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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.

김찬우
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    Is your question simply "is there such a thing as a continuous time series"? If so then yes. In fact you can use a discrete time observations to estimate continuous processes using filtering. Examples of this are actual weather forecasts, control systems, etc. It would be a bit interesting to consider predicting the posterior of the heads probability on the 45.5th throw, but theoretically possible if you consider the continuous "throw number" as an artifact of the filtering process and the weight applied to previous observations. – AdamO Mar 09 '21 at 20:21
  • Thank you very much for your reply!! My approach was a bit more theoretical, but I think I would use the same technic once I start simulating something. – 김찬우 Mar 11 '21 at 10:11

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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.)

rasmodius
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  • Thank you very much for your comment. Just to be clear, so does this become essentially running a unknown bandit infinite time? There are two kinds of bandits either good or bad. If it is good, the Poisson arrival rate of a good signal is higher. I don't know which type my bandit is, but I can run it infinite times. Each time I run a bandit, I collect the length of the time interval to get a signal. Then based on its distribution, I form posterior of having a good type bandit. – 김찬우 Mar 09 '21 at 12:05
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    To be honest, I don't know what you mean by bandit. I think if you elaborate your comment more, you can post it as a question. There you can be more specific about your problem and explain the details. – rasmodius Mar 09 '21 at 15:59
  • Your updated answer is tremendously helpful. I just needed sometime to clarify what my problem actually is. I will post a new one, once it becomes clear. Would you mind recommending a nice reference on this? – 김찬우 Mar 11 '21 at 10:10
  • I like the book "Bayesian Probability Theory Applications in the Physical Sciences", by von der Linden, Dose & von Toussaint. If you find it, you can check the derivation of the Poisson distribution and the section called stochastic processes. But you can easily find many others look for binomial and Poisson, and also from Bayesian parameter estimation. I suspect your problem might be (or at least related) to what is known as Cox process. – rasmodius Mar 11 '21 at 18:20
  • Sorry I just did! Thanks :) – 김찬우 Mar 17 '21 at 12:56