Start Date: 03/24/2019
Course Type: Common Course
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This course describes Bayesian statistics, in which one's inferences about parameters or hypotheses are updated as evidence accumulates. You will learn to use Bayes’ rule to transform prior probabilities into posterior probabilities, and be introduced to the underlying theory and perspective of the Bayesian paradigm. The course will apply Bayesian methods to several practical problems, to show end-to-end Bayesian analyses that move from framing the question to building models to eliciting prior probabilities to implementing in R (free statistical software) the final posterior distribution. Additionally, the course will introduce credible regions, Bayesian comparisons of means and proportions, Bayesian regression and inference using multiple models, and discussion of Bayesian prediction. We assume learners in this course have background knowledge equivalent to what is covered in the earlier three courses in this specialization: "Introduction to Probability and Data," "Inferential Statistics," and "Linear Regression and Modeling."
Welcome! Over the next several weeks, we will together explore Bayesian statistics.
In this module, we will work with conditional probabilities, which is the probability of event B given event A. Conditional probabilities are very important in medical decisions. By the end of the week, you will be able to solve problems using Bayes' rule, and update prior probabilities.
Please use the learning objectives and practice quiz to help you learn about Bayes' Rule, and apply what you have learned in the lab and on the quiz.
This course describes Bayesian statistics, in which one's inferences about parameters or hypotheses
|Bayesian statistics||Bayesian statistics, named for Thomas Bayes (1701–1761), is a theory in the field of statistics in which the evidence about the true state of the world is expressed in terms of "degrees of belief" known as Bayesian probabilities. Such an interpretation is only one of a number of interpretations of probability and there are other statistical techniques that are not based on 'degrees of belief'. One of the key ideas of Bayesian statistics is that "probability is orderly opinion, and that inference from data is nothing other than the revision of such opinion in the light of relevant new information."|
|Bayesian statistics||The formulation of statistical models using Bayesian statistics has the unique feature of requiring the specification of prior distributions for any unknown parameters. These prior distributions are as integral to a Bayesian approach to statistical modelling as the expression of probability distributions. Prior distributions can be either hyperparameters or hyperprior distributions.|
|Bayesian statistics||Bayesian inference is an approach to statistical inference that is distinct from frequentist inference. It is specifically based on the use of Bayesian probabilities to summarize evidence.|
|Bayesian econometrics||where formula_12 and which is the centerpiece of Bayesian statistics and econometrics. It has the following components:|
|Bayesian model of computational anatomy||Several methods based on Bayesian statistics have emerged for submanifolds and dense image volumes.|
|Foundations of statistics||Bandyopadhyay & Forster describe four statistical paradigms: "(1) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) the Akaikean-Information Criterion-based statistics".|
|Bayesian estimation of templates in computational anatomy||Several methods based on Bayesian statistics have emerged for submanifolds and dense image volumes.|
|Charles Lawrence (mathematician)||He developed a tutorial on Bayesian statistics and Gibbs sampling, as well as the introduction courses in Bayesian statistics at Brown University.|
|Bayesian statistics||The general set of statistical techniques can be divided into a number of activities, many of which have special Bayesian versions.|
|Latent variable||Bayesian statistics is often used for inferring latent variables.|
|Ronald Fisher||Although a prominent opponent of Bayesian statistics, Fisher was the first to use the term "Bayesian".|
|IMDb||The formula_2 in this formula is equivalent to a Bayesian posterior mean (See Bayesian statistics).|
|Bayesian statistics||Statistical graphics includes methods for data exploration, for model validation, etc. The use of certain modern computational techniques for Bayesian inference, specifically the various types of Markov chain Monte Carlo techniques, have led to the need for checks, often made in graphical form, on the validity of such computations in expressing the required posterior distributions.|
|Bayesian statistics||The Bayesian design of experiments includes a concept called 'influence of prior beliefs'. This approach uses sequential analysis techniques to include the outcome of earlier experiments in the design of the next experiment. This is achieved by updating 'beliefs' through the use of prior and posterior distribution. This allows the design of experiments to make good use of resources of all types. An example of this is the multi-armed bandit problem.|
|Wishart distribution||The following variance computation could be of help in Bayesian statistics:|
|Prediction interval||Seymour Geisser, a proponent of predictive inference, gives predictive applications of Bayesian statistics.|
|Foundations of statistics||Statistics later developed in different directions including decision theory (and possibly game theory), Bayesian statistics, exploratory data analysis, robust statistics and nonparametric statistics. Neyman–Pearson hypothesis testing contributed strongly to decision theory which is very heavily used (in statistical quality control for example). Hypothesis testing readily generalized to accept prior probabilities which gave it a Bayesian flavor.|
|Charles Lawrence (mathematician)||In the past several years, based on the statistical algorithm development by Lawrence and his collaborators, several programs have also been publicly available and widely used, such as the Gibbs Motif Sampler, the Bayes aligner, Sfold, BALSA, Gibbs Gaussian Clustering, and Bayesian Motif Clustering. His work in Bayesian Statistics won the Mitchell Prize for outstanding applied Bayesian statistics paper in 2000.|
|Estimation theory||It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics). It is then necessary to define the Bayesian probability|
|Posterior predictive distribution||In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values.|