This thesis considers extensions of the standard independent hidden Markov model approach previously used by TNO for modelling printer nozzles. These extensions introduce parametrised transition probabilities and incorporate interactions between neighbouring nozzles to better capture the real-world printing process. The aim of this thesis is to investigate Bayesian methods to infer their model parameters. Although not the goal of this thesis, accurate parameter estimation paves the way for diagnosing nozzle malfunctions, ultimately improving printer performance.

The first method is a sampling algorithm with adaptive proposal distributions. We then present two variational inference (VI) algorithms, which aim to minimise the divergence between an approximate and true posterior of the unknown parameters. The first VI method makes use of model-specific approximations, while the latter is more flexible. The adaptive sampler, however, remains the most general of the three algorithms in addition to being asymptotically unbiased.

On synthetic data, all methods were capable of producing good estimates of the model parameters. When a good initialisation is available, the sampling method may be faster than the VI approaches; however, if no good start is known, the other methods may be preferred. The runtimes of the algorithms appeared to grow linearly with the number of nozzles. In addition, a linear scaling with the number of time steps appeared plausible.

While the methods performed well on the toy models studied here, their efficacy must be confirmed on more complex systems and demonstrated in real-world applications. If runtimes prove prohibitive, sub-sampling approaches or stochastic variational inference methods could be investigated.

Quantum random walks are the quantum analogs of classical random walks and appear to be promising tools to design fast quantum algorithms. Therefore it is important to study their time-related features and see how these differ compared to the classical case. For the discrete-time quantum walk on the line it has been shown that the probability to be absorbed by an absorbing boundary equals 2/π in contrast to the classical case where this probability equals 1, hence a quantum walk may continue forever without getting absorbed. It is also shown that mixing times of discrete-time quantum walks on the hypercube scale with n, the dimension of the hypercube, which is faster than O(n log(n)) for the classical random walk. So the quantum walk might offer a slight speed-up compared to the classical case. Finally, it is shown that the mixing times of the continuous-time quantum walk on a 2-layer multiplex graph depend on the eigenvalue gaps of the corresponding Laplacian matrix L. When the strength of the connections between the layers of a multiplex graph becomes very large, the eigenvalues of the Laplacian matrix converge. Thus the mixing times of the continuous-time quantum walk on 2-layer multiplex graphs converge.