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<!DOCTYPE html>
<html>
<head>
<title>Introduction to Bayesian Inference in R</title>
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class: center, middle
# Introduction to Bayesian Inference in R
## Robert Dodier
This document and associated material is on Github:
[robert-dodier/bayesian-inference-r-2016](https://github.com/robert-dodier/bayesian-inference-r-2016)
---
# What is Bayesian inference?
* Bayesian inference = laws of probability applied to any proposition
* Proposition = statement about uncertain variables
* Variables = observable quantities, model parameters, latent (hidden) variables, hypotheses, etc
* Fundamental operation is to compute conditional probabilities
* i.e. probability of an interesting proposition ...
* given (conditional on) some data (i.e. whatever is fixed, either by observation or assumption)
* Bayesian inference is a framework within which we construct a way to handle any problem involving uncertain propositions
---
# What to do about uninteresting variables
* What about variables which are neither interesting nor given??
* Laws of probability dictate the Right Thing To Do:
* namely, integrate joint probability of "interesting" and "not interesting" over "not interesting"
* formally,
`\(P(\mathrm{interesting} | \mathrm{given}) = \int P(\mathrm{interesting}, \mathrm{uninteresting} | \mathrm{given}) d\mathrm{uninteresting}\)`
(where integral is a summation if uninteresting is discrete)
* This is the motivation for the computational stuff we'll see later
---
# Some problems involving uncertain propositions
* Missing data
* Forecasts
* Diagnosis
* Model selection
* Model parameters
* "What if" scenarios
---
# Reasonable, non-Bayesian approaches
* Missing data: plug in estimate of missing variable
* Forecasts: output forecast = best guess
* Diagnosis: output diagnosis = best guess
* Model selection: choose "the best" model
* Model parameters: choose best parameters
* "What if" scenarios: plug in example inputs into model and get output
---
# Approaches as enriched by a Bayesian perspective
* Missing data: consider all possible values of missing variable
* Forecasts: consider all possible output values
* Diagnosis: consider all possible diagnoses
* Model selection: consider all possible models
* Model parameters: consider all possible parameters
* "What if" scenarios: consider all possible results
---
# A closer look at the model parameters problem
* Goal is to construct posterior distribution of parameters given data
* In some cases an exact distribution of parameters can be constructed exactly or approximately
* In other cases, can't construct it explicitly but can construct an algorithm to sample from posterior
* ... then to produce e.g. forecasts, sample from posterior and use parameters to generate output
* ... and the result is a posterior distribution over the output
---
# Linear regression example
* We have a small data sample
* We'll assume the data are described by a linear model `\(y = a_0 + a_1 x + \mathrm{noise}\)`
* We'd like to compute a forecast at a given point
* ... taking uncertainty in the model parameters into account
---
---
---
---
---
There are many possible lines, some more plausible, some less
---
Let's set our goal to compute the posterior distribution `\(p(y | x, \mathrm{data})\)`
* For that, we need to average over the variables we don't know, namely `\(a_0\)` and `\(a_1\)`.
* Specifically we need to compute
`\(\int \int p(y | x, a_0, a_1) p(a_0, a_1 | \mathrm{data}) p(a_0, a_1) da_0 da_1\)`
---
# Computing integrals -- (1) symbolic
* Stuff like: `\(\int_0^x \alpha e^{-\alpha t} \beta e^{-\beta (x - t)} dt = \frac{\alpha \beta}{\beta - \alpha} (e^{-\alpha x} - e^{-\beta x})\)`
* Gives insight if assumptions are satisfied
* Limited applicability
* In R, maybe rsympy or maybe ryacas which are glue code for computer algebra systems
---
# Computing integrals -- (2) quadrature formulas
* stats::integrate adaptive subdivision based on rules exact for polynomials
---
# Linear regression example using quadrature
`py.given.x` computes `\(p(y|x, \mathrm{data})\)`
```r
py.given.x <-
function (y, x) {
integrate (
function (a.1) {
sapply (a.1,
function (a.1) {
integrate (function (a.0) {
py.given.x.a0.a1.sigma.e (y, x, a.0, a.1, 0.25) *
P2 (a.0, a.1)},
-Inf, Inf)$value})},
0, Inf)}
```
Integrand is `\(p(y | x, a_0, a_1) p(a_0, a_1 | \mathrm{data}) p(a_0, a_1)\)`
---
# Calculating a posterior distribution via quadrature
```r
yy <- seq (from=0.4, to=2.1, by=0.01)
sapply (yy, function (y) {py.given.x (y, 0.75) $ value}) -> pp
I <- sum(pp)*0.01
pp <- pp/I
plot (x=yy, y=pp, type="l")
```
---
---
# About quadrature formulas
* Works well in 1 dimension, tolerable in a few more
* Unworkable in many dimensions (a common application scenario)
---
# Computing integrals -- (3) simple Monte Carlo
* runif, rnorm, rgamma, etc generate pseudorandom numbers
* Approximate integral as average over samples: `\(\int f(x) p(x) dx \approx \frac{1}{n} \sum_i f(x_i)\)`
---
# Linear regression example using simple Monte Carlo
```r
py.given.x.via.monte.carlo <-
function (y, x) {
mean (mapply (function (a.0, a.1) {
py.given.x.a0.a1.sigma.e (y, x, a.0, a.1, 0.25) *
L(a.0, a.1, 0.25) *
psigma.e(0.25)},
rnorm (1000, mean=0, sd=1),
rgamma (1000, shape=2, scale=1)))}
```
---
# Calculating a posterior distribution via simple Monte Carlo
```r
pp <- sapply (yy, function (y) {py.given.x.via.monte.carlo (y, 0.75)})
I <- sum(pp)*0.01
pp <- pp/I
```
---
---
# About simple Monte Carlo
* Easy to apply, may be inefficient
* Limited applicability (difficulty of constructing sampling distribution)
* ... this is the motivation for Markov chain Monte Carlo
---
# Computing integrals -- (4) Markov chain Monte Carlo
* More general than ordinary Monte Carlo
* If one can't explicitly construct sampling distribution,
construct Markov chain s.t. stationary distribution is desired distribution
* For this Markov chain, averages over samples are equal to averages over the desired distribution
* From current sample, jump somewhere else; maybe accept the new sample. Repeat many times.
