Hi,

​

Suppose we want to estimate the distribution of entities in one dimension x and how the distribution evolves over time. For the sake of example the dimension can represent the spatial position of the entities, e.g. in east-west direction. Unfortunately the entities do not have identifiers, so we can not tell if two measurements x_1 and x_2 belong to the same entity or not if they arrive at different times.

​

The observability of the entities is limited which means that current x positions are not known at all times. Measurements of x arrive with varying temporal density. Thus, binning by time would lead to some time bins having few, others having a lot of data points.

​

If we assume that the total number fof entities is constant, how could we estimate the distribution over x, given t, in such a way that the temporal density is accounted for? I’d imagine a system where measurements in times of low density have a longer time-to-live, i.e. contribute to the density estimation for longer times compared to times when a lot of measurements arrive.

​

Is this a stochastic filtering problem? I’d be grateful for any hints what papers or textbook chapters to read to understand this issue better. My experience in the field is mostly limited to classification and regression with only limited exposure to density estimation and unsupervised learning. So, apologies if the question is trivial, I currently just don’t know where to start reading.

​

Thanks!