memory length, while incremental training (Figure \[fig:trainingSchematic\]b) is structurally reminiscent of an infinite impulse response (IIR) filter where the previous output is fed back into the filter. With batch training, the hand-picked data set may not contain all the transitions of interest, so some transitions may not be trained at all. The incremental approach applies training and forgetting simultaneously, retaining infrequently observed transitions longer.

To illustrate the effect of forgetting on the distributions, we have trained a single state of the Markov chain repeatedly using alternating transitions $\gamma_{0}=0\to\gamma_{1}=1$ and $\gamma_{0}=0\to\gamma_{1}=0$. This is analogous to flipping an unbiased coin numerous times and observing heads every other flip, from which we expect an increasingly narrow distribution for $p_{0}$ peaking near 0.5 (Figure \[fig:forgettingFactor\]a). This result would be preferred if the pattern were never expected to change, but such concentration in the distribution hinders its ability to change and is therefore undesirable. Using 15% forgetting ($\lambda=0.85$) gives a distinctly broader distribution lifted off the horizontal axis (Figure \[fig:forgettingFactor\]b). The distribution—and therefore its expected value—shifts laterally with each alternate observation, even after many iterations; this extra mobility reflects greater adaptability. The value $\lambda=0.85$ is much more forgetful than would be used in practice; Section 