The Bayesian method, also known as Bayesian method, is an inferential rule that is commonly used whenever an update is required within any probability estimate (Cohen et al. 12-57). The estimates are done on hypotheses whenever additional evidence is in the process of acquisition. This technique is majorly used in statistics and more especially mathematical statistics. In most of the cases involving the usage of Bayesian method requires, that it be derived through statistical methods. The derivation is usually automatic and ensures that the method well in combination with other competing methods that could have otherwise been used. This paper seeks to discuss the Bayesian method as applied in mathematical statistics. Either, it describes the algorithm that is used in the method clearly stating the input and output representations found within this method. The paper then discusses various measures that must be read out whenever this method is applied in updating various phenomena with an evaluation at the end of the possible results that its application might yield for researchers. It narrows down to the application in genetics to study how mimicry to evolution is studied through Bayesian model.

The Bayesian model has over the years been applied in wireless networks to connect low power sensors. These sensors have the ability to closely sense the activities of individuals and those of social interests. These sensor data provide contextual activities like location, surrounding and weather that are deduced using techniques such as markov models, ontology, clustering, Bayesian approaches and decision trees. The Bayesian networks are mainly used to features that define contexts. The process involves the usage of five major algorithms. The general Bayesian formula is as shown below:

The inference is used in deriving posterior probability through a sequence of two antecedents. The antecedents are the prior probability and the likelihood of a function that are both derived from of data under observation that is also probabilistic. In the formula above the elements are as explained below:

/- Is used to represent a conditional probability and in most cases it stands for ‘given’

H- Is a representation of any form of hypothesis whose probability stands to be affected by the data presented. In most cases, there are a number of hypotheses that do compete of which the most probable is chosen.

E- Is regarded as the evidence that usually acts a representation of the new data that were never used in the computation of the prior probabilities.

P(H) – This refers to the prior probability otherwise referred to as the probability of H before the observation of E takes place. It is a representation of previous probability estimates which states that a hypothesis is true before any other new data or evidence is given out.

P(H/E)- Represents the posterior probability. The posterior probability is the probability of H when E has been observed and given. It is usually a representation of the new evidence that the researchers need to know.

P(E/H)-. This stands for the probability of E being observed when H has been given. When it is given as a function of E when H is fixed, it is referred to as the likelihood of an occurrence taking place. The functional likelihood must never be confused with the function P(H/E) (which is a function of H rather than E). This function usually gives the compatibility of the new evidence as a function of the previous hypothesis.

P(E)- is also referred to as the marginal likelihood or even as the evidence model. It is a factor that is always the same in the hypotheses present.

Either, when different values of H are presented within any study, it is only the factors P(H) and P(E/H) that do affect the value of P(H/E). This is because both of these factors do appear in the numerator meaning that the posterior probability is always proportional to both the factors.

When applied in a network, the Bayesian method represents a relationship between disease and its symptoms in a probabilistic manner (Fox, 25-105). Within the network, computation of probabilities of different types of diseases present can be done. The nodes within the Bayesian nodes do represent quantities that are observable, variables that are latent, hypotheses or parameters that are unknown. The edges are always a representation of the conditional dependencies while the nodes that are not connected acting as a representation of the conditionally independent variable. The nodes are usually associated with the probability variables that are presented by the same nodes.

In genetics, the probabilistic paradigm can be used in the study of biological evolutions especially when it comes to mutations. This helps in yielding various successive generations through emulations of the reproduction to achieve appropriate fitness level. The mutation probability functions usually take up inputs that are of particular values for the parent nodes. This scenario can be illustrated as shown below: (Fox, 25-45) If the parent nodes are denoted by m Boolean variables, the probability function can then be represented in tabular form by 2m entries. Each entry has to be for the particular 2m combinations that are possible from the original parent nodes that can either be true or false. The same concept can also be applied in the case of an undirected or even a cyclic frame. Either, it can also be employed whenever graphs are used in the process of estimating various variables. This happens when the Markov networks analysis are being used.

Still in biological mutation, the Bayesian model can be used whenever a model selection is given in solving a problem (Marin et al, 32-67). Such problems require that two models are chosen on the basis of some data that have been observed. The Bayes factor integral can in this case be applied when likelihood of an occurrence taking is to be estimated. The occurrence whose likelihood is being estimated has to correspond to other occurrences of which it has to stand out as the maximum occurrence likely to take place. The test is thus referred to as the classical likelihood test ratio. In such as case, the model never depends on any set of single parameters. It integrates all the parameters within the area of study.

