follows analogously. $$\begin{alignedat}{2}f_{k,j}(p_{k})\, & =\, & \Gamma_{k}\, & \left(\Gamma_{k+1}f_{k,j}^{(1)}(p_{k})+(1-\Gamma_{k+1})f_{k,j}^{(0)}(p_{k})\right)\\
 & \ + & (1-\Gamma_{k})\, & f_{k,j-1}(p_{k})\\
g_{k,j}(q_{k})\, & =\  & (1-\Gamma_{k})\, & \left(\Gamma_{k+1}g_{k,j}^{(1)}(q_{k})+(1-\Gamma_{k+1})g_{k,j}^{(0)}(q_{k})\right)\\
 & \ + & \Gamma_{k}\, & g_{k,j-1}(q_{k})
\end{alignedat}
\label{eq:fractionalPosteriors}$$ Once the new distributions $f_{k,j}(p_{k})$ and $g_{k,j}(q_{k})$ have been found, forgetting is applied similarly to Equation \[eq:forgetting\], where Equations \[eq:fractionalPosteriors\] are used instead for the posterior distributions. The post-forgetting distributions are then stored.

### Effect of Training on Distribution Shape

To illustrate the connection between training data patterns and the shapes of $f_{k}(p_{k})$ and $g_{k}(q_{k})$, we have trained two Markov chains with the MERL Belady conference room data from March 22 to June 9 and sampled the distributions afterward. In Figure \[fig:Transition-probabilities\], two sets of distributions—one for $\to$ (a) and the other for $\to$ (b)—are shown for both strong forgetting ($\lambda=0.85$, solid) and no forgetting ($\lambda=1.0$, dashed). In Figure \[fig:Transition-probabilities\]a we see that both occupancy and vacancy at strongly imply vacancy at . In other words, early morning occupancy is very uncommon and usually brief. Because occupancy is rare at , the transition $\gamma_{2}=1_{2}\to\gamma_{3}=1_{3}$ (blue) is very weakly trained and has a very flat distribution. In Figure \[fig:Transition-probabilities\]b, we see that occupancy at is more varied, resulting in 