
Contents
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What is a function?
What is a function?
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Can we see another example?
Can we see another example?
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This function looks vaguely familiar…is this a linear function?
This function looks vaguely familiar…is this a linear function?
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So what does this have to do with regression analysis?
So what does this have to do with regression analysis?
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Can we see an example?
Can we see an example?
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Can you summarize the relationship between Var1 and Var3 with a straight line?
Can you summarize the relationship between Var1 and Var3 with a straight line?
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What’s the second equation written in black?
What’s the second equation written in black?
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So our goal is to find the signal within the data?
So our goal is to find the signal within the data?
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What exactly is science?
What exactly is science?
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How do we go about actually conducting science?
How do we go about actually conducting science?
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What flavor of science will we explore in this chapter?
What flavor of science will we explore in this chapter?
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Great! What will we be analyzing?
Great! What will we be analyzing?
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Grit?
Grit?
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Can you really measure a person’s grittiness?
Can you really measure a person’s grittiness?
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Can we have a peek at their dataset?
Can we have a peek at their dataset?
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How do I get started with my analysis?
How do I get started with my analysis?
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OK, so how do I find patterns between variables?
OK, so how do I find patterns between variables?
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Where were we?
Where were we?
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Is this a statistical model?
Is this a statistical model?
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Why is it incomplete? What exactly is a statistical model?
Why is it incomplete? What exactly is a statistical model?
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What assumptions do we make about how the Survivor data were generated?
What assumptions do we make about how the Survivor data were generated?
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Is that all there is to it?
Is that all there is to it?
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OK, where do we start?
OK, where do we start?
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And what is step 2?
And what is step 2?
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All of these prior distributions are vague. Couldn’t we have used existing Survivor data to help set our priors?
All of these prior distributions are vague. Couldn’t we have used existing Survivor data to help set our priors?
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What is step 3?
What is step 3?
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On to steps 4 and 5?
On to steps 4 and 5?
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How do we get started?
How do we get started?
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What is Gibbs sampling again?
What is Gibbs sampling again?
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How do we make this modification?
How do we make this modification?
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Brilliant! Can we get started now?
Brilliant! Can we get started now?
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Now can we see the Gibbs sampler in action?
Now can we see the Gibbs sampler in action?
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Goodness! That is a lot of subscripting to track and calculating to do!
Goodness! That is a lot of subscripting to track and calculating to do!
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And where does Bayes’ Theorem fit in?
And where does Bayes’ Theorem fit in?
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Are the Geman brothers responsible for this approach?
Are the Geman brothers responsible for this approach?
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Why the emphasis on marginal densities?
Why the emphasis on marginal densities?
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What do the results look like for all 10 trials?
What do the results look like for all 10 trials?
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How do we know if our MCMC results are any good?
How do we know if our MCMC results are any good?
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And how do we summarize our results?
And how do we summarize our results?
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What about the Bayesian credible intervals?
What about the Bayesian credible intervals?
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Can we do something similar for τ?
Can we do something similar for τ?
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So what is our linear equation?
So what is our linear equation?
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What would Bayes do?
What would Bayes do?
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Does the posterior predictive distribution help us assess model fit?
Does the posterior predictive distribution help us assess model fit?
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Do you get different results if you use a different prior?
Do you get different results if you use a different prior?
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Can we summarize this chapter?
Can we summarize this chapter?
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What’s next?
What’s next?
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17 The Survivor Problem: Simple Linear Regression with MCMC
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Published:May 2019
Cite
Abstract
While one of the most common uses of Bayes’ Theorem is in the statistical analysis of a dataset (i.e., statistical modeling), this chapter examines another application of Gibbs sampling: parameter estimation for simple linear regression. In the “Survivor Problem,” the chapter considers the relationship between how many days a contestant lasts in a reality-show competition as a function of how many years of formal education they have. This chapter is a bit more complicated than the previous chapter because it involves estimation of the joint posterior distribution of three parameters. As in earlier chapters, the estimation process is described in detail on a step-by-step basis. Finally, the posterior predictive distribution is estimated and discussed. By the end of the chapter, the reader will have a firm understanding of the following concepts: linear equation, sums of squares, posterior predictive distribution, and linear regression with Markov Chain Monte Carlo and Gibbs sampling.
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