class: center, middle, inverse, title-slide .title[ # Getting Started with Stan ] .author[ ### Dan Ovando ] .date[ ### Updated 2022-06-23 ] --- # Objectives for Workshop 1. Basics of Bayes - What is it and why to use it 2. Bayesian regression with `rstanarm` - Diagnosing with `shinystan` - Getting and plotting results with `tidybayes`/`ggdist` 3. Roll your own - Writing your own models in Stan - Not going to address `brms` --- class: center, middle, inverse ### follow along at [danovando.github.io/learn-stan/slides](https://danovando.github.io/learn-stan/slides) --- # Objectives for Workshop <br> <br> <br> .center[**To Bayes or not to Bayes should be a philosophical question, not a practical one**] --- class: center, middle, inverse # Basics of Bayes --- # Basics of Bayes Bayes Theorem: <br> <br> $$p(model|data) = \frac{p(data|model) \times p(model)}{p(data)} $$ --- # Basics of Bayes `$$prob(thing|data) \propto prob(data|thing)prob(thing)$$` <img src="https://imgs.xkcd.com/comics/bridge.png " width="70%" style="display: block; margin: auto;" /> `$$prob(crazy|jumping) \propto prob(jumping|crazy)prob(crazy)$$` .center[if `\(friends = CRAZY\)`, then stay on bridge and eat cookies!] .small[[xkcd](https://imgs.xkcd.com/comics/bridge.png)] --- # Bayesian vs. Frequentist .pull-left[ ### Bayes Data are fixed, parameters are random What is the probability of a model given the data? - e.g. conditional on my model and data, how likely is it that a stock is overfished? Clean way to bring in prior knowledge But, means you have to think about priors ] .pull-right[ ### Frequentist Data are random, parameters are fixed How likely are the observed data if a given model is true? What you probably learned in stats classes. Can't really say anything about the probability of a parameter. No need to think about priors ] --- # A Likelihood Refresher Likelihoods and model fitting are an entire course, but we need some common language to move forward. What's missing from this regression equation? `$$y_i = \beta{x_i}$$` -- `$$y_i = \beta{x_i} + \epsilon_i$$` What do we generally assume about the error term `\(\epsilon_i\)`? -- OLS assumes that errors are I.I.D; independent and identically distributed `$$\epsilon_i \sim normal(0,\sigma)$$` --- # A Likelihood Refresher Another way to write that model is a data-generating model `$$y_i \sim N(\beta{x_i},\sigma)$$` This means that the likelihood ( P(data|model)) can be calculated as `$$\frac{1}{\sqrt{2{\pi}\sigma^2}}e^{-\frac{(\beta{x_i} - y)^2}{2\sigma^2}}$$` Or for those of you that prefer to speak code `dnorm(y,beta * x, sigma)` Most model fitting revolves around finding parameters that maximize likelihoods! --- # I thought you said I needed a prior? What we just looked at is a regression estimated via maximum likelihood (would get same result by OLS). To go Bayes, you need to specify priors on all parameters. What are the parameters of this model? `$$y_i \sim normal(\beta{x_i},\sigma)$$` -- We can use these priors for the parameters `$$\beta \sim normal(0,2.5)$$` `$$\sigma \sim cauchy(0,2.5)$$` And so our posterior (no longer just likelihood) is proportional to `dnorm(y,beta * x, sigma) x dnorm(beta,0, 2.5) X dcauchy(sigma,0,2.5)` ??? why only proportional? notice a bit of a dirty secret here: the priors have to stop somewhere. --- # A Quick Note on Priors .pull-left[ Priors can be freaky: We've been taught that our science should be absolutely objective Now your telling me I *have* to include "beliefs"?? * Somebody pour me a good strong p-value. ] .pull-right[ <img src="https://imgs.xkcd.com/comics/frequentists_vs_bayesians.png" width="80%" style="display: block; margin: auto;" /> ] --- # A Quick Note on Priors "Best" case scenario - You can include the results of another study as priors in yours You usually know *something* - What's your prior on the average length of a whale? Does get harder for more esoteric parameters - A uniform prior is NOT UNINFORMATIVE (informative) Data quickly overwhelm priors When in doubt, check sensitivity to different priors - If your key results depend on priors, be prepared to defend them - See [Schad et al. (2019)](https://pubmed.ncbi.nlm.nih.gov/32551748/) --- # Breath. .pull-left[ If that all made you a little queasy, don't worry, we're back to code! This can seem like deep stuff, but I really find it easier than the frequentist stats I was trained on. The problem becomes more about making models of the world than remember what test goes with what kind of problem. I can't recommend these two books enough. [Statistical Rethinking](https://xcelab.net/rm/statistical-rethinking/) [Bayesian Models: A Statistical Primer for Ecologists](https://xcelab.net/rm/statistical-rethinking/) ].pull-right[ <img src="https://media.giphy.com/media/51Uiuy5QBZNkoF3b2Z/giphy.gif" width="80%" style="display: block; margin: auto;" /> ] --- # Basics of Bayes - MCMC Big reason that Bayes wasn't historically used as much was that except for some specific cases Bayesian models have no analytical solution (and debates about priors) - Frequentist can usually solve for the answer (given some very specific assumptions usually involving asymptotics) Started to change when computers made Markov Chain Monte Carlo (MCMC) practical - Monte Carlo - try lots of different numbers - Markov Chain - an elegant way of choosing those numbers --- # Basics of Bayes - MCMC .pull-left[ MCMC can be shown to always converge! Sounds like magic, but basically says, "If you try every number in existence, you'll