diff --git a/chapter-04-classification.md b/chapter-04-classification.md
index 7a5b716..afe49fb 100644
--- a/chapter-04-classification.md
+++ b/chapter-04-classification.md
@@ -11,7 +11,7 @@ use_math: true
 
 Though linear regression can be applied in the case of binary qualitative
 responses, difficulties arise beyond two levels. For example, choosing a coding
-scheme is problematic and different coding scheme can yield wildly different
+scheme is problematic, and different coding schemes can yield very different
 predictions.
 
 ### Logistic Regression
@@ -21,489 +21,201 @@ $$ y $$ belongs to a particular category rather than modeling the response
 itself.
 
 Logistic regression uses the [logistic function][glossary-logistic-function] to
-ensure a prediction between $$ 0 $$ and $$ 1 .$$ The logistic function takes the
-form
+ensure predictions lie between $$ 0 $$ and $$ 1 $$. The logistic function takes
+the form
 
 $$ \normalsize p(X) = \frac{e^{\beta_{0} + \beta_{1}X}}{1 + e^{\beta_{0} +
 \beta_{1}X}} .$$
 
-This yields a probability greater than $$ 0 $$ and less than $$ 1 .$$
+This yields a probability greater than $$ 0 $$ and less than $$ 1 $$.
 
-The logistic function can be rebalanced to yield
+The logistic function can be rearranged to yield
 
 $$ \normalsize \frac{p(X)}{1 - p(X)} = e^{\beta_{0} + \beta_{1}X} $$
 
-$$ \frac{p(X)}{1 - p(X)} $$ is known as the [odds][glossary-odds] and takes on a
-value between $$ 0 $$ and infinity.
+$$ \frac{p(X)}{1 - p(X)} $$ is known as the [odds][glossary-odds] and takes on
+values between $$ 0 $$ and infinity.
 
 As an example, a probability of 1 in 5 yields odds of $$ \frac{1}{4} $$ since
-$$ \frac {0.2}{1 - 0.2} = \frac{1}{4} .$$
+$$ \frac{0.2}{1 - 0.2} = \frac{1}{4} .$$
 
-Taking a logarithm of both sides of the logistic odds equation yields an
-equation for the [log-odds][glossary-log-odds] or [logit][glossary-logit],
+Taking a logarithm of both sides yields the [log-odds][glossary-log-odds] (or
+[logit][glossary-logit]):
 
-$$ \normalsize \mathrm{log} \bigg \lgroup \frac{p(X)}{1 - p(X)} \bigg \rgroup =
+$$ \normalsize \log \bigg( \frac{p(X)}{1 - p(X)} \bigg) =
 \beta_{0} + \beta_{1}X $$
 
-Logistic regression has a logit that is linear in terms of $$ X . $$
+Logistic regression has a logit that is linear in terms of $$ X $$.
 
-Unlike linear regression where $$ \beta_{1} $$ represents the average change in
-$$ Y $$ with a one-unit increase in $$ X $$, for logistic regression, increasing
-$$ X $$ by one-unit yields a $$ \beta_{1} $$ change in the log-odds which is
-equivalent to multiplying the odds by $$ e^{\beta_{1}} .$$
+Unlike linear regression, where $$ \beta_{1} $$ represents the average change in
+$$ Y $$ for a one-unit increase in $$ X $$, in logistic regression, increasing
+$$ X $$ by one unit results in a $$ \beta_{1} $$ change in the log-odds. This is
+equivalent to multiplying the odds by $$ e^{\beta_{1}} $$.
 
-The relationship between $$ p(X) $$ and $$ X $$ is not linear and because of
-this $$ \beta_{1} $$ does not correspond to the change in $$ p(X) $$ given
-one-unit increase in $$ X $$. However, if $$ \beta_{1} $$ is positive,
-increasing $$ X $$ will be associated with an increase in $$ p(X) $$ and,
-similarly, if $$ \beta_{1} $$ is negative, an increase in $$ X $$ will be
-associated with a decrease in $$ p(X) $$. How much change will depend on the
-value of $$ X $$.
+The relationship between $$ p(X) $$ and $$ X $$ is not linear. Thus,
+$$ \beta_{1} $$ does not correspond directly to the change in $$ p(X) $$ for a
+one-unit increase in $$ X $$. However:
+- If $$ \beta_{1} > 0 $$, increasing $$ X $$ increases $$ p(X) $$
+- If $$ \beta_{1} < 0 $$, increasing $$ X $$ decreases $$ p(X) $$
+
+---
 
