Bayesian Networks (BNs), also known as Bayesian Belief Networks (BBNs) and Belief Networks, are probabilistic graphical models that use a directed acyclic graph (DAG) to represent a set of random variables along with their conditional interdependencies. These networks are useful in exploring and displaying causal relationships between key factors and system outcomes in a simple, easy-to-understand manner.

Initially developed as a formal method of analysing decision strategies under uncertain conditions in artificial intelligence research, BNs have since been found to be applicable to a wide range of problems of varying size and complexity, where uncertainty is inherent in the system. Although they have only recently begun to be used in environmental modelling, they have the potential to be highly useful in this field.

Bayesian networks use the Bayes' Theorem, also known as the Bayes' rule or the Bayes' law, in their operation. The prior probability represents the likelihood that an input parameter will be in a specific state in Bayes' theorem, while the conditional probability calculates the likelihood of a parameter's state given the states of input parameters affecting it. The posterior probability is the probability of a parameter being in a specific state given the input parameters, the conditional probabilities, and the rules governing how the probabilities are calculated.

Once nodes are updated in accordance with Bayes' Rule, the network is solved, as shown by the equation:

P(A|B) = (P(B|A) * P(A)) / P(B)

Where:

- P(A) is the parameter A's prior distribution

- P(B) is the parameter B's prior distribution

- P(A|B) is the posterior distribution, the probability of A given new data B

- P(B|A) is the likelihood function, the probability of B given existing data A.

In addition to their simple causal graphical structure, Bayesian networks have other appealing properties that make them particularly useful for data analysis and decision-making. They can be easily extended and modified, can incorporate missing data using Bayes' theorem, and have been shown to have good predictive accuracy. Additionally, they can be understood without much mathematical background.

BUILDING A BAYESIAN NETWORK:

The first steps in building a Bayesian network are to define the model's objectives and end-users. The proposed model should have a clear operational meaning for the modeller, 'domain' experts, and model users. The model's temporal and spatial scales should also be included in the objective. Where possible, this process should be carried out in a participatory setting to ensure that the full range of issues and potential inputs to the model are identified.

To help define focus issues and scales clearly, a conceptual model or influence diagram can be created. However, the conceptual model often needs modification to avoid becoming overly complex. The goal of a model should be to describe key system features rather than provide an accurate representation of reality.

Advantages and Disadvantages of Bayesian Networks:

Advantages:

- Graphical and visual networks provide a model for visualising the structure of probabilities as well as developing designs for new models.

- Relationships determine the type of relationship and whether or not it exists between variables.

- Computations efficiently solve complex probability problems.

- A Bayesian Network graph is readable by both computers and humans; both can interpret the information, unlike some networks that humans cannot read, such as neural networks.

Disadvantages:

Constructing a network is a costly endeavor, particularly when dealing with high-dimensional data. Complicating matters is the challenge of interpreting this network; distinguishing between causes and effects necessitates the use of copula functions.

Despite these obstacles, the Bayesian network has proved itself a valuable tool across a range of applications. Some of the most notable include:

1. Spam Filtering:

Gmail, among other email providers, uses a Bayesian spam filter to weed out unwanted and unsolicited emails.

2. Turbo Codes:

The creation of high-performance forward error correction codes is made possible thanks to Bayesian Networks. Such codes are commonly found in 3G and 4G mobile networks.

3. Image Processing:

Mathematical operations via Bayesian Networks assist in the conversion of images to digital format. These networks can also be used to enhance images.

4. Biomonitoring:

Quantifying chemical concentration in medical patients is made easier with Bayesian Networks. Indicators are used to measure blood and tissue.

5. Gene Regulatory Network (GNR):

Cell DNA segments interact with other cell contents through protein and RNA expression products in GNRs. Bayesian Networks are used to analyze predicted behavior.

In summary, Bayesian Networks are valuable tools despite the associated challenges. Their applications have been instrumental in fields such as spam filtering, mobile network creation, image processing, biomonitoring, and analysis of Gene Regulatory Networks.

References:

a) Neapolitan, R. E. Learning Bayesian networks.

b) Pearl, J. (1987). Evidential reasoning using stochastic simulation of causal models. Artificial Intelligence.

c) Jigsaw Academy. Bayesian Network Blog.

d) Wikipedia.