find the solution" The trick is finding an efficient way to explore the parameter space. ' - Something that jumps around with purpose ] .pull-right[ <img src="http://giphygifs.s3.amazonaws.com/media/DpN771RFKeUp2/giphy.gif" style="display: block; margin: auto;" /> ] --- # Basics of Bayes - MCMC ```r set.seed(42) x <- 1:500 #make up independent data true_param <- .5 # true relationship between x and y y <- true_param * x # simulate y steps <- 1e4 # number of MCMC steps to take param <- rep(0, steps) # vector of parameter values old_fit <- log(sum((y - (x * param[1]))^2)) # fit at starting guess jumpyness <- 1 # controls random jumping around for (i in 2:steps){ proposal <- rnorm(1,param[i - 1], 0.2) # propose a new parameter new_fit <- log(sum((y - (x *proposal))^2)) # calculate fit of new parameter rand_accept <- log(runif(1,0,jumpyness)) # adjust acceptance a bit if (new_fit < (old_fit - rand_accept)){ # accept in proportion to improvment param[i] <- proposal } else { param[i] <- param[i - 1] } } ``` --- # Basics of Bayes - MCMC <img src="slides_files/figure-html/unnamed-chunk-7-1.svg" style="display: block; margin: auto;" /> --- # Enter Stan Stan (primarily) uses a very elegant method called Hamiltonian Monte Carlo with a No-U-turn sampler (NUTs) to help Bayesian models converge quickly with relatively clear diagnostics. We don't have time to go into it, but trust me, it's cool. See [Monnahan et al. 2018](https://besjournals.onlinelibrary.wiley.com/doi/full/10.1111/2041-210X.12681) --- # Enter Stan <img src="https://besjournals.onlinelibrary.wiley.com/cms/asset/5ae32b4d-686e-4ea2-9d8a-402aba0a02fe/mee312681-fig-0003-m.jpg" width="50%" style="display: block; margin: auto;" /> --- # Why Stan? .pull-left[ - There are lots of Bayesian software options out there - BUGs, JAGS, NIMBLE, greta - Stan is my personal favorite - Blazing fast. - Amazing documentation and community - Robust & unique diagnostic tools - Good enough for Gelman - Available in lots of languages The Bayesian standard in social sciences, medicine etc. Modern Bayesian textbooks built around it ] .pull-right[  ] --- # Why Stan?  --- # Why Stan? .pull-left[ FYI, it's named after a person, [Stanislaw Ulam](https://en.wikipedia.org/wiki/Stanislaw_Ulam), inventor of Monte Carlo process, so it's "Stan", not "stan","STAN", "S.T.A.N." Cons... - No easy way for discrete latent parameters (maybe coming? maybe in `cmdstan`) - Less familiar in some parts of ecology? - ? ] .pull-right[  ] --- class: center, middle, inverse # Bayesian regression with `rstanarm` --- # Bayesian regression with `rstanarm` Bayesian methods traditionally required writing your own code of the model. Made running say a regression pretty annoying. - No one wants to run JAGS when `lm(y ~ x)` is on the table `rstanarm` was developed to reduce the "it's a pain" reason for not using Bayesian regressions - Modification of classic "applied regression modeling" (`arm`) package --- # Bayesian regression with `rstanarm` We're doing to play with the `gapminder` dataset <table> <thead> <tr> <th style="text-align:left;"> country </th> <th style="text-align:left;"> continent </th> <th style="text-align:right;"> year </th> <th style="text-align:right;"> lifeExp </th> <th style="text-align:right;"> pop </th> <th style="text-align:right;"> gdpPercap </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> Afghanistan </td> <td style="text-align:left;"> Asia </td> <td style="text-align:right;"> 1952 </td> <td style="text-align:right;"> 28.801 </td> <td style="text-align:right;"> 8425333 </td> <td style="text-align:right;"> 779.4453 </td> </tr> <tr> <td style="text-align:left;"> Afghanistan </td> <td style="text-align:left;"> Asia </td> <td style="text-align:right;"> 1957 </td> <td style="text-align:right;"> 30.332 </td> <td style="text-align:right;"> 9240934 </td> <td style="text-align:right;"> 820.8530 </td> </tr> <tr> <td style="text-align:left;"> Afghanistan </td> <td style="text-align:left;"> Asia </td> <td style="text-align:right;"> 1962 </td> <td style="text-align:right;"> 31.997 </td> <td style="text-align:right;"> 10267083 </td> <td style="text-align:right;"> 853.1007 </td> </tr> <tr> <td style="text-align:left;"> Afghanistan </td> <td style="text-align:left;"> Asia </td> <td style="text-align:right;"> 1967 </td> <td style="text-align:right;"> 34.020 </td> <td style="text-align:right;"> 11537966 </td> <td style="text-align:right;"> 836.1971 </td> </tr> <tr> <td style="text-align:left;"> Afghanistan </td> <td style="text-align:left;"> Asia </td> <td style="text-align:right;"> 1972 </td> <td style="text-align:right;"> 36.088 </td> <td style="text-align:right;"> 13079460 </td> <td style="text-align:right;"> 739.9811 </td> </tr> <tr> <td style="text-align:left;"> Afghanistan </td> <td style="text-align:left;"> Asia </td> <td style="text-align:right;"> 1977 </td> <td style="text-align:right;"> 38.438 </td> <td style="text-align:right;"> 14880372 </td> <td style="text-align:right;"> 786.1134 </td> </tr> </tbody> </table> --- # Bayesian regression with `rstanarm` Let's start with a simple model: predict log(life expectancy) as a function of log(gdp) <img src="slides_files/figure-html/unnamed-chunk-10-1.svg" style="display: block; margin: auto;" /> --- # Bayesian regression with `rstanarm` ```r simple_lifexp_model <- rstanarm::stan_glm(lifeExp ~ gdpPercap, data = gapminder, refresh = 1000, chains = 1) ``` ``` ## ## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1). ## Chain 1: ## Chain 1: Gradient