A Bayesian network (BN) is a probabilistic directed acyclic graph representing a set of random variables and their dependence on one another. BNs play an important role in machine learning as they can be used to calculate the probability of a new piece of data being sorted into an existing class by comparison with training data.

Each variable requires a finite set of mutually exclusive (independent) states. A node with a dependent is called a parent node and each connected pair has a set of conditional probabilities defined by their mutual dependence. Each node depends only on its parents and has conditional independence from any node it is not descended from. Using this definition, and taking n to be the number of nodes in the set of training data, the joint probability of the set of all nodes, {X_{1}, X_{2}, · · · X_{n}}, is defined for any graph as

**P(X _{i}) = ∏^{n}_{i=1} P(X_{i}|π_{i})**

where π_{i} refers to the set of parents of X_{i}. Any conditional probability between two nodes can then be calculated.

An argument for the use of BNs over other methods is that they are able to “smooth” data models, making all pieces of data usable for training. However, for a BN with * m* nodes, the number of possible graphs is exponential in

*; a problem which has been addressed with varying levels of success. The bulk of the literature on learning with BNs utilises model selection. This is concerned with using a criterion to measure the fit of the network structure to the original data, before applying a heuristic search algorithm to find an equivalence class that does well under these conditions. This is repeated over the space of BN structures. A special case of BNs is the dynamic (time-dependent) hidden Markov model (HMM), in which only outputs are visible and states are hidden. Such models are often used for speech and handwriting recognition, as they can successfully evaluate which sequences of words are the most common.*

**n**Quantum Bayesian networks (QBNs) and hidden quantum Markov models (HQMMs) have been demonstrated theoretically, but there is currently no experimental research. The format of a HMM lends itself to a smooth transition into the language of open quantum systems. * Clark et al.* claim that open quantum systems with instantaneous feedback are examples of HQMMs, with the open quantum system providing the internal states and the surrounding bath acting as the ancilla, or external state. This allows feedback to guide the internal dynamics of the system, thus conforming to the description of an HQMM.

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