# It seems that the examples about Forward–backward_algorithm on wiki does not make sense when interpret to sunny rainy story.

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?

$$\mathbf {O_{1}} ={\begin{pmatrix}0.9&0.0\\0.0&0.2\end{pmatrix}}~~\mathbf {O_{2}} ={\begin{pmatrix}0.9&0.0\\0.0&0.2\end{pmatrix}}~~\mathbf {O_{3}} ={\begin{pmatrix}0.1&0.0\\0.0&0.8\end{pmatrix}} ...$$

• Don't forget to include links or references, so people know what you are talking about. – Graham Kemp Jul 25 '19 at 3:41

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.