A thorough discussion of the similarities and differences between quantum and classical probability, as well as my personal take on the conceptual basis of quantum theory, can be found in

• J. Rau, On quantum vs. classical probability, Annals of Physics 324, 2622 (2009) [doi][arXiv][pdf]
• J. Rau, Measurement-based quantum foundations, Foundations of Physics 41, 380 (2011) [doi][arXiv][pdf]

Various techniques for estimating a quantum state based on incomplete or noisy measurement data are discussed in

• J. Rau, Evidence procedure for efficient quantum-state tomography, Physical Review A 82, 012104 (2010) [doi][arXiv][pdf]
• J. Rau, Inferring the Gibbs state of a small quantum system, Physical Review A 84, 012101 (2011) [doi][arXiv][pdf]
• J. Rau, Assessing thermalization and estimating the Hamiltonian with output data only, Physical Review A 84, 052101 (2011) [doi][arXiv][pdf]
• J. Rau, Appearance of Gibbs states in quantum-state tomography, Physical Review A 90, 062114 (2014) [doi][arXiv][pdf]

Probabilistic considerations may pertain not only to the quantum states themselves but also to their time evolution. When the latter is not known with certainty, it may be modeled with random matrices. One example is considered in

• J. Rau, Ballistic transport through chaotic cavities: Parametric correlations and the weak localization peak in a Brownian-motion model, Physical Review B 51, 7734 (1995) [doi][arXiv][pdf]