 #### Estimating Regression Coefficients
 
-Logistic regression uses a strategy called [maximum
-likelihood][glossary-maximum-likelihood] to estimate regression coefficients.
+Logistic regression uses [maximum likelihood][glossary-maximum-likelihood] to
+estimate coefficients.
+
+We choose $$ \beta_{0} $$ and $$ \beta_{1} $$ such that predicted probabilities
+$$ \hat{p}(x_i) $$ align closely with observed classes.
 
-Maximum likelihood plays out like so: determine estimates for $$ \beta_{0} $$
-and $$ \beta_{1} $$ such that the predicted probability of $$ \hat{p}(x_{i}) $$
-corresponds with the observed classes as closely as possible. Formally, this
-yield an equation called a [likelihood function][glossary-likelihood-function]:
+This yields the [likelihood function][glossary-likelihood-function]:
 
-$$ \normalsize \ell(\beta_{0}, \beta_{1}) = \displaystyle
-\prod_{i:y_{i}=1}p(X_{i}) \times \displaystyle \prod_{j:y_{j}=0}(1-p(X_{j})) .
-$$
+$$ \normalsize \ell(\beta_{0}, \beta_{1}) =
+\prod_{i:y_{i}=1} p(X_{i}) \times
+\prod_{j:y_{j}=0} (1 - p(X_{j})) .$$
 
-Estimates for $$ \beta_{0} $$ and $$ \beta_{1} $$ are chosen so as to maximize
-this likelihood function.
+We select parameters that maximize this function.
 
-Linear regression's least squares approach is actually a special case of maximum
-likelihood.
+Linear regression's least squares is a special case of maximum likelihood.
 
-Logistic regression measures the accuracy of coefficient estimates using a
-quantity called the [z-statistic][glossary-z-statistic]. The z-statistic is
-similar to the t-statistic. The z-statistic for $$ \beta_{1} $$ is represented
-by
+Logistic regression evaluates coefficient accuracy using the
+[z-statistic][glossary-z-statistic]:
 
 $$ \normalsize \textrm{z-statistic}(\beta_{1}) =
 \frac{\hat{\beta}_{1}}{\mathrm{SE}(\hat{\beta}_{1})} $$
 
-A large z-statistic offers evidence against the null hypothesis.
+A large z-statistic provides evidence against the null hypothesis.
 
-In logistic regression, the null hypothesis
+The null hypothesis is:
 
 $$ \normalsize H_{0}: \beta_{1} = 0 $$
 
-implies that
+which implies:
 
-$$ \normalsize p(X) = \frac{e^{\beta_{0}}}{1 + e^{\beta_{0}}} $$
+$$ \normalsize p(X) =
+\frac{e^{\beta_{0}}}{1 + e^{\beta_{0}}} $$
 
-and, ergo, $$ p(X) $$ does not depend on $$ X . $$
+Thus, $$ p(X) $$ does not depend on $$ X $$.
 
-#### Making Predictions
+---
 
-Once coefficients have been estimated, predictions can be made by plugging the
-coefficients into the model equation
+#### Making Predictions
 
-$$ \normalsize \hat{p}(X) = \frac{e^{\hat{\beta_{0}} + \hat{\beta_{1}}X}}{1 +
-e^{\hat{\beta}_{0} + \hat{\beta}_{1}X}} . $$
+Predictions are made using:
 
-In general, the estimated intercept, $$ \hat{\beta}_{0} , $$ is of limited
-interest since it mainly captures the ratio of positive and negative
-classifications in the given data set.
+$$ \normalsize \hat{p}(X) =
+\frac{e^{\hat{\beta}_{0} + \hat{\beta}_{1}X}}
+{1 + e^{\hat{\beta}_{0} + \hat{\beta}_{1}X}} .$$
 
-Similar to linear regression, [dummy variables][glossary-dummy-variable] can be
-used to accommodate qualitative predictors.
+The intercept $$ \hat{\beta}_{0} $$ is often less interpretable, as it mainly
+captures the baseline class balance.
 