---
# Markov chain Monte Carlo example
We'll work with a function proportional to the distribution we want to sample;
namely the product of the likelihood function and the prior for the parameters.
Here is the likelihood function for the linear regression example:
```r
L <- function (a.0, a.1, sigma.e) {
py.given.x.a0.a1.sigma.e (data.y[1], data.x[1], a.0, a.1, sigma.e) *
py.given.x.a0.a1.sigma.e (data.y[2], data.x[2], a.0, a.1, sigma.e) *
py.given.x.a0.a1.sigma.e (data.y[3], data.x[3], a.0, a.1, sigma.e)}
```
and here is the prior:
```r
pa.0 <- function (a.0) {exp(-0.5*a.0^2)/sqrt(2*pi)}
pa.1 <- function (a.1) {a.1*exp(-a.1)}
```
---
---
Proposal distribution which is just a Gaussian bump:
```r
Q <- function (a.0, a.1) {rnorm (2, mean=c(a.0, a.1), sd=0.05)}
```
Generate a new proposed point:
```r
p1 <- Q (p0[1], p0[2])
```
Compute acceptance ratio:
```r
r <- L (p1[1], p1[2]) / L (p0[1], p0[2])
```
Accept or reject proposal:
```r
if (r > 1 || r > runif(1)) {p0 <- p1}
```
---
Generate a sequence of samples given a starting point `xy0`:
```r
mcmc.sequence <- function (n, p0) {
a.0 <- vector (length=n)
a.1 <- vector (length=n)
a.0[1] <- p0[1]
a.1[1] <- p0[2]
for (i in 2:n) {
p1 <- Q (p0[1], p0[2])
r <- L (p1[1], p1[2]) / L (p0[1], p0[2])
if (r > 1 || r > runif(1)) {p0 <- p1}
a.0[i] <- p0[1]
a.1[i] <- p0[2]
}
list (x=a.0, y=a.1)
}
```
---
"Burn-in" sequence -- initial samples not representative of distribution
---
Running the chain longer yields more representative samples
---
---
# About Markov chain Monte Carlo
* MCMC slower than ordinary MC due to autocorrelation, but more general
* Writing a sampler is a distraction that obscures the model
---
# Stan, a package for Bayesian inference
* Stan has its own modeling language, primarily declarative
* User states how variables are related, Stan figures out calculations
* Inference via "no U-turn sampling" MCMC
* Also implements variational inference and optimization functions
---
# Linear regression model in Stan
```
model {
real mu;
for (i in 1:3) {
mu = alpha[1] + alpha[2] * x[i];
y[i] ~ normal(mu, sigma_e);
}
}
generated quantities {
real y_pred;
y_pred = normal_rng(alpha[1] + alpha[2] * new_x, sigma_e);
}
```
---
# Making inferences via RStan
```r
library (rstan)
stan.data <- list (x=c(0.1, 0.5, 0.6), y=c(0.35, 0.9, 1.1), sigma_e=0.25, new_x=0.75)
my.stan.model <- stan (file="model.stan", data=stan.data, iter=2000, chains=4)
traceplot (my.stan.model, pars = c("alpha"), inc_warmup = T)
y_pred_struct <- extract (my.stan.model, 'y_pred')
y_pred <- unlist (y_pred_struct, use.names=F)
hist (y_pred, freq=F, main="Density of y|x via Stan")
```
---
Samples from the Markov chain for the parameters
---
---
# What use is a probability?
* Probability calculations are part of a bigger picture of decision analysis
* Rational approach is to make decision on basis of expected utility
* Utility = quantify value or preference
* Probability = quantify belief
* Expected utility = utility integrated wrt probability distribution
* Instead of using a probability by itself, consider expected gain/loss
* e.g. P(change service|data) = 0.2; what should service provider do?
* answer has to depend on what service provider stands to gain or lose
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