The positive drift in the log of the forward coefficients likely results from cumulative numerical errors over time, which is a common issue when repeatedly applying the log-sum-exp trick. Since log-probabilities should remain non-positive, any increase above zero suggests small inaccuracies are building up with each iteration. To mitigate this, consider normalizing the forward coefficients at each time step. For instance, you could re-center the log probabilities by subtracting the maximum log coefficient at each step. This normalization maintains relative values without altering the sequence, helping to control numerical errors and keep the values within the expected range.