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. Information uncertainity must be continuous that way the possible outcomes can be seen in a pattern
#2. The uncertainity must be increasing with number of possible outcomes so that various uncertainities are covered that way
#3. Uncertainity must be additive that way if there are any independent outcomes those are covered too7E2. 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 = 0.7
p <- c(p, 1-p)
-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(0.2, 0.25, 0.25, 0.3)
-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(0.33, 0.33, 0.33)
-sum(p * log(p))## [1] 1.0975767M1. 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?
#Akaike Information Criterion (AIC) : It is defined as sum of deviance of in sampling training data (Dtrain) and twice the number of parameters (p) follows:
# AIC = Dtrain + 2p
#Widely Applicable Information Criterion (WAIC)- It is defined as the difference between sum of average likelihood of observations (log point wise predictive density (lppd) and the penality term, mathematically defined as follows:
# WAIC(y, Θ) = -2(lppd - varΘ logp(yi|Θ))
#As the name suggests, WAIC is more general. WAIC does not assume Gaussian posterior , however, it does assume that the posterior distirbution is close to multivariate Gaussian and. And priors are probably overwhelmed or flat by likelihood7M2. Explain the difference between model selection and model comparison. What information is lost under model selection?
# Model selection is the task of selecting a statistical model from a set of candidate models, given data.
# Model comparison we choose a model based on multiple models for causal inference and metrics.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.
# like any experiment we would like to control as much variation as possible. different observation n would result in different IC even with the same model set up. Thus, models become not compariable, 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, that's why 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 allow less variation in effective parameters, which limiting the ability of parameters to fit closely to the data itself, thus prevent overfitting. 7M6. Provide an informal explanation of why overly informative priors result in underfitting.
#Overly informative priors resulting in underfitting. Similar to that of above statemen, overly informative priors limit the effective parameter sampling range. Thus it is very important how the priors are chosen. if the true distribution does not overlap with the prior given, the model would yield very off results.