# The the complete inference isthe posterior distribution of

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Last updated: September 23, 2019

The dataset that will be used is Pima Indians Diabetes Database of National Institute of Diabetes andDigestive and Kidney Diseases can be extracted by https://archive.ics.uci.edu/ml/datasets/Pima+Indians+Diabetes. Its refer to 786 pregnant women living near Phoenix, Arizona, USA.Attribute Information:1. Number of times pregnant;2.

Plasma glucose concentration a 2 hours in an oral glucose tolerance test;3. Diastolic blood pressure (mm Hg);4. Triceps skin fold thickness (mm);5.

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2-Hour serum insulin (mu U/ml);6. Body mass indexweightinkg(heightinm) 2 ;7. Diabetes pedigree function;8. Age (years);9. Outcome variable (0 or 1).In the frequentist paradigm the probability is interpreted as long-run relative frequency. In frequentiststatistics, the parameter p is a fixed but unknown constant, not a random variable.

The sampling distributionmeasures how the statistic varies over all possible samples, given the unknown fixed parameter value. Thisdistribution does not have anything to do with the actual data that occurred. Frequentist statistics uses thesampling distribution of the statistic to perform inference on the parameter. From a Bayesian perspective,this is a backwards form of inference.

This contrasts with Bayesian statistics where the complete inference isthe posterior distribution of the parameter given the actual data that occurred:g(pdata).The fundamental idea behind maximum likelihood estimation is that a good choice for the estimate ofa parameter of interest is the value of the parameter that makes the observed data most likely to haveoccurred. To do this, we need to establish some sort of function that gives us the probability for the data,and we need to ???nd the value of the parameter that maximizes this probability.

This function is called the”likelihood function” in classical statistics, and it is essentially the product of sampling densities-probabilitydistributions-for each observation in the sample.The basic elements of Bayesian inferential approach are introduced through the basic problem of learningabout a population proportion. Moreover we will compare this approach with the frequentist inference.Before taking data, one has beliefs about the values of the proportion and one models his or her beliefs interms of prior distribution. After data have been observed, one update one’s beliefs about the proportion bycomputing the posterior distribution. One summarizes this probability distribution to perform inferences.

Also, one may be interesting in predicting the likely outcomes of the news sample taken from the population.1In the present study, we are interested in learning about the Gestational diabetes develops during pregnancy.According to the National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK) ,women with gestational diabetes have a 35 to 60 percent chance of developing type 2 diabetes within 20 years.Gestational diabetes occurs in about 2 to 10 percent of all pregnancies.The incidence of obesity and gestational diabetes (GDM) is rising worldwide. Then in this study we willfocus on the BMI (body mass index) since many studies has proven that there are strong association betweenBMI and the risk of diabetes in pregnancy.Let p represent the proportion of the women with gestational diabetes in Pima Indians Diabetes Database.We are interested in learning about the location of p .

The value of the proportion p is unknown. In theBayesian viewpoint, a person’s beliefs about the uncertainty in this proportion are represented by a probabilitydistribution placed on this parameter. This distribution reflects the person’s subjective prior opinion aboutplausible values of p.

A random sample of Pima Indians Diabetes Database will be taken to learn about this proportion. Basedon the previous information researched about the gestational diabetes it will help us in constructing a priordistribution. We also believe, that the women overweight can contribute to the risk of diabetesin pregnancy, but in which proportion of the p?Many of the commands in R base package can be used in this setting. The probability distribution commandssuch as dbinom and dbeta , and simulation commands, such as rbeta , rbinom and sample , are helpful insimulating draws from the posterior and predictive distribution.

Also we illustrate the special R commandspdisc , histprior , and discint in the LearnBayes package, which are helpful in constructing prior andcomputing and summarize a posterior.