evaluation took 8.4e-05 seconds ## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.84 seconds. ## Chain 1: Adjust your expectations accordingly! ## Chain 1: ## Chain 1: ## Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 1: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 1: ## Chain 1: Elapsed Time: 0.060138 seconds (Warm-up) ## Chain 1: 0.159642 seconds (Sampling) ## Chain 1: 0.21978 seconds (Total) ## Chain 1: ``` --- # Bayesian Regression using `rstanarm` ```r summary(simple_lifexp_model) ``` ``` ## ## Model Info: ## function: stan_glm ## family: gaussian [identity] ## formula: lifeExp ~ gdpPercap ## algorithm: sampling ## sample: 1000 (posterior sample size) ## priors: see help('prior_summary') ## observations: 1704 ## predictors: 2 ## ## Estimates: ## mean sd 10% 50% 90% ## (Intercept) 54.0 0.3 53.6 54.0 54.3 ## gdpPercap 0.0 0.0 0.0 0.0 0.0 ## sigma 10.5 0.2 10.3 10.5 10.7 ## ## Fit Diagnostics: ## mean sd 10% 50% 90% ## mean_PPD 59.5 0.3 59.0 59.5 59.9 ## ## The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')). ## ## MCMC diagnostics ## mcse Rhat n_eff ## (Intercept) 0.0 1.0 1142 ## gdpPercap 0.0 1.0 1099 ## sigma 0.0 1.0 1085 ## mean_PPD 0.0 1.0 1011 ## log-posterior 0.1 1.0 321 ## ## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1). ``` --- # Digging into `rstanarm` `rstanarm` function regression functions start with `stan_<model typs>` - `stan_glm` - `stan_glmer` - `stan_gamm4` - `stan_lm` In most ways the syntax etc. works the same as the ML versions: going from a standard binominal glm to a Bayesian as as simple as `glm(diagnosis ~ traits, data = data, family = binomial())` `stan_glm(diagnosis ~ traits, data = data, family = binomial())` --- # Digging into `rstanarm` This isn't the class to learn `glm`s / hierarchical modeling (see the books I keep plugging) What we will go over are some of the stan-specific options and diagnostics that apply across any model type you're using, but starting from this simple regression. --- # Key Stan options .pull-left[ `iter`: # of iterations per chain `warmup`: # of iter to use in warmup `chains`: # of chains to run `cores`: # of parallel cores to run chains on `max_treedepth`: controls how far the model will look for a new proposal before giving up <!-- - higher values allow model to explore a flatter posterior --> `adapt_delta`: the target rate that new proposals are accepted <!-- - higher means the model takes smaller steps --> `seed` sets the seed for the run ] .pull-right[ ```r simple_lifexp_model <- rstanarm::stan_glm( log(lifeExp) ~ log(gdpPercap), data = gapminder, iter = 2000, warmup = 1000, chains = 1, cores = 1, prior = normal(), control = list(max_treedepth = 15, adapt_delta = 0.8), seed = 42 ) ``` ] --- # iter, warmup, and chains The # of kept iterations you keep is `iter` - `warmup` - No need to thin! (though you can) - Only real reason is to reduce memory use of saved models? Balance between long enough to tune HMC / explore posterior, short enough to be fast. `warmup` defaults to half of `iter` Sometimes slower is faster: giving stan enough iterations to warmup properly can actually be faster than trying to go with small number of iterations. Best practice with any kind of MCMC is to run multiple chains - If working, should all converge to the same place - 4 is a good number - Chains can be run in parallel, but have startup costs - If model is really fast, parallel may be slower --- # `treedepth` and `adapt_delta` `treedepth` controls how many steps the NUTS sampler will take before giving up on finding a new value (at which time it will go back to where it started). Increasing this can helpful if the posterior is really flat. `adapt_delta` controls how small the step sizes are. The higher `adapt_delta` is, the higher the target acceptance rate is, and therefore the smaller the step size. - Basically, if the posterior has lots of small nooks and crannies, a low `adapt_delta` might result in step sizes that jump right over those points. - This will produce a "divergence" warning, which is not good (we'll come back to this) - Stan will suggest increasing `adapt_delta` if that happens --- # Setting Priors ```r simple_lifexp_model <- rstanarm::stan_glm( log(lifeExp) ~ log(gdpPercap), data = gapminder) ``` I thought you said Bayes requires priors? I don't see any in that regression. `rstanarm` has some pretty clever selectors for [weakly informative priors](https://mc-stan.org/rstanarm/articles/priors.html) --- # Setting Priors ```r rstanarm::prior_summary(simple_lifexp_model) ``` ``` ## Priors for model 'simple_lifexp_model' ## ------ ## Intercept (after predictors centered) ## Specified prior: ## ~ normal(location = 4.1, scale = 2.5) ## Adjusted prior: ## ~ normal(location = 4.1, scale = 0.58) ## ## Coefficients ## ~ normal(location = 0, scale = 2.5) ## ## Auxiliary (sigma) ## Specified prior: ## ~ exponential(rate = 1) ## Adjusted prior: ## ~ exponential(rate = 4.3) ## ------ ## See help('prior_summary.stanreg') for more details ``` --- # Setting Priors You can adjust these, and Stan recommends explicitly setting them since defaults may change. see `?rstanarm::priors` and the vignette [here](https://mc-stan.org/rstanarm/articles/priors.html) ```r lifexp_model <- stan_glm( log(lifeExp) ~ log(gdpPercap), data = gapminder, refresh = 0, prior = normal(autoscale = TRUE), # prior on the model coefficients prior_intercept = normal(autoscale = TRUE), # prior for any intercepts prior_aux = exponential(autoscale = TRUE) # in the case prior