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I would like to know if there is a way to build a confidence interval, for a random variable which has a Bernoulli distribution, based on its history. I mean if the order of its states is 11100 (i.e. lets consider its 5 last states), the confidence interval should be different from the one with history 00111, because in the 2nd case "on" states are the latest ones, and the center of the interval must be closer to 1, than in the first case. Thank you in advance.

Dionis Beqiraj
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    Try asking this on Cross Validated or on Math SE. This seems more theoretical rather than programming related. http://stats.stackexchange.com, http://math.stackexchange.com – rayryeng Apr 12 '15 at 02:03
  • How are the trials dependent? Sounds like a Markov process... – A. Donda Apr 12 '15 at 15:18
  • Yes, it might be a Markov chain but the problem is that I don't know the probabilities matrix of this process. And further more, I don't know how to build a confidence interval for a Markov chain. I'm looking for something that can predict the system behavior based in a sample I have. I'm not sure if I should use Bootstrap, because it works when the distribution of the variable is unknown. In my case, the variable I have is a boolean one (so it has a Bernoulli distribution hasn't it?). Thank you for answering. – Dionis Beqiraj Apr 12 '15 at 17:05
  • @rayryeng I've already asked there. Thank you anyway :) – Dionis Beqiraj Apr 12 '15 at 17:16

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If you have the ability to observe sufficiently many replicates of the process, or to observe a single replicate for a sufficiently long time, and to assume that it is a finite-order Markov process, you can estimate the transition probabilities the Markov process. This will tell you, for example, what the probability of getting "TRUE" is, based on the history. This may not be what you are seeking: I am not quite sure what you mean by a "confidence interval".

AAK
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  • Yes, I understand what you mean, but I can't estimate those probabilities. But, I can generalize it by using a formula as I explain in this comment here [http://math.stackexchange.com/questions/1231598/dependent-bernoulli-trials-confidence-interval]. Can I predict the system outcome now ? – Dionis Beqiraj Apr 13 '15 at 18:09
  • The post you reference seems to suggest that you have a model of the process in mind. With a model, you can think about estimating that model and asking it to make forecasts. Without a model, your question isn't concrete enough to have a definite answer. Let me give an example: suppose your Bernoulli process is iid. Then the probability that the next state will be 1 given that the preceding 5 were 11000 is exactly the same as it would be if the preceding 5 states were 00111. You have some model in mind but need to make it explicit before your question has an answer. – AAK Apr 13 '15 at 20:17
  • The real process I have maybe is too random to be predicted so I'm going to look for other solutions. Anyway, thank you for trying to help me! :) – Dionis Beqiraj Apr 13 '15 at 20:42