would be to use beta distribution as a likelihood function. Bete distribution is of the exponential family, but the conjugate prior is "complicated". People would usually use optimization, Bayesian quadrature, or MCMC instead of using conjugacy instead.
For a non-Bayesian approach, you could use the method of moments to estimate the parameters of the beta distribution
$$\begin{align} \alpha &= \bar{x}\, \Big( \frac{\bar{x}(1- \bar{x})}{s^2} - 1 \Big) \\ \beta &= (1 - \bar{x})\, \Big( \frac{\bar{x}(1- \bar{x})}{s^2} - 1 \Big) \end{align}$$
or, for the more complicated model, logistic regression with probabilities as the dependent variable.
When you estimate the distribution for the probabilities, you can use it to deduce what would be the expected counts of successes from a binomial distribution with such probabilities, or the distribution of the counts. You just need to plug in the probabilities to the binomial distribution and you can plot them, integrate over them, or do Monte Carlo sampling by first sampling $p \sim \mathsf{Beta}(\alpha, \beta)$ and then $y \sim \mathsf{Binom}(p, N)$.
