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. Uncertainty should be measured on a continuous scale with spacing between adjacent values being consistent.
#2. Uncertainty should increase as the number of possible events increases.
#3. Uncertainty should be additive for independent events.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 Dtrain+2p where Dtrain is the in-sample training deviance and p is the number of free parameters estimated in the model
# WAIC is a generalized version of the AIC to singular statistical model.
# WAIC is the most general. To move from WAIC to DIC, we must assume that the posterior distribution is approximately multivariate Gaussian.7M2. Explain the difference between model selection and model comparison. What information is lost under model selection?
# Model selection is the selection of the model with the lowest value of the information criterion and discarding all other models with higher values. This practice loses information about the relative model accuracy. This practice loses information about relative model accuracy contained in the differences among information criterion values7M3. 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.
# Inforrmation criterion estimates prediction error based on sum of deviance, not on average deviance. It relatively evaluates model performance of statistical models trained on the same set of data. If models are trained on different number of observations, model with higher number of observations will have higher deviance, thereby leading to incorrect comparison. 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 decreases for WAIC. In WAIC, with more concentrated priors, the log-likelihood will become more concentrated and variance will decrease (pWAIC). It will not affect PSIS because PSIS uses a single data point and average over samples. 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.