The chapter began with the problem of overfitting, a universal phenomenon by which models with more parameters fit a sample better, even when the additional parameters are meaningless. Two common tools were introduced to address overfitting: regularizing priors and estimates of out-of-sample accuracy (WAIC and PSIS). Regularizing priors reduce overfitting during estimation, and WAIC and PSIS help estimate the degree of overfitting. Practical functions compare in the rethinking package were introduced to help analyze collections of models fit to the same data. If you are after causal estimates, then these tools will mislead you. So models must be designed through some other method, not selected on the basis of out-of-sample predictive accuracy. But any causal estimate will still overfit the sample. So you always have to worry about overfitting, measuring it with WAIC/PSIS and reducing it with regularization.
Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Problems are labeled Easy (E), Medium (M), and Hard(H).
Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.
7E1. State the three motivating criteria that define information entropy. Try to express each in your own words.
#1. The uncertainty should be in a continuous manner so that a pattern can be sought.
#2. The uncertainty should be increased with the number of outcomes.
#3. The uncertainty should be addictive so that independent outcomes can be included.7E2. Suppose a coin is weighted such that, when it is tossed and lands on a table, it comes up heads 70% of the time. What is the entropy of this coin?
library(rethinking)## Loading required package: rstan## Warning: package 'rstan' was built under R version 3.6.3## Loading required package: StanHeaders## Loading required package: ggplot2## Warning: package 'ggplot2' was built under R version 3.6.3## rstan (Version 2.21.2, GitRev: 2e1f913d3ca3)## For execution on a local, multicore CPU with excess RAM we recommend calling
## options(mc.cores = parallel::detectCores()).
## To avoid recompilation of unchanged Stan programs, we recommend calling
## rstan_options(auto_write = TRUE)## Do not specify '-march=native' in 'LOCAL_CPPFLAGS' or a Makevars file## Loading required package: parallel## Loading required package: dagitty## Warning: package 'dagitty' was built under R version 3.6.3## rethinking (Version 2.01)##
## Attaching package: 'rethinking'## The following object is masked from 'package:stats':
##
## rstudentp <- c(0.7, 1 - 0.7)
(H <- -sum(p * log(p)))## [1] 0.61086437E3. Suppose a four-sided die is loaded such that, when tossed onto a table, it shows “1” 20%, “2” 25%, “3” 25%, and “4” 30% of the time. What is the entropy of this die?
p <- c(0.2, 0.25, 0.25, 0.3)
(H <- -sum(p * log(p)))## [1] 1.3762277E4. Suppose another four-sided die is loaded such that it never shows “4”. The other three sides show equally often. What is the entropy of this die?
p <- c(1/3, 1/3, 1/3)
(H <- -sum(p * log(p)))## [1] 1.0986127M1. Write down and compare the definitions of AIC and WAIC. Which of these criteria is most general? Which assumptions are required to transform the more general criterion into a less general one?
#AIC is defined as Dtrain+2p: Dtrain is the in-sample training deviance and p is the number of free parameters estimated in the model.
#WAIC is defined as −2(lppd−pWAIC)=−2(∑Ni=1logPr(yi)−∑Ni=1V(yi)): Pr(yi) is the average likelihood of observation i in the training sample and V(yi) is the variance in log-likelihood for observation i in the training sample.
#Comparing those 2, WAIC is more general than AIC. To bridge the gap from WAIC to AIC, we need to assume the posterior distribution is close to multivariate Gaussian and the priors are either flat or overwhelmed by the likelihood.7M2. Explain the difference between model selection and model comparison. What information is lost under model selection?
#Model selection chooses to retain the model with the lowest information criterion value and to discard all other models with higher values. This will result in losing information about relative model accuracy contained in the differences among information criterion values.
#Model averaging uses Bayesian information criteria to construct a posterior predictive distribution and leverages the uncertainty in multiple models. This does not lose information like model selection.7M3. When comparing models with an information criterion, why must all models be fit to exactly the same observations? What would happen to the information criterion values, if the models were fit to different numbers of observations? Perform some experiments, if you are not sure.
#Information criteria are based on deviance, and that is accrued over observations without being divided by the number of observations. So a model with more observations will have a higher deviance and thus worse accuracy according to information criteria. It wouldn't be accurate to contrast models fit to different numbers of observations.7M4. What happens to the effective number of parameters, as measured by PSIS or WAIC, as a prior becomes more concentrated? Why? Perform some experiments, if you are not sure.
#As prior becomes more concentrated, the effective number of parameters decreases. Because the two are negatively related to each other: WAIC(y, Θ) = -2(lppd - varΘ logp(yi|Θ)).7M5. Provide an informal explanation of why informative priors reduce overfitting.
#Informative priors reduce overfitting because they constrain the flexibility of the model, and make it less likely for extreme parameter values to be assigned high posterior probability. 7M6. Provide an informal explanation of why overly informative priors result in underfitting.
##Overly informative priors result in underfitting because they constrain the flexibility of the model which makes it less likely for the valid parameter values to be assigned high posterior probability.