What i have learned about Bayesian analysis
I am not too familiar with Bayesian method of data analysis, when I started reading this paper I found myself in unchartered waters, the first questions came to my mind, What is Bayesian statistics? Since Bayesian methods of data analysis based on Bayesian statistics. Bayesian statistics is; named after Thomas Bayes; a theory in statistics in which the evidence about the true state of an event is determined based on Bayesian probabilities. The main idea in Bayesian statistics is that the inferences from data is updated based on the findings of new relevant information.
Bayesian analysis can be divided into the following steps
In this step the current knowledge about the parameters of the model is determined by placing a probability distribution on those parameters and written in the form
Step2-Update model parameters:
When new data become available (y). The information they have about the model parameters is updated and expressed in a likelihood form which is proportional to the probability or distribution of the data collected given the model parameters. The format can be written in the below format.
Combine step 1 with step 2 to produce a new probability distribution called posterior distribution. The posterior is proportional to the prior times the likelihood and can be represented more accurately in the below format:
The graph below summarize and offer a visual representation of the above steps.
Bayesian analysis can solve many of the problems of conventional data analysis, but Bayesian statistic has problems of its own:
· Require intensive computations, especially when integration done over uncertain parameters. The invention of computers made such computations possible but before that Bayesian methods more often were not feasible.
· Require specifying prior probability distributions, which most of the time are unknown. Such assumption can be subjective, my assumption might be different than yours.
· Bayes theorem might be true for random variables such X and Y, but statisticians against the notion of treating hypothesis and parameters as random variables.
Gelman, A., Carlin, J.B., Stern, H.S. & Rubin, D.B. Bayesian Data Analysis (Chapman & Hall/CRC, Boca Raton, Florida, USA, 1995).
MacKay, D.J.C. Information Theory, Inference, and Learning Algorithms (Cambridge Univ. Press, Cambridge, UK, 2003).
Jaynes, E.T. Probability Theory: The Logic of Science (Cambridge Univ. Press, Cambridge, UK, 2003).
Hacking, I. The Emergence of Probability (Cambridge Univ. Press, Cambridge, UK, 1975).