Bayesian updating with continuous prior in continuous time I am considering an example where a person flips his (unfair) coin to examine what is the probability of getting head.
I could find some posts saying that the posterior distribution follows Beta distribution in discrete time.
Would there be a formula for continuous time as well?
Thank you.
 A: The process of tossing a coin repeatedly is obviously a process that is discrete in time. You cannot toss a coin continuously. However, if your problem is not exactly that of a coin and the random variable assumes either true or false values, then you can go to a continuous description using a Poisson process. I suggest that you check how, from a discrete binomial distribution, you can go to a continuous description and obtain the Poisson distribution. In the Poisson case, your problem is characterized by the length of the time interval and the rate at which you obtain a success (i.e. number of heads per unit time).
UPDATE: I'd like to be more specific about your question. The repetition of coin toss follows a binomial distribution. This represents a series of coin tosses, each at a different (discrete) time step. The conjugate prior of a binomial distribution is a Beta distribution. Now, if you go to the continuum limit, as I said you can use a Poisson distribution. In this case, the conjugate prior (time-continuous as you call it, however the prior refers to the parameters of the distribution, not the time, so either I don't know what you mean or this is just not a good name for the prior) is a Gamma distribution. (Note that if you want to approach the problem from the continuum perspective, you need Poisson & Gamma, you cannot simply use the prior from the continuum case over the discrete case.)
