In the Forward-backward algorithm, this Transition probabilities:
$\mathbf {T} ={\begin{pmatrix}0.7&0.3\\0.3&0.7\end{pmatrix}}$
could be viewed as the probabilities of the weather at t+1, for instance, $T_{11}$ could be viewed as
the weather has a 70% chance of staying the same
the emission probabilities
$\mathbf {B} ={\begin{pmatrix}0.9&0.1\\0.2&0.8\end{pmatrix}}$
could also be interpreted as
90% people use umbrella when it is rainy
in this context, what does following observations mean?
$\mathbf {O_{1}} ={\begin{pmatrix}0.9&0.0\\0.0&0.2\end{pmatrix}}~~\mathbf {O_{2}} ={\begin{pmatrix}0.9&0.0\\0.0&0.2\end{pmatrix}}~~\mathbf {O_{3}} ={\begin{pmatrix}0.1&0.0\\0.0&0.8\end{pmatrix}} ...$
To be precise, $\mathbf T_{11}$ is the conditional probability that: the weather stays in state 1 (when it is in state 1). It is that we have $\mathbf T_{22}=\mathbf T_{11}$ which indicates that the event that the weather stays the same is independent of the current state of the weather.
The diagonal elements of $\mathbf O_j$ present the conditional probabilities for observing event $j$ given each system state. These events are given in the array, $\{$umbrella, umbrella, no-umbrella, umbrella, umbrella$\}$.
So $\mathbf O_1$ presents us with the conditional probability for observing an umbrella would be 0.9 when given a rainy day, and 0.2 when given a sunny day.