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I have to infer the probability of a system failing from observations. Since probabilities are bounded between 0 and 1, they are sometimes modeled using Beta distribution. While the traditional Kalman filter formulation works in the Gaussian space, can I use the same approach but use Beta and Binomial distributions? Are there any existing work in this domain?

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    $\begingroup$ The KF deals with conditional means and variances. Are those your primary focus or are you looking to do some kind of updating of something else? $\endgroup$
    – Glen_b
    Commented Jan 17, 2023 at 21:49
  • $\begingroup$ I'm trying to track the status of a system (represented through the probability of failure) over time. Noisy sensors track the status of the system at irregular intervals, and I want to update my belief on the system's status by incorporating the observations from the sensors. I'm thinking of modeling the system state using a Beta dist. and update the system's state using Binomial likelihoods to incorporate sensor observations. $\endgroup$
    – PPR
    Commented Jan 17, 2023 at 22:07
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    $\begingroup$ It seems like you're seeking a fast Bayesian update of parameter estimates of the underlying beta (parameterized in some suitable way). This might be possible at least for some models because these distributions are exponential family (with implications for the sufficient statistics). Once you define your state you'll need to have some way of evolving the state (relating states over time), of course. I'd suggest first figuring out how you intend your betas to be related over time. $\endgroup$
    – Glen_b
    Commented Jan 17, 2023 at 22:16
  • $\begingroup$ Thanks a lot for your insights @Glen_b. Your insights are very helpful. What methods can I use to model the temporal evolution of the system state? To elaborate, how can I formulate the prediction step? Can I use a time series model formulation (an autoregressive model or a simple regression formulation) to model the prediction step? Have you encountered any literature in this domain (i.e., recursive state estimation using Beta and Binomial formulation)? Thanks again for your input. $\endgroup$
    – PPR
    Commented Jan 17, 2023 at 22:32
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    $\begingroup$ 1. Step back from thinking in pure KF terms; indeed if you're saying "how do I formulate a prediction step" you're about 10 steps along the road I was asking you to focus on step 1 of. You might not end up with something that looks very close to a Kalman filter (especially since you say you want to focus on likelihood). 2. It's not for me to tell you what your model that relates the parameters of beta over time should be. E.g. you know what you're modelling, while I do not. You presumably have some level of application-area/subject-matter knowledge, while I have no idea what that area even is $\endgroup$
    – Glen_b
    Commented Jan 18, 2023 at 2:04

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