Monte Carlo Simulation Bayesian Statistics

Bayesian Statistics

There are many areas within the field of Statistics which utilise Monte Carlo simulation techniques. In modern Statistics there has been great interest and emphasis placed upon the research and application of Bayesian Statistics. This area of Statistics provides an alternative framework for thinking about the world around us. In classical Statistics real-world phenomena can be modelled with the characteristics of these phenomena adequately described by some probability distribution of a parameter θ. Parameters are often meaningful quantities from the phenomena, for example the height of members of a population. Classical Statistics fixes such parameters within the model and lets observations drive any statistical inference regarding the real-world process. Within the Bayesian framework these parameters are viewed as unknown and random, but about which belief can be specified. Given evidence or observations from the real-world process these beliefs can be updated and improved to provide a better understanding of the process. This method of updating beliefs is derived from Bayes Theorem and hence gives rise to the name Bayesian Statistics.

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In the application of this basic Bayesian idea initial beliefs about parameters of the model are represented via a probability distribution known as the prior distribution P(θ). The probability or chance of making specific observations from the real-world phenomena given the model specified is known as the likelihood P(Data | θ). Updated beliefs having observed the data specify a probability distribution known as the posterior distribution P(Data | θ). These quantities are linked in Bayes Theorem as follows:

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There are a few main priorities within Bayesian Inference. Firstly it is important to use appropriate choices of prior distributions to correctly convey beliefs about the real-world process. For example if there is a large amount of uncertainty present in a belief about a parameter then this would lead to a choice of prior distribution which reflects this. Alternatively given that the updated beliefs represented by the posterior distribution are largely dependent upon the prior distribution it is common to use prior distributions which enable simpler analyses. This results in the idea of conjugacy, ie choices of certain prior distributions in certain contexts will yield posterior distributions from the same family. Situations in which conjugacy arises are highly desirable. The other main priority within Bayesian Inference is clearly to obtain posterior distributions and compute meaningful statistics from these. Conjugacy means that this process is quite simple and hence desirable. However the vast majority of situations involve posterior distributions which are difficult to both calculate and analyse. It is this aspect of Bayesian Statistics in which Monte Carlo simulation techniques have become so important.


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