Effect of Emission Probabilities on Posterior Estimation in Hidden Markov Models

Question

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.

Answer 1

This answer assumes you have the right emission probabilities. It's not that you don't know them, but rather that they are not ideal for your application.

Let $K_t$ be the $3x3$ state transition matrix. Let $p(x_{t-1}|y_{1:t-1})$ be a row vector. Let $p(y_t|x_t)$ be the emission probabilities for different $x_t$ values given a specific value of $y_t$. Generally,

$$ p(x_t|y_{1:t}) \propto p(x_{t-1}|y_{1:t-1}) K_t \circ p(y_t|x_t). $$

In your case $p(y_t|x_t)$ is constant in $x_t$ given a specific $y_t$. Which means

$$ p(x_t|y_{1:t}) \propto p(x_{t-1}|y_{1:t-1}) K_t. $$

So for each time point, your emission probabilities will not play a role at all.

PS: $\circ$ is the element-by-element multiplication or Hadamard product.