Bayesian Inference Based On Single Parameter Exponential Family

By Bayesian data analysis, we mean practical methods for making inferences from data using probability models for quantities we observe and for quantities about which we wish to learn. The essential characteristic of Bayesian methods is their explicituse of probability for quantifying uncertainty in inferences based on statistical data analysis.Based on Bayesian inference with single-parameter,we study at binomial model and normal model.We devide normal model situation in two parts;that is ‘Normal distribution with known variance but unknown mean’ and ‘normal distribution with known mean but unknown variance’ and discuss about them in details.We will talk about the connotation and development of Bayesian inference in the first chapter.In the second chapter,we will research binomial model and normal model.In the third one,single-parameter Bayesian data analysis will be used in football point spreads.And the last chapter is the summary of the research. The data come from the point spread and actual game outcome for 672 professional football games played during the 1981, 1983, and 1984.(Much of the 1982season was canceled due to a labor dispute.)After inspection,we are sure the samples have an approximate normal distribution with mean approximately zero and variance unknown.The process contain some steps as follow.We print a scatterplot of the 672 samples,and then get rid of the point whose point spread equals to zero,making the model concise.After caculating the probabilities of favorite wins, favorite wins|x=3.5, favorite wins by more than the point spread, favorite wins by more than the point spread|x=3.5,we find some differences.Further research,we define d,which is equals to actual outcome minus point spread,and print scatterplot and histogram.It’s obvious that most of the points are near the line y=0 according to the scatterplot.Based on the histogram,it’s easy to find that normal model fits the sample very much.Through further discussion,we assume the data points di,i=1,2,…n,are independent samplesfrom a N(0,) distribution.It lays a foundation to next steps.It’s easy to know follows to inverse- expanding from former knowledge.After theory,we use software R to compute the values.We find that the central 95%posterior interval for is [172.8547,214.1064],and thus of ’s is [13.1,14.6],with our goal to estimate finished.During deducting it,we obtain that the fair point spread is approximately 3.5。...