sigma ) ``` --- # Setting Priors Suppose I have a firm belief GDP is completely uncorrelated with life expectancy. ```r sp_lifexp_model <- stan_glm( log(lifeExp) ~ log(gdpPercap), data = gapminder, refresh = 0, prior = normal(0,0.025), # prior on the model coefficients prior_intercept = normal(autoscale = TRUE), # prior for any intercepts prior_aux = exponential(autoscale = TRUE) # in the case prior sigma ) ``` --- .pull-left[ <br> <br> ```r plot(sp_lifexp_model, pars = "log(gdpPercap)") + labs(title = "Strong Prior") ``` <img src="slides_files/figure-html/unnamed-chunk-18-1.svg" style="display: block; margin: auto;" /> ].pull-right[ <br> <br> ```r plot(lifexp_model, pars = "log(gdpPercap)") + labs(title = "Weak Prior") ``` <img src="slides_files/figure-html/unnamed-chunk-19-1.svg" style="display: block; margin: auto;" /> ] --- # Potential Exercise - Priors Try out a couple different kinds of priors - How informative do they have to be before they start substantially affecting results? - How does this change as you reduce the sample size of the data? - Look at `?rstanarm::priors` - Test out the effect of different distributions for the priors --- # Diagnosing `stan` fits `rstanarm` is just calling Stan in the background. This means that all the same diagnostics apply whether you're using `rstan`, `rstanarm`, `brms`, or any of the other Stan packages. The problem: things like `lm` will always¹ "work" Numerical optimizers like HMC have no such guarantee * You need to make sure the algorithm has converged Stan has numerous built-in diagnostics to help you do this. .footnote[ [1] except when it doesn't --- # Key Stan Diagnostics - Divergences Divergences are the big one to watch out for. Divergences are a warning that the model has missed some part of the parameter space (they're a handy product of the math behind HMC). Chains with large numbers of divergences probably have unreliable results. Simplest solution is to increase `adapt_delta`, which will slow down the model some. If divergences persist, try increasing the warmup period, and if that fails, you probably need to think about the structure of your model. A few divergences may still pop up once in a while: This doesn't automatically mean that your model doesn't work, but you should explore further to see what's going on. Thinning a model with a few divergences out of thousands of iterations can help. --- # Key Stan Diagnostics - `max_treedepth` The `max_treedepth` parameter of a Stan model controls the maximum number of steps that the NUTS algorithm will take in search of a new proposal. Getting lots of `max_treedepth exceeded` warnings means that the algorithm hit the max number of steps rather than finding where the trajectory of the parameter space doubles back on itself. This is more of an efficiency problem than a fundamental model fitting problem like divergences. Solutions include increasing `max_treedepth`, and trying a longer warmup period, and reparameterizing the model. --- # More Key `stan` Diagnostics * R-hat Measures whether all the chains have mixed for each parameter. Parameters with high R-hat values probably haven't converged * Bulk ESS Measures the effective sample size of each parameter. Too low suggests you might need more iterations, a better parameterization, or some thinning * Energy E-BFMI is a measure of the warmup success. A warning here suggests you might need a model reparameterization or just need a longer warmup. **THESE ARE ALL ROUGH APPROXIMATIONS SEE [here](https://mc-stan.org/misc/warnings.html) FOR MORE DETAILS** --- # Diagnosing Stan models <!-- This is one of the nicest features of Bayesian analysis: the diagnostics are more or less the same no matter what kind of model you're fitting. --> Stan has numerous built in functions for looking at these diagnostics, and will even include helpful suggestions about them! They generally stat with `rstan::check_<something>` ```r rstan::check_hmc_diagnostics(lifexp_model$stanfit) ``` ``` ## ## Divergences: ## ## Tree depth: ## ## Energy: ``` --- # Diagnosing with shinystan The built-in functions are great for diagnosing large numbers of models, make plots, etc. Stan also comes with a built in model visualizer called `shinystan` that allows you to explore all standard model diagnostics, as well as lots of other features ```r rstanarm::launch_shinystan(lifexp_model) ``` --- # An Unhappy HMC ```r sad_model <- stan_glmer( log(lifeExp) ~ log(gdpPercap) + (year | country), data = gapminder, cores = 4, chains = 4, refresh = 0, adapt_delta = 0.2, iter = 2000, warmup = 400 ) ``` --- # Happy HMC, Sad Model What happens if we successfully run a bad model? ```r library(gapminder) library(rstanarm) hist(log(gapminder$gdpPercap / 1e4)) gapminder$thing <- log(gapminder$gdpPercap / 1e4) bad_model <- rstanarm::stan_glm(thing ~ year, family = gaussian(link = "log"), data = gapminder) ``` --- class: inverse, center, middle # Analyzing Results --- # Analyzing Results So far we've covered the bare bones basics of how to fit and care for a Stan model. The objective of all that model fitting is to look at our results and make inference! We're now going to look at how to do that with stan models. Similar to the diagnostics, these apply to any kind of stan model. --- # Where are my results?? For `rstanarm` models, a lot of the standard methods for getting summary results from regressions work (e.g. `broom::tidy()`) ```r lifexp_model %>% broom.mixed::tidy() ``` ``` ## # A tibble: 2 × 3 ## term estimate std.error ## <chr> <dbl> <dbl> ## 1 (Intercept) 2.86 0.0235 ## 2 log(gdpPercap) 0.147 0.00284 ``` I'm not going to focus on those here. Instead we're going to look at methods for extracting and visualizing results from any stan model. --- # `rstan::extract` Our simple model has two parameters: `(Intercept)` and `log(gdpPercap)` Let's pull out the HMC draws for each one (note that with `rstanarm` you have to use the stanfit object) ```r rstan::extract(lifexp_model$stanfit, permute = TRUE) %>% listviewer::jsonedit() ``` <div id="htmlwidget-9ed1f3c02d4bbb93f309" style="width:504px;height:288px;" class="jsonedit html-widget"></div> <script type="application/json" 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-- --- # `tidybayes` `tidybayes` is a great package for getting results out of Stan models (and any kind of Bayesian model actually, even JAGS!) ```r tidybayes::tidy_draws(lifexp_model) %>% select(1:5) %>% head() %>% knitr::kable(digits = 2, format = "html") ``` <table> <thead> <tr> <th style="text-align:right;"> .chain </th> <th style="text-align:right;"> .iteration </th> <th style="text-align:right;"> .draw </th> <th style="text-align:right;"> (Intercept) </th> <th style="text-align:right;"> log(gdpPercap) </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 2.86 </td> <td style="text-align:right;"> 0.15 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2 </td> <td style="text-align:right;"> 2.85 </td> <td style="text-align:right;"> 0.15 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 2.85 </td> <td style="text-align:right;"> 0.15 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 4 </td> <td style="text-align:right;"> 2.85 </td> <td style="text-align:right;"> 0.15 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 5 </td> <td style="text-align:right;"> 2.87 </td> <td style="text-align:right;"> 0.15 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 6 </td> <td style="text-align:right;"> 6 </td> <td style="text-align:right;"> 2.89 </td> <td style="text-align:right;"> 0.14 </td> </tr> </tbody> </table> --- # more `tidybayes` `spread_draws` and `gather_draws` are more commonly used - No idea if these names will changes to reflect `tidyr 1.0` ```r tidybayes::gather_draws(lifexp_model,`(Intercept)`,`log(gdpPercap)`) ``` ``` ## # A tibble: 8,000 × 5 ## # Groups: .variable [2] ## .chain .iteration .draw .variable .value ## <int> <int> <int> <chr> <dbl> ## 1 1 1 1 (Intercept) 2.86 ## 2 1 2 2 (Intercept) 2.85 ## 3 1 3 3 (Intercept) 2.85 ## 4 1 4 4 (Intercept) 2.85 ## 5 1 5 5 (Intercept) 2.87 ## 6 1 6 6 (Intercept) 2.89 ## 7 1 7 7 (Intercept) 2.86 ## 8 1 8 8 (Intercept) 2.81 ## 9 1 9 9 (Intercept) 2.92 ## 10 1 10 10 (Intercept) 2.88 ## # … with 7,990 more rows ``` --- # Bayes + Tidyverse ```r tidybayes::gather_draws(lifexp_model,`(Intercept)`,`log(gdpPercap)`) %>% ggplot(aes(.value, fill = factor(.chain))) + geom_density(alpha = 0.75) + facet_wrap(~.variable, scales = "free") + theme_minimal() + theme(legend.position = "top") ``` <img src="slides_files/figure-html/unnamed-chunk-28-1.svg" style="display: block; margin: auto;" /> --- # [`tidybayes`](https://github.com/mjskay/tidybayes) & [`ggdist`](https://mjskay.github.io/ggdist/) Lots of examples, explore on your own. ```r lifexp_model %>% as.data.frame() %>% ggplot(aes(`log(gdpPercap)`)) + stat_halfeye() ``` <img src="slides_files/figure-html/unnamed-chunk-29-1.svg" style="display: block; margin: auto;" /> --- # Statistical Tests - Bayesian vs. Frequentist <img src="imgs/rethinking-golem.png" width="1699" height="450" style="display: block; margin: auto;" /> .footnote[McElreath -Statistical Rethinking ] --- # Statistical Tests with Stan Suppose I show you the following results from `lm` >The estimated coefficient of log(gdpPercap) was 0.146 (95% CI 0.14-0.15) How would you interpret this? -- <img src="https://media.giphy.com/media/K8zzqui9viWT6/giphy.gif" width="80%" style="display: block; margin: auto;" /> ??? More of less, if you repeated the same experiment many times, 95% of the time the CI from this model would contain the true value --- # Statistical Tests with Stan Remember, in a Bayesian world, we've estimated a posterior probability: `$$P(model|data)$$` This sounds a lot more like what we want! Bayesian models allow us say things like > Conditional on the data and the model, there is a 95% probability that the coefficient of log(gdpPercap) is between 0.14 and 0.15 --- # Statistical Tests with Stan In a Bayesian world, most statistical tests go from mathematical exercises to data wrangling! Let's augment our model a bit to play with this idea. ```r lifexp_model <- stan_glm(log(lifeExp) ~ log(gdpPercap) + country, data = gapminder, refresh = 0, adapt_delta = 0.95, iter = 15000, cores = 4) ``` --- # Statistical Tests with Stan Suppose we wanted to know the probability that the effect of `log(gdpPercap)` was greater than 0? ```r lifexp_model %>% tidybayes::tidy_draws() %>% summarise(`Prob. log(gdp) effect is > 0` = mean(`log(gdpPercap)` > 0)) ``` ``` ## # A tibble: 1 × 1 ## `Prob. log(gdp) effect is > 0` ## <dbl> ## 1 1 ``` --- # Statistical Tests with Stan Suppose we wanted to estimate the mean difference in the intercepts of Europe and the Americas? ```r lifexp_model %>% tidybayes::gather_draws(`country.