Is it viable to use an HMM to evaluate how well a catalogue is used?
Ad Question 1: Assuming that your assumptions on how the catalogue is used -- that is the choice of the next cell only depends on the current (or constantly many preceeding) cell(s), not the (full) history -- then yes, you can use a Markov Chain to model it.

However, you do not seem to need the "Hidden" part; this is only useful if you have (probabilistic) output in the states which you observe. In contrast, you want to observe the students' cell states directly.

Imagine you can not observe which cells students are in but only how much they like the current cell; this could be implemented by notorious "Was this useful to you?" buttons; in general, assume every students gives feedback from $\{1, \dots, k\}$ in every cell but you do not know which cell they are in. Then you can use a HMM to estimate their cell sequence given their feedback sequence.

Ad Question 2: Let me illustrate the use continuing above train of thought. For $n=k=3$, we have the following Markov chain:


[source]

$p_{i,j}$ is the probability to transition from state $i$ to state $j$; note that $p_{i,0}=0$ for all $i$ and those edges have been left out¹. Let $p_{i,\bot}$ the probability of terminating in state $i$ (we leave out another dummy state for clarity). $q_{i,l}$ is the probability to emit $l$ when entering (or, equivalently, leaving) state $i$; this models our feedback. Of course, we require $\sum_j p_{i,j} + p_{i,\bot} = \sum_l q_{i,l} = 1$ for all $i$.

Note that the state sequence $0,1,2$ has probability $p_{0,1} \cdot p_{1,2}$ -- that is a central property of Markov chains. The output sequence $1,2$, on the other hand, has probability $q_{0,1}\cdot p_{0,1}\cdot q_{1,2} + q_{0,1}\cdot p_{0,2}\cdot q_{2,2} + q_{0,1}\cdot p_{0,3}\cdot q_{3,2}$.

In a real world scenario, we would have to train our Markov chain. Let us assume we have some sequences of both state and output²:

$\qquad \begin{align} &amp;(0,1), (1,1), (2,1), (3,1) \\
                      &amp;(0,2), (1,3), (3,1) \\
                      &amp;(0,1), (3,3), (1,2)
\end{align}$

Now we just count how often each transition and output happened and set our probabilities to the relative frequencies³:


[source]

Now you can do the interesting stuff. Ignoring the output, you can determine the path with highest probability, you can determine the output sequence with highest probability but most importantly, and here the hidden part comes into play, you can find the most likely state sequence given an output sequence with Viterbi algorithm.

Bottom line: yes, you can use Markov chains resp. hidden Markov models (depending on your needs) to model your scenario. Wether the fundamental assumption -- the probability for the next cell only depends on the current cell -- makes sense is another question entirely.




We need a starting state, but we can make it a dummy by never going back. We can ignore its output or introduce a dummy output that is only and always emitted in state $0$.
If we have only output sequences, we can use the forward-backward algorithm.
Due to our low number of observations, we have a number of impossible transitions/emissions. Usually you assume that everything is possible and has just not been observed due to low probability. We can compensate for this by adding $1$ to all counts; this way, all sequences have probability $\gt 0$.