  We propose a new algorithm for learning a bridged diffusion process using
score-matching methods. Our method relies on reversing the dynamics of the
forward process and using this to learn a score function, which, via Doob's
$h$-transform, gives us a bridged diffusion process; that is, a process
conditioned on an endpoint. In contrast to prior methods, ours learns the score
term $\nabla_x \log p(t, x; T, y)$, for given $t, Y$ directly, completely
avoiding the need for first learning a time reversal. We compare the
performance of our algorithm with existing methods and see that it outperforms
using the (learned) time-reversals to learn the score term. The code can be
found at https://github.com/libbylbaker/forward_bridge.