*`, regex = TRUE) %>% mutate(country = str_replace(.variable, "country",'')) %>% left_join(gapminder %>% select(country, continent) %>% unique()) %>% group_by(.draw, continent) %>% summarise(mi = mean(.value)) %>% group_by(.draw) %>% summarise(`Continent Difference` = mi[continent == "Europe"] - mi[continent == "Americas"]) -> continent_delta head(continent_delta,5) ``` ``` ## # A tibble: 5 × 2 ## .draw `Continent Difference` ## <int> <dbl> ## 1 1 -0.00894 ## 2 2 -0.00837 ## 3 3 -0.00217 ## 4 4 -0.00182 ## 5 5 -0.00431 ``` --- # Statistical Tests with Stan <img src="slides_files/figure-html/unnamed-chunk-35-1.svg" style="display: block; margin: auto;" /> ??? We basically just replaced that crazy flowchart of statistical tests with the `tidyverse` --- # Potential Exercise - Penguins! Using `rstanarm` answer the most pressing scientific question of our time: On average, how much chonkier are Gentoo penguins than Adelie penguins? For extra fun, what is the probability that Gentoo penguins are on average 1400g heavier than Adelie? The press awaits your answer (with uncertainty) --- # Potential Exercise - Penguins! <img src="slides_files/figure-html/unnamed-chunk-36-1.svg" style="display: block; margin: auto;" /> --- # Model Comparison with `loo` We often use things like AIC to pick between models - What is AIC measuring? -- Stan has built in functionality for leave-one-out cross validation `loo` - Usable on any `stanfit` object Lots of options besides `loo` - WAIC, BIC, DIC We don't have time to dig into this, but see documentation [here](https://mc-stan.org/loo/) --- # Model Comparison with `loo` `elpd_diff` measures how much worse a model is than the preferred model - Should be at least 2 times `se_diff` to be meaningful ```r model_a <- stan_glm(log(lifeExp) ~ log(gdpPercap), data = gapminder, refresh = 0) model_b <- stan_glm(log(lifeExp) ~ log(gdpPercap):continent, data = gapminder, refresh = 0, iter = 2000) loo::loo_compare(loo(model_a), loo(model_b)) ``` ``` ## elpd_diff se_diff ## model_b 0.0 0.0 ## model_a -116.1 18.0 ``` ??? The difference in ELPD is much larger than several times the estimated standard error of the difference again indicating that the --- class: center, middle, inverse # Writing models in Stan --- # Writing models in Stan `rstanarm` is great and helps make Bayesian regressions just as easy as `lm/glm/gamm` etc. Hopefully you've also seen that Bayesian models can make statistical inference MUCH simpler. Lots of times though, we want to move beyond linear regression. Bayesian estimation through HMC is particularly good at estimating really complex models. For that, you may need to write your own stan code - [brms](https://paul-buerkner.github.io/brms/) is a great package for writing non-linear models in Stan without writing the Stan code. - Not going to cover here: I recommend learning the real deal for complex problems --- # Stan Stan is a programming language written in C++, with a lot of "sugar" to make your life easier. But, it is still a compiled language. This means that unlike R / Python, you can't just run it line by line in an IDE. This makes it much faster, but also a little tougher to get going. But, the good news is that you can use R for all your pre-and-post model wrangling, and Stan for just the model fitting. --- # Anatomy of a .stan file Best way is to write a Stan model is create a new .stan file. Each stan model is broken into blocks ```stan functions{ // store user declared functions } data{ // declare data being passed to the model } parameters{ // declare parameters to be estimated } transformed parameters{ // apply transformations to parametres (e.g log) } model{ // define the statistical parameters } generated quantities{ // calculate things like posterior predictives } ``` --- # General Stan syntax For the most part, most things work like they do in R! - e.g. `for`, `if`, `while` work exactly the same - `x[1,1:5]` gives the first five columns of the first row of x - Things are indexed at 1! Major differences - It's C++, so everything has to end with `;` - You have to explicitly declare types and dimensions - This is usually the hardest part for people - `//` instead of `#` for comments --- # Data Types R is very permissive about data types and sizes - real times an integer, no problem - Matrix times an array, no problem - Automatically creates storage of the right size for you Most languages are not so forgiving, including Stan - Uses strong, static typing - You have to explicitly declare variable types (real, integer, logical, etc) - You have to explicitly declare their size - This and the need to compile seems to be what throws people the most --- # Data Types The main ones you will deal with are - `real`: your basic continuous numbers - `integer`: integers! mostly used for indexing ```stan real x; // a real number int n; // an integer vector[n] z; // a vector of length n int index[n] // an array of n 1-D objectes (I know, I know) ``` From there, you can store things in matrices or arrays --- # Data Types: matrices Core matrix classes are - `vector` for column vectors - `row_vector` for row vectors - `matrix` for 2D matrices Matrices **can only store real numbers** (we'll come back to this) By default, Stan performs matrix operations (e.g. multiplication) on matrix objects - `.