variation distance between the ground truth probabilities $\Pr[X_{t+1}|X_t]$ and $\Pr[Y_t|X_t]$ and their estimations $\widehat{\Pr}[X_{t+1}|X_t]$ and $\widehat{\Pr}[Y_t|X_t]$ using various methods. The result is shown in Figure \[fig:sim\_exp1\]. As we can see, the proposed method indeed works best, obtaining almost perfect recovery when sample size is above $10^8$. The CPD based method does not work as well since it cannot obtain accurate estimates of the third-order statistics that it needs. Initialized by CPD, EM improves from CPD but the performance is still far away from the proposed method. NMF is not working well since it does not have identifiability in this case.

Application: Hidden Topic Markov Model
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Analyzing text data is one of the core application domains of machine learning. There are two prevailing approaches to model text data. The classical bag-of-words model assumes that each word is *independently* drawn from certain multinomial distributions. These distributions are different across documents, but can be efficiently summarized by a small number of *topics*, again mathematically modeled as distributions over words; this task is widely known as *topic modeling* [@hofmann2001unsupervised; @blei2003latent]. However, it is obvious that the bag-of-words representation is oversimplified. The $n$-gram model, on the other hand, assumes that words are conditionally dependent 