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) Be measured on a continuous scale such that the spacing between adjacent values is consistent.
# (2) Capture the size of the possibility space such that its value scales with the number of possible outcomes.
# (3) Be additive for independent events such that it does not matter how the events are divided.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?
p <- 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(0.20, 0.25, 0.25, 0.30)
(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 D_train + 2p, where D_train is the in-sample training deviance and p is the number of free parameters estimated in the model.
# WAIC is defined as −2(lppd − p_WAIC) = −2(∑logPr(y_i) − ∑V(y_i)), where Pr(y_i) is the average likelihood of observation i in the training sample and V(y_i) is the variance in log-likelihood for observation i in the training sample.
# Both definitions involve two components: an estimate or analog of the in-sample training deviance, and an estimate or analog for the number of free parameters estimated in the model.# WAIC is the most general. To transform from WAIC to AIC, we need to assume that the posterior distribution is approximately multivariate Gaussian and the priors are 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 is to select the model with the lowest information criterion value and to discard all other models with higher values.
# Model comparison uses multiple models to understand how the variables included influence prediction and affect implied conditional independencies in a causal model. In model comparison, information is preserved to help us make judgments and decisions about data and models.
# Model selection loses information about relative model accuracy contained in the differences among information criterion values.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 from the deviance of all observations without being divided by the number of observations (sum and not an average). A model with a more observations will have a higher deviance and worse accuracy according to the information criterion, so it is not reasonable to compare models with different number of observations.# To confirm this, the WAIC can be calculated for models with different number of observations.
library(rethinking)## Loading required package: rstan## Loading required package: StanHeaders## Loading required package: ggplot2## 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## rethinking (Version 2.13)##
## Attaching package: 'rethinking'## The following object is masked from 'package:stats':
##
## rstudentdata(Howell1)
d <- Howell1[complete.cases(Howell1), ]
d200 <- d[sample(1:nrow(d), size = 200, replace = FALSE), ]
d300 <- d[sample(1:nrow(d), size = 300, replace = FALSE), ]
d400 <- d[sample(1:nrow(d), size = 400, replace = FALSE), ]
m200 <- map(
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * log(weight)
),
data = d200,
start = list(a = mean(d200$height), b = 0, sigma = sd(d200$height))
)
m300 <- map(
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * log(weight)
),
data = d300,
start = list(a = mean(d300$height), b = 0, sigma = sd(d300$height))
)
m400 <- map(
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * log(weight)
),
data = d400,
start = list(a = mean(d400$height), b = 0, sigma = sd(d400$height))
)
(model.compare <- compare(m200, m300, m400))## Warning in compare(m200, m300, m400): Different numbers of observations found for at least two models.
## Model comparison is valid only for models fit to exactly the same observations.
## Number of observations for each model:
## m200 200
## m300 300
## m400 400## Warning in ic_ptw1 - ic_ptw2: longer object length is not a multiple of shorter
## object length
## Warning in ic_ptw1 - ic_ptw2: longer object length is not a multiple of shorter
## object length## WAIC SE dWAIC dSE pWAIC weight
## m200 1212.253 21.77327 0.0000 NA 3.322575 1.000000e+00
## m300 1876.338 28.84335 664.0842 30.96956 3.589118 6.250953e-145
## m400 2423.033 27.79046 1210.7800 29.77638 3.011042 1.209113e-263# From the three models compared with 200, 300 and 400 observations, we can see that the WAIC increases with the number of observations. There is also a warning about the number of observations being different.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.
# The effective number of parameters decrease, as a prior becomes more concentrated.
# For PSIS, the model becomes less flexible as the prior become more concentrated.
# For WAIC, the p_WAIC is a measure of the variance in the log-likelihood for each observation in the training sample. As the prior becomes more concentrated, the likelihood will also become more concentrated and the variance will decrease.d <- Howell1[complete.cases(Howell1), ]
d$height.log <- log(d$height)
d$height.log.z <- (d$height.log - mean(d$height.log)) / sd(d$height.log)
d$weight.log <- log(d$weight)
d$weight.log.z <- (d$weight.log - mean(d$weight.log)) / sd(d$weight.log)
m_not_concentrated <- map(
alist(
height.log.z ~ dnorm(mu, sigma),
mu <- a + b * weight.log.z,
a ~ dnorm(0, 10),
b ~ dnorm(1, 10),
sigma ~ dunif(0, 10)
),
data = d
)
m_concentrated <- map(
alist(
height.log.z ~ dnorm(mu, sigma),
mu <- a + b * weight.log.z,
a ~ dnorm(0, 0.10),
b ~ dnorm(1, 0.10),
sigma ~ dunif(0, 1)
),
data = d
)
WAIC(m_not_concentrated, refresh = 0)## WAIC lppd penalty std_err
## 1 -102.7375 55.63443 4.265661 36.45425WAIC(m_concentrated, refresh = 0)## WAIC lppd penalty std_err
## 1 -102.5639 55.66322 4.381273 36.63186# p_WAIC decreases with the more concentrated prior.7M5. Provide an informal explanation of why informative priors reduce overfitting.
# Informative priors reduce overfitting because they constrain the flexibility of the model. They make it less likely for extreme parameter values to be assigned high posterior probability. In simpler terms, informative priors reduce overfitting by forcing the model to learn less from the sample data.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 too much. They make it less likely for "correct" parameter values to be assigned high posterior probability. In simpler terms, overly informative priors result in underfitting by preventing the model from learning enough from the sample data.