-#### Multiple Logistic Regression
+[Dummy variables][glossary-dummy-variable] can be used for qualitative predictors.
 
-Using a strategy similar to that employed for linear regression, [multiple
-logistic regression][glossary-multiple-logistic-regression] can be generalized
-as
+---
 
-$$ \normalsize log \bigg \lgroup \frac{p(X)}{1 - p(X)} \bigg \rgroup = \beta_{0}
-+ \beta_{1}X_{1} + \ldots + \beta_{p}X_{p} $$
+#### Multiple Logistic Regression
 
-where $$ X = (X_{1}, X_{2}, \ldots, X_{p}) $$ are $$ p $$ predictors.
+Generalization:
 
-The log-odds equation for multiple logistic regression can be expressed as
+$$ \normalsize \log \bigg( \frac{p(X)}{1 - p(X)} \bigg) =
+\beta_{0} + \beta_{1}X_{1} + \cdots + \beta_{p}X_{p} $$
 
-$$ \normalsize p(X) = \frac{e^{\beta_{0} + \beta_{1}X_{1} + \ldots +
-\beta_{p}X_{p}}}{1 + e^{\beta_{0} + \beta_{1}X_{1} + \ldots + \beta_{p}X_{p}}} $$
+Equivalent probability form:
 
-Maximum likelihood is also used to estimate $$ \beta_{0}, \beta_{1}, \ldots,
-\beta_{p} $$ in the case of multiple logistic regression.
+$$ \normalsize p(X) =
+\frac{e^{\beta_{0} + \beta_{1}X_{1} + \cdots + \beta_{p}X_{p}}}
+{1 + e^{\beta_{0} + \beta_{1}X_{1} + \cdots + \beta_{p}X_{p}}} $$
 
-In general, the scenario in which the result obtained with a single predictor
-does not match the result with multiple predictors, especially when there is
-correlation among the predictors, is referred to as
-[confounding][glossary-confounding]. More specifically, confounding describes
-situations in which the experimental controls do not adequately allow for ruling
-out alternative explanations for the observed relationship between the
-predictors and the response.
+---
 
-#### Logistic Regression For More Than Two Classes
+Confounding occurs when relationships between predictors distort results.
 
-Though multiple-class logistic regression is possible, [discriminant
-analysis][glossary-discriminant-analysis] tends to be the preferred means of
-handling multiple-class classification.
+---
 
 ### Linear Discriminant Analysis
 
-While logistic regression models the conditional distribution of the response $$
-Y $$ given the predictor(s) $$ X , $$ [linear discriminant
-analysis][glossary-linear-discriminant-analysis] takes the approach of modeling
-the distribution of the predictor(s) $$ X $$ separately in each of the response
-classes , $$ Y $$, and then uses [Bayes' theorem][glossary-bayes-theorem] to
-invert these probabilities to estimate the conditional distribution.
-
-Linear discriminant analysis is popular when there are more than two response
-classes. Beyond its popularity, linear discriminant analysis also benefits from
-not being susceptible to some of the problems that logistic regression suffers
-from:
-
-- The parameter estimates for logistic regression can be surprisingly unstable
-  when the response classes are well separated. Linear discriminant analysis
-  does not suffer from this problem.
-- Logistic regression is more unstable than linear discriminant analysis when $$
-  n $$ is small and the distribution of the predictors $$ X $$ is approximately
-  normal in each of the response classes.
-
-#### Classification With Bayes' Theorem
-
-Assuming a qualitative variable $$ Y $$ that can take on $$ K \geq 2 $$
-distinct, unordered values, the [prior probability][glossary-prior-probability]
-describes the probability that a given observation is associated with the kth
-class of the response variable $$ Y . $$
+[Linear discriminant analysis][glossary-linear-discriminant-analysis] models
+$$ X $$ given $$ Y $$ and applies
+[Bayes' theorem][glossary-bayes-theorem].
 