*` or `./` for elementwise operations - use `'` to transpose, e.g. `x'` will convert x from a column vector to a row vector --- # Data Types: arrays Matrices are by definition special classes of 2D objects - Think of them as your basic math building blocks Arrays are a little more permissive: think of them more like lists in R - Can store arbitrary numbers of objects within individual slots (all of the same type) ```stan array[3, 4] real a; ``` creates a 3 x 4 array with a 1D real number in each slot Think of arrays as storage, matrix as math. --- # Arrays & Integers Arrays are the only object type that can store sequences of integers Suppose that you want to store a sequence of integers to identify a set of columns in a matrix ```stan int n; // number of integers array[n] integer a; // an array with n slots each with an integer in them matrix[2,8] x; // a 2 x 8 matrix x[1,a]; // the entries of x at positions a in row 1 ``` --- # Loops etc. Good news! `for`, `if`, `while` etc all work exactly like they do in R You don't need to declare loop indices. ```stan for (i in 1:10){ if (i < 2){ } // close if else { while (i < 6){ } // close while } // close else } // close for ``` --- # Writing models in Stan We'll practice this by filling in a model ```stan real x; x = 2 + 2; ``` --- # The Data <img src="slides_files/figure-html/unnamed-chunk-44-1.svg" style="display: block; margin: auto;" /> --- # data block Anything being passed from R to stan goes in here Remember you need to declare both type and dimensions! ```stan data{ int<lower = 1> n; //number of observations vector[n] spawners; // vector of spawners vector[n] recruits; // vector of recruits } ``` --- # The Model We're going to fit a simple Beverton-Holt Spawner-Recruit model to these data. `$$log(recruits) \sim N\left(\frac{(0.8 \times r0 \times h \times s)}{ (0.2 \times s0 \times (1 - h) +(h - 0.2) \times s)},\sigma\right)$$` <img src="slides_files/figure-html/unnamed-chunk-46-1.svg" style="display: block; margin: auto;" /> --- # transformed data block All objects must be declared in the first part of a block ```stan transformed data{ vector[n] log_recruits; // log recruitment log_recruits = log(recruits); } ``` --- # parameter block The parameter block is where you declare things the model will estimate. Notice that Stan lets you declare bounds in here! ```stan parameters{ real<lower = 0.2, upper = 1> h; //steepness real log_r0; // unfished recruitment real log_s0; // unfished spawners real<lower = 0> sigma; // observation error } ``` --- # transformed parameter block Consider a population model: The biomass over time if just a transformation of the parameters. By default Stan returns draws from things in the transformed parameters ```stan transformed parameters{ vector[n] recruits_hat; vector[n] log_recruits_hat; real r0; real s0; r0 = exp(log_r0); s0 = exp(log_s0); recruits_hat = (0.8 * r0 * h * spawners) ./ (0.2 * s0 * (1 - h) +(h - 0.2) * spawners); log_recruits_hat = log(recruits_hat); } ``` --- # model block This is where you write your *statistical* model Things created in `model` block are only visible in `model` block - Trade-off between output usability/clarity Note naturally vectorized performance! **You can't put priors on transformed parameters** .footnote[without some voodoo] ```stan model{ log_recruits ~ normal(log_recruits_hat - 0.5 * sigma^2, sigma); // bias correction sigma ~ cauchy(0,2.5); log_s0 ~ normal(15,2); log_r0 ~ normal(8,2); h ~ beta(6,2); } ``` --- # generated quantities This is mostly used for generating posterior predictives. Note the R-like syntax of most things Note *non-vectorized* performance ```stan generated quantities{ vector[n] pp_rhat; for (i in 1:n) { pp_rhat[i] = exp(normal_rng(log_recruits_hat[i] - 0.5 * sigma^2, sigma)); } } ``` --- # Fitting the Model Once you've written the model, you just need to pass it to `rstan::stan` ```r chains <- 4 inits <- list(s0 = max(flound$ssb) * 1.5, r0 = max(flound$recruits), h = 0.7) inits <- lapply(1:chains, function(x) map(inits, jitter,1)) bh_fit <- rstan::stan( file = here::here("src","bh_model.stan"), data = list(spawners = flound$ssb, recruits = flound$recruits, n = length(flound$ssb)), init = inits, chains = chains, cores = 1, iter = 10000, refresh = 0 ) ``` --- # A quick note on `inits` Stan and HMC seem pretty magic at times, but they are still numerical optimiziers - If you start them close to the right values, things will be easier You can pass initial parameter values to Stan using the `init` command - By default Stan does some wizardry behind the scenes to come up with good inits `init` takes the form of a "list of lists" - One list per chain - Inside each list per chain a list of initial parameter values - Can provide initial values for as few or as many parameters as you want, missing parameters get Stan's defaults - A reason that centering and scaling thing can be helpful --- # A quick note on `inits` Here's a way that I generate a list of starting conditions with slightly different values Basically, a loop of loops over the elements ```r inits <- list(s0 = max(flound$ssb) * 1.5, r0 = max(flound$recruits), h = 0.7) inits <- lapply(1:chains, function(x) lapply(inits, jitter,10)) inits ``` ``` ## [[1]] ## [[1]]$s0 ## [1] 873111.3 ## ## [[1]]$r0 ## [1] 796376541 ## ## [[1]]$h ## [1] 0.5973038 ## ## ## [[2]] ## [[2]]$s0 ## [1] 711511.4 ## ## [[2]]$r0 ## [1] 725593277 ## ## [[2]]$h ## [1] 0.8043471 ## ## ## [[3]] ## [[3]]$s0 ## [1] 831628 ## ## [[3]]$r0 ## [1] 728288638 ## ## [[3]]$h ## [1] 0.7381813 ## ## ## [[4]] ## [[4]]$s0 ## [1] 913759.6 ## ## [[4]]$r0 ## [1] 916196317 ## ## [[4]]$h ## [1] 0.8292789 ``` --- # Some weird things in passing data I lost days of my life to these. If you want to pass a factor to Stan, you need to first convert it to an integer ``` data = list(x = as.integer(my_factor)) ``` In R, a scalar (one number) is a vector of length 1. This makes Stan mad. ``` data { int<lower=1> N; real