I am using the Forward Algorithm to compute the most likely state at a given time, given a series of observations (Pointwise Maximum A Posteriori Estimation). While studying the performance of this estimation algorithm, I noticed that the emission probabilities play a crucial role in how well the algorithm performs. If the emission probabilities are very similar to each other, for example, given 2 observation symbols and 3 hidden states, if the emission probability matrix is \begin{equation} \begin{bmatrix} 0.6 &0.6 &0.6 \\ 0.4 &0.4 &0.4 \end{bmatrix} \end{equation} the algorithm performs very badly and the estimates are simply based on which states occurs more frequently (based on the transition probability matrix). This is expected because we are not getting any information from the observations.
My Question: How do I justify this behavior theoretically? How does a difference in emission probabilities affect the accuracy of a PMAP estimator? Any insights will be deeply appreciated.