-The [density function][glossary-density-function] of $$ X $$ for an observation
-that comes from the kth class is defined as
-
-$$ \normalsize f_{k}(X) = \mathrm{Pr}(X=x|Y=k) . $$
-
-This means that $$ f_{k}(X) $$ should be relatively large if there's a high
-probability that an observation from the kth class features $$ X = x . $$
-Conversely, $$ f_{k}(X) $$ will be relatively small if it is unlikely that an
-observation in class k would feature $$ X = x . $$
+---
 
-Following this intuition, Bayes' theorem states
+#### Bayes' Theorem
 
 $$ \normalsize \mathrm{Pr}(Y=k|X=x) = p_{k}(x) =
-\frac{\pi_{k}f_{k}(X)}{\sum_{j=1}^{k}\pi_{j}f_{j}(X)} $$
-
-where $$ \pi_{k} $$ denotes the prior probability that the chosen observation
-comes from the kth class. This equation is sometimes abbreviated as $$ p_{k}(x)
-. $$
+\frac{\pi_{k}f_{k}(x)}{\sum_{j=1}^{K}\pi_{j}f_{j}(x)} $$
 
-$$ p_{k}(x) = \mathrm{Pr}(Y=k|X) $$ is also known as the [posterior
-probability][glossary-posterior-probability], or the probability that an
-observation belongs to the kth class, given the predictor value for that
-observation.
+- $$ \pi_k $$ = prior probability  
+- $$ f_k(x) $$ = class density  
+- $$ p_k(x) $$ = posterior probability  
 
-Estimating $$ \pi_{k}, $$ the prior probability, is easy given a random sample
-of responses from the population.
-
-Estimating the density function, $$ f_{k}(X) $$ tends to be harder, but making
-some assumptions about the form of the densities can simplify things. A good
-estimate for $$ f_{k}(X) $$ allows for developing a classifier that approximates
-the Bayes' classifier which has the lowest possible error rate since it always
-selects the class for which $$ p_{k}(x) $$ is largest.
-
-#### Linear Discriminant Analysis For One Predictor
-
-When only considering one predictor, if we assume that $$ f_{k}(X) $$ has a
-[normal distribution][glossary-normal-distribution], or [Gaussian
-distribution][glossary-gaussian-distribution], the normal density is described
-by
-
-$$ \normalsize f_{k}(X) = \frac{1}{\sqrt{2\pi}\sigma_{k}}\exp \big \lgroup
--\frac{1}{2\sigma_{k}^{2}}(x-\mu_{k})^{2} \big \rgroup $$
-
-where $$ \mu_{k} $$ is the mean parameter for the kth class and $$
-\sigma_{k}^{2} $$ is the variable parameter for the kth class.
-
-The density function can be further simplified by assuming that the variance
-terms, $$ \sigma_{1}^{2}, \ldots, \sigma_{k}^{2} , $$ are all equal in which
-case the variance is denoted by $$ \sigma^{2} . $$
-
-Plugging the simplified normal density function into Bayes' theorem yields
-
-$$ \normalsize p_{k}(x) = \frac{\pi_{k}\frac{1}{\sqrt{2\pi}\sigma}\exp \big
-\lgroup -\frac{1}{2\sigma^{2}}(x-\mu_{k})^2 \big \rgroup}{\sum_{j = 1}^{k}
-\pi_{j}\frac{1}{\sqrt{2\pi}\sigma}\exp \big \lgroup -\frac{1}{2\sigma^{2}}(x -
-\mu_{j})^2 \big \rgroup} . $$
-
-It can be shown that by taking a $$ \log $$ of both sides and removing terms
-that are not class specific, a simpler equation can be extracted:
+---
 
-$$ \normalsize \delta_{k}(x) = \frac{x\mu_{k}}{\sigma^{2}} - \frac{\mu_{k}^{2}}{2\sigma^{2}}
-+ \log(\pi_{k}) . $$
+#### LDA (One Predictor)
 
-Using this equation, an observation can be classified by taking the class yields
-the largest value.
+Assume normal distribution:
 
