Constructing a Velocity Field

Given a normal distribution on with small enough , it is clear that the marginal probability

This is indeed the basic premise of any scientific measurement (i.e. the original distribution can be inferred from imperfect measurements which have sufficiently high precision ).

Based on this fact we design a marginal probability path given on observation written as

with and . It is clear that for this will closely approximate the original distribution of the population, denoted as .

Thus the problem becomes determining a time trajectory which connects these two end points in order to determine and thus approximate .

The question is then to ask: What is the velocity field for the marginal probability path?

A natural answer might be exactly the expectation of the velocity field which corresponds to that is

The marginal velocity field described by generates exactly the marginal probability trajectory described by . To verify this, we see if they satisfy the continuity equation:

Intuitively, this follows from the fact that and are but transformations by linear operators. To verify this,

as desired.