smaller than the number observed states $N$. As we explained in the introduction, even for this seemingly easier case, it is not known that we can guarantee unique recovery of the HMM parameters *just from the pair-wise co-occurrence probability*. What is known is that the tensor CPD approach is able to guarantee identifiability given exact triple-occurrence probability. We will demonstrate in this section that it is much harder to obtain accurate triple-occurrence probability comparing with the co-occurrence probability. As a result, if the sufficiently scattered assumption holds for the emission probability, the estimated parameters obtained from our method are always more accurate than those obtained from tensor CPD.

Fixing $N=100$ and $K=20$, the transition probabilities are synthetically generated from a random exponential matrix of size $K\times K$ followed by row-normalization; for the emission probabilities, approximately 50% of the entries in the $N\times K$ random exponential matrices are set to zero before normalizing the columns, which is shown to satisfy the sufficiently scattered condition with very high probability [@huang2015principled]. We let the number of HMM realizations go from $10^6$ to $10^8$, and compare the estimation error for the transition matrix and emission matrix by the aforementioned methods. We show the total 