and $g_{0}(q_{k})=1$ as before. The distribution $f_{k,j}(p_{k})$ does not change from $f_{k,j-1}(p_{k})$ unless the space was occupied, and $g_{k,j-1}(q_{k})$ is also left alone unless the space was vacant. In other words, to update the distributions for a state, a transition out of that state must have been observed.

### Forgetting Factor

As training proceeds, the distributions $f_{k}(p_{k})$ and $g_{k}(q_{k})$ become increasingly narrow and converge toward delta functions, the oldest and newest training data exerting equal but ever-decreasing influence on the model; even the newest training data becomes diluted. This is acceptable for batch training, where the history length is chosen explicitly, but not for incremental training, where eventually the distributions cannot change at all. We introduce a forgetting factor $\lambda$ to gradually discount older training data and allow the Markov chain to retain its flexibility. Linear forgetting is implemented using $$\begin{aligned}f_{k,j}^{\prime}(p_{k}) & =\lambda f_{k,j}(p_{k})+(1-\lambda)f_{0}(p_{k})\\
g_{k,j}^{\prime}(q_{k}) & =\lambda g_{k,j}(q_{k})+(1-\lambda)g_{0}(q_{k}),
\end{aligned}
\label{eq:forgetting}$$ where $f_{0}(p_{k})=1$ and $g_{0}(q_{k})=1$, and $f_{k,j}(p_{k})$ and $g_{k,j}(q_{k})$ are the posterior distributions that have just been trained before application of forgetting.[^7]

There is no direct equivalence between forgetting factors and batch training history length; batch training (Figure \[fig:trainingSchematic\]a) is analogous to a finite impulse response (FIR) filter with a defined 