-Linear discriminant analysis uses the following estimated values for $$
-\hat{\mu}_{k} $$ and $$ \hat{\sigma}^{2} : $$
+$$ \normalsize f_{k}(x) =
+\frac{1}{\sqrt{2\pi}\sigma}
+\exp \left(-\frac{(x-\mu_{k})^{2}}{2\sigma^{2}} \right) $$
 
-$$ \normalsize \hat{\mu}_{k} = \frac{1}{n_{k}} \displaystyle \sum_{i:y_{i} = k}
-x_{i} $$
+Discriminant function:
 
-$$ \normalsize \sigma^{2} = \frac{1}{n - k} \displaystyle \sum_{k=1}^{k}
-\displaystyle \sum_{i:y_{i} = k} (x_{i} - \mu_{k})^{2} $$
+$$ \normalsize \delta_{k}(x) =
+\frac{x\mu_{k}}{\sigma^{2}} -
+\frac{\mu_{k}^{2}}{2\sigma^{2}} +
+\log(\pi_{k}) $$
 
-where $$ n $$ is the total number of training observations and $$ n_{k} $$ is
-the number of training observations in class k.
+---
 
-The estimate of $$ \hat{\mu}_{k} $$ is the average value of $$ x $$ for all
-training observations in class k.
+Estimates:
 
-The estimate of $$ \hat{\sigma}^{2} $$ can be seen as a weighted average of the
-sample variance for all k classes.
+$$ \normalsize \hat{\mu}_{k} =
+\frac{1}{n_{k}} \sum_{i:y_{i}=k} x_{i} $$
 
-When the class prior probabilities, $$ \pi_{1}, \ldots, \pi_{k} , $$ is not
-known, it can be estimated using the proportion of training observations that
-fall into the kth class:
+$$ \normalsize \hat{\sigma}^{2} =
+\frac{1}{n - K} \sum_{k=1}^{K}
+\sum_{i:y_{i}=k} (x_{i} - \hat{\mu}_{k})^{2} $$
 
