# Is there a way of path reconstruction using only the history of belief state?

Given a history of belief states, is there a common method that backtracks the most likely path of ending up in the current belief state?

I have a Markov model which calculates belief states after every step. The belief state is a representation of the most likely states one could be in. A belief state may look like this:

$$b=[1,0,0,0,0],$$ where I am in the state $$s_0$$ with 100% certainty.

I can store the belief state history like $$b_0, b_1, b_2,...b_n$$.

Is there a common way to represent and estimate the most likely sates one has been in?

A naive approach could be to just look for the state with the highest value per belief state and take that as the node along the reverse-path. But I am not confident enough, if that is a common and a good practice as it is not considering the fuzziness, which comes with a belief state. But then again, if I would take all states that are bigger than 0, I might not know which state leads to which state and if that transition is even possible.

The belief state in a POMDP is a distribution over the hidden state given all past actions and observations, i.e., at time $$k$$, the belief state is $$b_k(s_k) \triangleq P(s_k \mid a_{0:k-1}, z_{1:k})$$, where $$a$$ and $$z$$ are the actions and observations, respectively.
What you are asking about and calling "backtracking" boils down to the question: "given that at time $$k$$, I know the history of actions $$a_{0:k-1}$$ and observations $$z_{1:k}$$, what is the distribution over states at some past time step $$t, that is, $$P(s_t\mid a_{0:k-1},z_{1:k})$$"?
This is commonly known as Bayesian smoothing. You might be familiar with Bayesian filtering, which aims to recover the belief state as defined in the first paragraph - an estimate of the current state given actions and observations up to the current time. Smoothing also uses information gained after time $$t$$ to estimate the state at time $$t$$, and hence it is only possible after those actions and observations become known. There is also prediction, where you estimate the distribution over the state in the future.