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. Make sure to include plots if the question requests them. 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.
# First motivating criteria: uncertainty should be measured on a continuous scale with consistency of spacing between adjacent values;
# Second motivating criteria: uncertainty should be increasing with the number of different possible outcomes;
# Third motivating criteria: uncertainty should be additive for independent events no matter how the events are separated. 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?
p1 = c(0.7, 1 - 0.7)
e = -sum(p1*log(p1))
e## [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?
p2 = c(0.2, 0.25, 0.25, 0.3)
e = -sum(p2*log(p2))
e## [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?
p3 = c(1/3, 1/3, 1/3)
e = -sum(p3 * log(p3))
e## [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: Akaike Information Criterion. It is defined as the sum of deviance of in sampling training data and twice the number of free parameters.It estimates K-L Distance in theory.
# AIC = D_train + 2p = −2lppd + 2p
# WAIC: Widely Applicable Information Criterion. It is defined as the difference between the sum of average likelihood of observations and the penalty term.
# WAIC is the most general criteria. There are following assumptions: 1. Posterior distribution is Gaussian; 2. The posterior to be either flat or overwhelmed by likelihood and the sample size N is much greater than the number of parameters k.7M2. Explain the difference between model selection and model comparison. What information is lost under model selection?
# Model Selection: Choose the model with the lowest information criterion value and discard other models with higher values.
# Model Comparison: Utilize and analyze multiple models in making predictions.
# Under model selection, information about relative model accuracy contained in the differences among criterion values and casual inference would lose. 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.
# The reason is that deviance is the measure of model performance.
# If the models were fit to different numbers of observations, models with more observations will have higher deviance, which may cause less accuracy. Therefore, all models need to be fit to exactly the same observations for fairness.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 a prior becomes more concentrated, the effective number of parameters decreases, it is because the observed likelihood will become more concentrated. 7M5. Provide an informal explanation of why informative priors reduce overfitting.
# As informative priors makes the model more restrictive and reduce its flexibility, it will reduce overfitting. Outliers or extreme parameter values do not have too much influence on the model and thus will not be assigned high posterior probability. Thus, informative priors can help the model learn not learn too much from them. 7M6. Provide an informal explanation of why overly informative priors result in underfitting.
# As overly informative priors limit the model too much, it will let the model lean not enough from sample data. As the correct parameter values are less likely to be assigned with a high posterior probability, and therefore limit the flexibility of the model by too much.