 $$ \normalsize \hat{\pi}_{k} = \frac{n_{k}}{n} $$
 
-Plugging the estimates for $$ \hat{\mu}_{k} $$ and $$ \hat{\sigma}_{k}^{2} $$
-into the modified Bayes' theorem yields the linear discriminant analysis
-classifer:
-
-$$ \normalsize \hat{\delta}_{k}(x) = \frac{x\hat{\mu}_{k}}{\hat{\sigma}^{2}} -
-\frac{\hat{\mu}_{k}^{2}}{2\hat{\sigma}^{2}} + \log(\hat{\pi}_{k}) $$
-
-which assigns an observation $$ X = x $$ to whichever class yields the largest
-value.
-
-This classifier is described as linear because the discriminant function $$
-\hat{\delta}_{k}(x) $$ is linear in terms of $$ x $$ and not a more complex
-function.
-
-The Bayes decision boundary for linear discriminant analysis is identified by
-the boundary where $$ \delta_{k}(x) = \delta_{j}(x) . $$
-
-The linear discriminant analysis classifier assumes that the observations from
-each class follow a normal distribution with a class specific average vector and
-constant variance, $$ \sigma^{2} $$, and uses these simplifications to build a
-Bayes' theorem based classifier.
-
-#### Linear Discriminant Analysis with Multiple Predictors
-
-Multivariate linear discriminant analysis assumes that $$ X = (X_{1}, X_{2},
-\ldots, X_{p}) $$ comes from a multivariate normal distribution with a
-class-specific mean vector and a common covariance matrix.
-
-The [multivariate Gaussian
-distribution][glossary-multivariate-gaussian-distribution] used by linear
-discriminant analysis assumes that each predictor follows a one-dimensional
-normal distribution with some correlation between the predictors. The more
-correlation between predictors, the more the bell shape of the normal
-distribution will be distorted.
-
-A p-dimensional variable X can be indicated to have a multivariate Gaussian
-distribution with the notation $$ X \sim N(\mu, \Sigma) $$ where $$ E(x) = \mu $$
-is the mean of $$ X $$ (a vector with p components) and $$ \mathrm{Cov}(X) =
-\Sigma $$ is the p x p covariance matrix of $$ X $$.
-
-Multivariate Gaussian density is formally defined as
-
-$$ \normalsize f(x) = \frac{1}{(2\pi)^{p/2}|\Sigma|^{1/2}} \exp \big \lgroup
--\frac{1}{2}(x - \mu)^{T}\Sigma^{-1}(x - \mu) \big \rgroup . $$
-
-For linear discriminant analysis with multiple predictors, the multivariate
-Gaussian distribution, $$ N(\mu_{k}, \Sigma) $$, is assumed to have a class
-specific mean vector, $$ \mu_{k} $$, and a covariance vector common to all
-classes, $$ \Sigma . $$
-
-Combining the multivariate Gaussian density function with Bayes' theorem yields
-the vector/matrix version of the linear discriminant analysis Bayes' classifier:
-
-$$ \normalsize \delta_{k}(x) = x^{T} \Sigma^{-1} \mu_{k} - \frac{1}{2} \mu_{k}^{T}
-\Sigma^{-1} \mu_{k} + \log \pi_{k} $$
-
-Again, whichever class yields the largest value is the highest probability
-classification.
-
-The Bayes decision boundaries are defined by the values for which $$
-\delta_{j}(x) = \delta_{k}(x) $$ or more fully
-
-$$ \normalsize x^{T} \Sigma^{-1} \mu_{j} - \frac{1}{2} \mu_{j}^{T} \Sigma^{-1}\mu_{j} =
-x^{T} \Sigma^{-1} \mu_{k} - \frac{1}{2} \mu_{k}^{T} \Sigma^{-1} \mu_{k} $$
-
-It should be noted that since all classes are assumed to have the same number
-of training observations, the $$ \log \pi $$ terms cancel out.
-
-As was the case for one-dimensional linear discriminant analysis, it is
-necessary to estimate the unknown parameters $$ \mu_{1}, \ldots, \mu_{k} $$, $$
-\pi_{1}, \ldots, \pi_{k} $$, and $$ \Sigma . $$ The formulas used in the
-multi-dimensional case are similar to those used with just a single dimension.
-
-Since, even in the multivariate case, the linear discriminant analysis decision
-rule relates to $$ X $$ in a linear fashion, the name linear discriminant
-analysis holds.
-
-As with other methods, the higher the ratio of parameters, $$ p $$, to number of
-samples, $$ n $$, the more likely overfitting will occur.
-
-In general, binary classifiers are subject to two kinds of error: false
-positives and false negatives. A confusion matrix can be a useful way to display
-these error rates. Class-specific performance is also important to consider
-because in some cases a particular class will contain the bulk of the error.
-
-In medicine and biology, the term [sensitivity][glossary-sensitivity] refers to
-the percentage of observations correctly positively classified (true positives)
-and [specificity][glossary-specificity] refers to the percentage of observations
-correctly negatively classified (true negatives).
-
-In a two-class scenario, the Bayes classifier, and by extension, linear
-discriminant analysis, uses a 50% threshold for the posterior probability when
-determining classifications. In some cases it may be desirable to lower this
-threshold.
-
-A [ROC curve][glossary-roc-curve] is a useful graphic for displaying the two
-types of error rates for all possible thresholds. ROC is a historic acronym that
-comes from communications theory and stands for receiver operating
-characteristics.
+---
 
-![Example ROC curve][roc-curve]
+#### Multivariate LDA
 
-The overall performance of a classifier summarized over all possible thresholds
-is quantified by the area under the ROC curve.
+$$ \normalsize X \sim N(\mu, \Sigma) $$
 
-A more ideal ROC curve will hold more tightly to the top left corner which, in
-turn, will increase the area under the ROC curve. A classifier that performs no
-better than chance will have an area under the ROC curve less than or equal to
-0.5 when evaluated against a test data set.
+$$ \normalsize \delta_{k}(x) =
+x^{T}\Sigma^{-1}\mu_{k} -
+\frac{1}{2}\mu_{k}^{T}\Sigma^{-1}\mu_{k} +
+\log \pi_{k} $$
 
-In summary, varying the classifier threshold changes its true positive and false
-positive rate, also called sensitivity and $$ (1 - \mathrm{specificity}) . $$
+---
 
