Editing Forward algorithm (section)

==Example==
This example on observing possible states of weather from the observed condition of seaweed. We have observations of seaweed for three consecutive days as dry, damp, and soggy in order. The possible states of weather can be sunny, cloudy, or rainy. In total, there can be <math>3^3=27</math> such weather sequences. Exploring all such possible state sequences is computationally very expensive. To reduce this complexity, Forward algorithm comes in handy, where the trick lies in using the conditional independence of the sequence steps to calculate partial probabilities, <math>\alpha(x_t) = p(x_t,y_{1:t}) = p(y_t|x_t)\sum_{x_{t-1}}p(x_t|x_{t-1})\alpha(x_{t-1})</math> as shown in the above derivation. Hence, we can calculate the probabilities as the product of the appropriate observation/emission probability, <math>p(y_t|x_t)</math> ( probability of state <math>y_t</math> seen at time t from previous observation) with the sum of probabilities of reaching that state at time t, calculated using transition probabilities. This reduces complexity of the problem from searching whole search space to just using previously computed <math>\alpha</math>'s and transition probabilities.