Imagine a coin with a time-varying probability of coming up heads. (For example, perhaps the probability follows a random walk that is constrained to live in [0, 1]. And say we have some information about how quickly we expect the true probability to vary over time (analogous to the standard deviation of the random walk described above).

At discrete time points, we are given data about the outcome of a given coin flip. The goal is to come up with a robust procedure to estimate that probability over time.

This sounds like a task for a Kalman filter. But the literature there seems to assume that the observations will be normally distributed around a linear function of the state. That feels quite violated by the assumptions above.

What's a reasonable way to proceed?

Imagine a coin with a time-varying probability of coming up heads. (For example, perhaps the probability follows a random walk that is constrained to live in $[0, 1]$. And say we have some information about how quickly we expect the true probability to vary over time (analogous to the standard deviation of the random walk described above).

At discrete time points, we are given data about the outcome of a given coin flip. The goal is to come up with a robust procedure to estimate that probability over time.

This sounds like a task for a Kalman filter. But the literature seems to assume that the observations will be normally distributed around a linear function of the state. That feels quite violated by the assumptions above.

What's a reasonable way to proceed?