杜克大学

Bayesian Statistics

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课程 4（共 5 门，Specialization Statistics with R）

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."

从本节课中

The Basics of Bayesian Statistics

<p>Welcome! Over the next several weeks, we will together explore Bayesian statistics. <p>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.</p><p>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.

Inference for a Proportion: Frequentist Approach3:32

Inference for a Proportion: Bayesian Approach7:32

Effect of Sample Size on the Posterior2:21

与讲师见面

Mine Çetinkaya-Rundel
Associate Professor of the Practice
Department of Statistical Science
David Banks
Professor of the Practice
Statistical Science
Colin Rundel
Assistant Professor of the Practice
Statistical Science
Merlise A Clyde
Professor
Department of Statistical Science

In the previous video, we worked through the RU-486 example with four successes,

pregnancies, in 20 trials, subjects.

Our prior distribution was uniform,

except that we placed a higher probability mass at p equals 0.50.

The likelihood of the data was calculated using the binomial distribution.

And peaked at p is equal to 0.20.

Putting together the prior and the likelihood, we obtained a posterior

distribution with the highest posterior probability at p is equal to 0.20.

While there's a peak at this value,

there's still some uncertainty in the posterior,

as other models also have some probability mass in the posterior distribution.

Next, let's take a look at what the posterior distribution would look

like if we had more data.

Suppose our sample size was 40 instead of 20,

and the number of successes was 8 instead of 4.

Note that we're still maintaining the 20% ratio between the sample size and

the number of successes.

We'll start with the same prior distribution.

Calculate the likelihood of the data which is also centered at 0.20, but is less

variable than the original likelihood we had with the smaller sample size.

And finally put these two together to obtain the posterior distribution.

The posterior also has a peak at p is equal to 0.20, but the peak is taller.

In other words, there is more mass on that model, and less on the others.

To illustrate the effect of the sample size even further,

we're going to keep increasing our sample size.

Still keeping the 20% ratio between the sample size and the number of successes.

So let's consider a sample with 200 observations and 40 successes.

Once again, we're going to use the same prior and

the likelihood is again centered at 20% and

almost all of the probability mass in the posterior is at p is equal to 0.20.

The other models do not have 0 probability mass, but

they're posterior probabilities are very close to zero.

So as you can see, as more data are collected,

the likelihood ends up dominating the prior.

This is why, while a good prior helps,

a bad prior can be overcome with a large sample.

However, it's important to note that this will only work as long as we don't place

a zero probability mass on any of the models in the prior.