The advantage towards the usage of Bayes factor in biological mutation is that it includes a penalty for the inclusion of many elements within the model (Cohen, 45-65). The penalty takes place automatically and in a natural way without being determined. This feature helps guide against over fitting the model with too many elements and parameters. In the event that an explicit version of the likelihood does not avail itself or even considered to be too costly when it is evaluated in a numerical manner, a Bayesian model based on approximations can be used to undertake model selection. The model selection has to take place within the Bayesian framework with the approximate Bayesian estimates of the original Bayesian factor assumed to be biased.

Various measures have to be considered before the application of Bayesian models in biological mutation is undertaken (Fox, 12-65). The measures are to be taken in a hierarchical manner depending on the models. The measures apply particular whenever there are scale variables that are estimated to be at higher levels within the predetermined hierarchy. This process is always referred to as the choosing of priors with care taken on what type of prior is to be chosen. Priors such as the Jeffrey’s prior, in most cases, do not work effectively due to its inability to normalize the whole scenario. This happens whenever the posterior distributions become improper. Either, any estimates made through the minimization of the expected loss will in most cases not be admissible.

There are also efficient algorithms that do exist whenever an inference is performed in the process of applying a Bayesian network (Marin et al. 35-69). The Bayesian networks that are modeled on the basis of sequential variables are referred to as dynamic Bayesian networks. These variables could either be in the form of speech signals or even sequence of the proteins. Either, the term influence diagrams refer to the Bayesian networks that are general in nature and represents solutions to decision problems that are under uncertainty. This scenario can be illustrated as shown below:

Suppose two events that cause grass to become wet exist and the events are either a sprinkler or rain, it can be analyzed through Bayesian representation in a network form (Cohen at al. 56-59). If, for example the rain has some direct effect on the usage of the sprinkler in that whenever it rains the sprinkler never works, then this situation is capable of being modeled with a network as follows. The variables are assumed to have two possible outcome values which are T (Meaning Truth) and F (Meaning False). The joint probability function of the two variables can thus be represented as:

The formula abbreviations are as explained below:

G Stands for grass being wet which translates to yes/no

S stands for the sprinkler being turned on which translates to yes/no

R stands for it is raining also translating to yes/no

The formula above can be used to answer various questions such as what is the probability that it will possibly rain given the fact that the grass are already wet. Such a question demands that a probability function that is conditional be used. Either, the formula can also be used to answer intervention questions on likelihood of an occurrence taking place. The predictions that are determined using this probability function can only be feasible if the variables are observed.

The application of the Bayesian model has proved to be very fruitful for most of the researchers as the process takes care of various factors and variables that can work against the realization of an appropriate result. Either, the model provides various modifications of the original version that can be applied under various conditions upon which the application of the original version might not be appropriate. Thus, therefore, means that the Bayesian model is the most appropriate model that can be used by researchers in different fields of operations to study a number of variables. Either, the obtained results can further be estimated using the improved version of the Bayesian model that also takes care of various penalties that guides against its misuse

There are also other areas in which the Bayesian network can be applied. It has been widely applied in the process of modeling knowledge under computational biology. The same concept has also been used bioinformatics especially in the process of regulating gene networks. Either, it has also been used to determine the structure of the protein and in the analysis of the gene expressions. In art, especially in the area of sports, Bayesian networks have been extensively used in betting and learning the data sets in the epistasis. Other areas in which the Bayesian network analysis are applied are in the area of medicine, bio-monitoring, classification of documents, retrieval of information, data fusion, processing images and in decision support systems among others.

## Works Cited.

Cohen, Henri & Claire Lefebvre. Handbook Of Categorization In Cognitive Science. Amsterdam: Elsevier, 2005. Internet Resource.

Marin, Jean-Michel & Christian P. Robert. Bayesian Core: A Practical Approach To Computational Bayesian Statistics. New York: Springer, 2007. Internet Resource.

Fox, Jean-Paul. Bayesian Item Response Modeling: Theory And Applications. New York, Ny: Springer, 2010. Internet Resource.