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Notice that the value of <math>\mathbf{\gamma_0}</math> is equal to <math>\mathbf{\hat{b}_{0:5}}</math> and that <math>\mathbf{\gamma_5}</math> is equal to <math>\mathbf{\hat{f}_{0:5}}</math>. This follows naturally because both <math>\mathbf{\hat{f}_{0:5}}</math> and <math>\mathbf{\hat{b}_{0:5}}</math> begin with uniform priors over the initial and final state vectors (respectively) and take into account all of the observations. However, <math>\mathbf{\gamma_0}</math> will only be equal to <math>\mathbf{\hat{b}_{0:5}}</math> when our initial state vector represents a uniform prior (i.e. all entries are equal). When this is not the case <math>\mathbf{\hat{b}_{0:5}}</math> needs to be combined with the initial state vector to find the most likely initial state. We thus find that the forward probabilities by themselves are sufficient to calculate the most likely final state. Similarly, the backward probabilities can be combined with the initial state vector to provide the most probable initial state given the observations. The forward and backward probabilities need only be combined to infer the most probable states between the initial and final points.
The calculations above reveal that the most probable weather state on every day except for the third one was "rain
==Performance ==
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