The standard Kalman filter uses a series of measurements observed over time, to decomposite the signal and noise.
However, when I'm modeling the distribution (pdf or cdf) of a variant, is there a similar tool?
For example, here's a random variable $X(month, PaymentDelay)$ describe how many days (PaymentDelay) people will pay the credit card bill after get informed.
This is a discrete variant valued on non-negative integer: $PaymentDelay \in [0,1,2, ...]$.
Each month there have an observation.
Define $\mu(month) = E(X(month,PaymentDelay))$, Kalman filter could be used to generate a more accurate estimation of $\mu$, to predict $\mu(month+1)$ from historical data.
However, if the target is the distribution $X$, is there a similar way to get an more accurate estimation on $\hat X(month+1, PaymentDelay)$ from historical observations?