y[N]; } ``` Suppose N = 1? To pass this to Stan from R ``` data = list(y = as.array(y)) ``` Note that ``` real y[N]; array[N] real y; ``` Are the same --- # Debugging Stan Debugging compiled languages is a skill that takes practice: you can't just go line-by-line and try things! There's a "check" button in the top-right corner of .stan files in RStudio - This will check if your file is *syntactically correct* `print(this_should_work);` prints out the content as the code runs - A good way to see if you accidentally have negative fish `algorithm = "Fixed_param"` is crazy handy - Forces Stan to keep sampling from the same initial parameters you pass it - Good way to test model behavior under specific parameters --- # Examining Fits Even though we're not using rstanarm anymore, the diagnostics are still the same. ```r check_hmc_diagnostics(bh_fit) ``` ``` ## ## Divergences: ## ## Tree depth: ## ## Energy: ``` We can also use `shinystan` to explore our fit --- # Looking at our fits Let's first examine fit of our estimated spawner recruit data. By default, `stan` returns draws from anything defined in `parameters`, `transformed parameters`, and `generated quantities` block Let's use `tidybayes` to get our fitted and posterior predictive recruits out. Note ability of `tidybayes` to deal with variables with more than one value ```r fitted_recruits <- tidybayes::gather_draws(bh_fit, recruits_hat[year], pp_rhat[year]) head(fitted_recruits,2) ``` ``` ## # A tibble: 2 × 6 ## # Groups: year, .variable [1] ## year .chain .iteration .draw .variable .value ## <int> <int> <int> <int> <chr> <dbl> ## 1 1 1 1 1 recruits_hat 288216770. ## 2 1 1 2 2 recruits_hat 291165098. ``` --- # Looking at our Fits Let's calculate the mean and 80% credible interval for each type of prediction over time ```r fitted_recruits <- fitted_recruits %>% group_by(year, .variable) %>% summarise(rhat = mean(.value), lower = quantile(.value, .055), upper = quantile(.value, .945)) %>% ungroup() %>% mutate(year = year + min(flound$year) - 1) ``` --- # Looking at our Fits And now plot! Why is the posterior predictive so much wider than the mean prediction? <img src="slides_files/figure-html/unnamed-chunk-57-1.svg" style="display: block; margin: auto;" /> --- # Potential Exercise * Extract the posterior of the steepness parameter *h* * Plot the posterior,and compare to the prior * Try and make the prior more/less informative and compare the prior/posterior plots --- # Using your model to make predictions Often in environmental problems we need to fit our model to one set of data, and predict on others. This is pretty easy using `rstanarm` and `brms`, but a bit more challenging using `stan`. The recommended option is to allow your model to take different data for the posterior predictive than the fitting data As an alternative, you can use the `algorithm = "Fixed_param"` option, and apply each draw from your posterior to your model. - See example [here](https://github.com/DanOvando/learn-stan/blob/master/scripts/generate-stan-predictions.R) --- # Challenges of Bayes Bayesian and Frequentist methods both make assumptions that may be more or less valid in different circumstances - Can be slower (but getting faster), especially for large models - But might not be the answer for your spatial-temporal model with 10,000 latent variables - Setting priors can be tough - Can't set multiple priors on the same thing - Make sure prior in the real world means the same thing as the prior in the small world (see [Choy et al. 2009](https://esajournals.onlinelibrary.wiley.com/doi/full/10.1890/07-1886.1)) - Can be abused... - No more significance so anything goes! - With strong enough priors anything is estimable... - Readers (reviewers) may be uncomfortable with it - Where's the regression table? **WHERE ARE THE P-VALUES** - Always clearly present your priors, and include sensitivity analyses - See Schad et al. (2019) "Toward a principled Bayesian workflow in cognitive science" --- # Where to learn more? Bayesian modeling and Stan are big and complex topics - They can be extremely useful in ecological modeling - They're not that scary Recommended Books - [Statistical Rethinking](https://xcelab.net/rm/statistical-rethinking/) - [Bayesian Models: A Statistical Primer for Ecologists](https://xcelab.net/rm/statistical-rethinking/) - [Bayesian Data Analysis](http://www.stat.columbia.edu/~gelman/book/) Online Resources - the Stan [community](https://discourse.mc-stan.org/) - Stan [documentation](https://mc-stan.org/rstan/) - [Example](https://github.com/stan-dev/example-models) models (including reworked BUGS examples) - [My tutorial](https://www.weirdfishes.blog/blog/fitting-bayesian-models-with-stan-and-r/) --- # Potential Exercise Assess Namibian hake! - Working alone or in groups try and fit a Schaefer model to the supplied Namibian hake data - Catches, CPUE - I'll check in every so often and ask for volunteers to describe their steps so far so we'll crowdsource our final model - If this is easy for you, try adding process error, using a Pella-Tomlinsion, etc. --- # Hake Data <img src="slides_files/figure-html/unnamed-chunk-58-1.svg" style="display: block; margin: auto;" /> --- # Starting tips Assume logistic growth `$$b_{t+1} = b_t \left(1 + r\left(1 - \frac{b_t}{k}\right)\right) - c_t$$` Assume CPUE is proportional to biomass per a constant q `$$cpue_t = qb_t$$` Assume log=normal errors, such that `$$log(cpue) \sim N(log(\hat{cpue}), \sigma_{obs})$$` Hint: what to do if `\(c_t >> b_t\)`? ---