class: center, middle, inverse, title-slide .title[ # <p>Teaching Bayesian Statistics at the Undergraduate Level</p> Posterior Simulation and Analysis ] .subtitle[ ## ICOTS workshop ] .author[ ### Mine Dogucu ] .date[ ### 2022-09-11 ] --- class: top # Motivation .pull-left[ ### Unit 1 <img src="img/unit_1.png" width="75%" style="display: block; margin: auto auto auto 0;" /> ] -- .pull-right[ ### Unit 3 <img src="img/unit_3.png" width="75%" style="display: block; margin: auto auto auto 0;" /> ] --- class: top # The snag The regression and classification models in Unit 3 get complicated: `$$\begin{split} f(\beta_0,\beta_1,\sigma \; | \; \vec{y}) & = \frac{\text{prior} \cdot \text{likelihood}}{ \int \text{prior} \cdot \text{likelihood}} \\ & \\ & = \frac{f(\beta_0) f(\beta_1) f(\sigma) \cdot \left[\prod_{i=1}^{n}f(y_i|\beta_0, \beta_1, \sigma) \right]} {\int\int\int f(\beta_0) f(\beta_1) f(\sigma) \cdot \left[\prod_{i=1}^{n}f(y_i|\beta_0, \beta_1, \sigma) \right] d\beta_0 d\beta_1 d\sigma} \\ & \\ & = \text{no thanks!} \\ \end{split}$$` --- class: top # Unit 2 **Chapter 6: _Approximating_ the Posterior** How can we *approximate* a posterior using Markov chain Monte Carlo (MCMC) techniques? -- <hr> **Chapter 7: MCMC Under the Hood** How does MCMC *work*? -- <hr> **Chapter 8: Posterior Inference and Prediction** How can we use our approximation for inference and prediction? --- class: top # Activity Let `\(\pi\)` be the proportion of films that pass the Bechdel test. Starting with a Beta(2, 2) **prior** model for `\(\pi\)`, and **data** that 9 of 20 films passed the test, our **posterior** model of `\(\pi\)` is Beta(11, 13): <!-- --> --- class: top # Pretend Now **PRETEND** that we weren't able to mathematically specify our posterior. Instead we can **approximate** this posterior using a Markov chain Monte Carlo sample: `$$\left\lbrace \pi^{(1)}, \pi^{(2)}, \ldots, \pi^{(N)} \right\rbrace$$` -- - the `\(\pi\)` values are *not independent*: `\(\pi^{(i)}\)` depends upon `\(\pi^{(i-1)}\)` depends upon `\(\pi^{(i-2)}\)` depends upon...(and so on) -- - the `\(\pi\)` values are *not* drawn from the Beta(11, 13) posterior -- - but *mathemagically*, when done well, the Markov chain sample will provide a reasonable *approximation* of the posterior --- class: middle # Activity goals - Get your hands on some MCMC samples. - Perform some MCMC *diagnostics* (can we trust our chain?). - Use an MCMC sample to *approximate* features of the posterior model. - Optional: learn how to simulate a model using `rstan` - Be patient with yourself! Many exercises require you to tap into your intuition. Solutions are there to help when wanted.