The first two sections are spent describing some of the general probmes that statisticians and researchers face is designing statistical tests and models. 1.3 Tools for golem engineering use models for several distinct purpose: designing inquiry extracting information from data making predictions this book focuses on the following tools towards these purposes: Bayesian data analysis model comparison multilevel models graphical causal models this book focuses mostly on code - how to do things (“golem engineering”) 1.

all models contain a “small world” and a “large world the small world is the self-contained logical world of the model the large world is the broader context in which one deploys a model Bayesian models learn from evidence that is optimal in the small world and must be demonstrated to perform well in the large world this chapter focuses on the small world 2.

this chapter teaches the basic skills for working with samples from the posterior distribution 3.1 Sampling from a grid-approximate posterior the following code generates a posterior distribution using grid approximation using the globe-tossing model from the previous chapter p_grid <- seq(from = 0, to = 1, length.

this chapter introduce linear regression as a Bayesian procedure under a probability interpretation (necessary for Bayesian word), linear reg. uses a Gaussian distribution for uncertainty about the measurement of interest 4.

correlation is very common in the real world multivariate regression is using more than one predictor variable to model an outcome reasons to use multivariate regression: a way of “controlling” for confounding variables multiple causation interaction between variables this chapter focuses on two thing multivariate models can help with: revealing spurious correlations revealing important correlations masked by hidden correlations of other variables this chapter will also discuss: multicolinearity categorical variables 5.

simpler models tend to be preferred this chapter deals with comparing explanatory value of a model and its complexity there are two general families: regularization use a scoring mechanism like information criteria this chapter introduces information theory to cover these metrics 6.

so far, we have assumed that each predictor has an independent association with the mean of the outcome now we will look at conditioning this estimate on another predictor using interactions fitting a model with interactions is easy, but understanding them can be harder 7.

estimation of posterior probability distributions using a stochastic process called *Markov chain Monte Carlo (MCMC)** estimation sample directly from the posterior instead of approximating curves (like the quadratic approximation) allows for models that do not assume a multivariate normality can use generalized linear and multilevel models use Stan to fit these models 8.

Gaussian distributions work well very frequently there are times it is better to use a different distribution if using a different distribution: use the distribution with the biggest entropy, because: the biggest entory is the widest and least informative distribution that still contrains the problem to values that are possible nature tends to produce emperical distributions with high entropy it works (pragmatic justification) this chapter is an introduction to generalized linear models (GLM) and the principle of maximum entropy GLMs are similar to the linear regressions we have used previously but they need not use Gaussian likelihoods any likelihood function can be used parameters can be atached to any of the parameters that describe its shape the principle of maximum entropy helps choose the likelihood functions choose the most conservative prior with the known constraints the following chapters look more closely as a few specific GLM types (counts, ordinal outcomes, mixtures) 9.

information if often thrown away by using ratios of counts instead of the counts themselves 10/20 and 1/2 are the same ratio, but the first has more information there is friction in using the count data instead of the proportions interpretation is less intuitive this chapter covers the two most popular count distributions: binomial regression: model a binary classification dead/alive, accept/reject 2 Poisson regression: models outcome without a known maximum a binomial models with a very large maximum and small probability per trial number of elephants in Kenya, number of people who apply to graduate school 10.

build more types of models by piecing together types we have already learned about will discuss ordered categorical models and zero-inflated/zero-augmented models mixtures are powerful, but interpretation is difficult 11.

multi-level models remember features of each cluster in the data as they learn about all of the clusters depending on the variation across clusters, the model pools information across clusters the pooling improves estimates about each cluster benefits of the multilevel approach: improved estimates for repeat sampling improved estimates for imbalance in sampling estimates of variation avoid averaging and retain variation multilevel regression should be the default approach this chapter starts with the foundations and the following two are more advanced types of multilevel models 12.

this chapter will show how to specify varying slopes in combination with varying intercepts enables pooling to improve estimates of how different units respond to or are influenced by predictor variables also improves the estimates of intercepts by “borrowing information across parameter types” “varying slope models are massive interaction machines” 13.

cover two common applications of Bayesian statistics: measurement error Bayesian imputation 14.1 Measurement error in the divorce and marriage data of states in the US, there was error in the measured variables (marriage and divorce rates) this data can be incorporated into the model the plot below shows the standard error in the measurement against median age of marriage and the population seems like smaller states have more error (smaller sample size) # Load the data.