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) Continuous measured scale so that the spacing between adjacent values is consistent.
# (2) Should increase with number of possible events.
# (3) Additive for independent events and also independent on how 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 Dtrain+2p where Dtrain is the in-sample training deviance and p is the number of free parameters estimated in the model
#
# DIC is defined as D¯+pD=D¯+(D¯+D̂ ) where D¯ is the average of the posterior distribution of deviance and D̂ is the deviance calculated at the posterior mean
#
# WAIC is defined as −2(lppd−pWAIC)=−2(∑Ni=1logPr(yi)−∑Ni=1V(yi)) where 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
#
# All three definitions involve two components: an estimate or analog of the in-sample training deviance (i.e., Dtrain for AIC, D¯ for DIC, and lppd for WAIC) and an estimate or analog for the number of free parameters estimated in the model (i.e., p in AIC, pD in DIC, and pWAIC in WAIC).
#
# WAIC is the most general, followed by DIC, and finally AIC . To move from WAIC to DIC, we must assume that the posterior distribution is approximately multivariate Gaussian. To move from DIC to AIC, we must further assume that the priors are flat or overwhelmed by the likelihood7M2. Explain the difference between model selection and model comparison. What information is lost under model selection?
#Model selection is choosing to retain the model with the lowest information criterion value and to discard all other models with higher values.This practice loses information about relative model accuracy contained in the differences among information criterion values; this is especially problematic when the selected model only outperforms its alternatives to a small degree.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 is based on deviance, different observations will result in different deviance.
# If the model were fit to different numbers of observations, model with more observations will have higher deviance, the result may be inaccurate. 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.
# When prior gets more concentrated, the effective parameters will decrease accordingly.
# The penalty term in WAIC and pWAIC, is the sum of variance in log-likelihood for each observation. The likelihood will reduce variance and pWAIC.7M5. Provide an informal explanation of why informative priors reduce overfitting.
# Informative priors constrain the flexibility of the model.
# Extreme parameter values will not be assigned high posterior probability.7M6. Provide an informal explanation of why overly informative priors result in underfitting.
# Overly information over constrain the flexibility of the model. Correct parameter is less likely to be assigned a high posterior probability.