Chapter 7 - Ulysses’ Compass

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.

Questions

7E1. State the three motivating criteria that define information entropy. Try to express each in your own words.

#1. First motivating criteria is the information uncertainity must be continuous that way the possible outcomes can be seen in a pattern
#2. Second motivating criteria is that the uncertainity must be increasing with number of possible outcomes so that various uncertainities are covered that way
#3. Third motivating criteria is that uncertainity must be additive that way if there are any independent outcomes those are covered too

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, 0.3)
entropy <- -sum(p * log(p))
entropy
## [1] 0.6108643

7E3. 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)
entropy <- -sum(p * log(p))
entropy
## [1] 1.376227

7E4. 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)
entropy <- -sum(p * log(p))
entropy
## [1] 1.098612

7M1. 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 an estimator of out-of-sample prediction error and estimates the quality of each model. AIC is a way to select model quality and evaluate overfitting in the model.
# WAIC is the generalized version of AIC onto singular statistical models. Also, WAIC is the most general criteria
# If we want to transform from WAIC to AIC, what's needed is 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. Therefore, we would lose information about relative model accuracy. Model averaging is using Bayesian information criteria to construct a posterior predictive distribution and leverages the uncertainty in multiple models. In this case, we would not lose information on its own.

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 criterion depends on the total deviance which is directly influenced by the number of observations. Therefore, when comparing models with different numbers of observations it will result in different Information Criterion. Model accuracy could also get lost in model comparison. As the conclusion, all models must be fit to exactly the same number 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.

#Prior and effective number of parameters are indirectly proportional. Therefore, as prior becomes more concentrated, the effective number of parameters decreases. This proportionality can be observed from the WAIC mathematical equation:
# WAIC(y, Θ) = -2(lppd - varΘ logp(yi|Θ)) 

7M5. Provide an informal explanation of why informative priors reduce overfitting.

#Informative priors restrains the range of parameters.So when such a model is applied, since the number of parameters these priors look for are limited there is no chance of an overfitting scenario.

7M6. Provide an informal explanation of why overly informative priors result in underfitting.

#Overly informative priors resulting in narrowing the range of parameters. Therefore, the number of parameters these priors look for are too limited and picky to develop a proper model.