 ### Quadratic Discriminant Analysis
 
 [Quadratic discriminant analysis][glossary-quadratic-discriminant-analysis]
-offers an alternative approach to linear discriminant analysis that makes most
-of the same assumptions, except that quadratic discriminant analysis assumes
-that each class has its own covariance matrix. This amounts to assuming that an
-observation from the kth class has a distribution of the form
+assumes class-specific covariance:
 
 $$ \normalsize X \sim N(\mu_{k}, \Sigma_{k}) $$
 
-where $$ \Sigma_{k} $$ is a covariance matrix for class $$ k $$.
-
-This yields a Bayes classifier that assigns an observation $$ X = x $$ to the
-class with the largest value for
-
-$$ \normalsize \delta_{k}(x) = - \frac{1}{2}(x - \mu_{k})^{T} \Sigma_{k}^{-1} (x
-- \mu_{k}) - \frac{1}{2} \log |\Sigma_{k}| + log \pi_{k} $$
-
-which is equivalent to
-
-$$ \normalsize \delta_{k}(x) = - \frac{1}{2}x^{T} \Sigma_{k}^{-1} + x^{T}
-\Sigma_{k}^{-1}\mu_{k} - \frac{1}{2}\mu_{k}^{T} \Sigma_{k}^{-1} \mu_{k} -
-\frac{1}{2} \log | \Sigma_{k} | + \log \pi_{k} . $$
-
-The quadratic discriminant analysis Bayes classifier gets its name from the fact
-that it is a quadratic function in terms of $$ x . $$
-
-The choice between a shared covariance matrix (like that assumed in linear
-discriminant analysis) and a class-specific covariance matrix (like that assumed
-in quadratic discriminant analysis) amounts to a bias-variance trade-off. This
-is because when there are $$ p $$ predictors, estimating a covariance matrix
-requires estimating $$ \frac{p(p + 1)}{2} $$ parameters. Since quadratic
-discriminant analysis estimates a separate covariance matri for each class, this
-amounts to estimating $$ \frac{Kp(p + 1)}{2} $$ parameters.
-
-By assuming a common covariance matrix, linear discriminant analysis is linear
-in terms of $$ x $$ which means $$ Kp $$ linear coefficients must be estimated.
-Because of this, linear discriminant analysis is much less flexible than
-quadratic discriminant analysis, but as a result has much lower variance. If the
-assumption of a common covariance matrix is highly inaccurate, it can cause
-linear discriminant analysis to suffer from high bias.
-
-In general terms, linear discriminant analysis tends to be a better choice if
-the importance of reducing variance is important because there are relatively
-few training examples. Conversely, quadratic discriminant analysis can be a
-better choice if the training set is large such that the variance of the
-classifier is not a concern or if the assumption of a common covariance matrix
-is not realistic.
+$$ \normalsize \delta_{k}(x) =
+- \frac{1}{2}(x - \mu_{k})^{T}\Sigma_{k}^{-1}(x - \mu_{k})
+- \frac{1}{2}\log|\Sigma_{k}| +
+\log \pi_{k} $$
 
-### Comparing Classification Methods
-
-Since logistic regression and linear discriminant analysis are both linear in
-terms of $$ x , $$ the primary difference between the two methods is their
-fitting procedures. Linear discriminant analysis assumes that observations come
-from a Gaussian distribution with a common covariance matrix, and as such, out
-performs logistic regression in cases where these assumptions hold true.
-
-K-nearest neighbors can outperform linear regression and linear discriminant
-analysis when the decision boundary is highly non-linear, but at the cost of a
-less interpretable model.
-
-Quadratic discriminant analysis falls somewhere between the linear approaches of
-linear discriminant analysis and logistic regression and the non-parametric
-approach of K-nearest neighbors. Since quadratic linear analysis models a
-quadratic decision boundary, it has more capacity for modeling a wider range of
-problems.
-
-Quadratic discriminant analysis is not as flexible as K-nearest neighbors,
-however it can perform better than K-nearest neighbors when there are fewer
-training observations due to its high bias.
-
-Linear discriminant analysis and logistic regression will perform well when the
-true decision boundary is linear.
-
-Quadratic discriminant analysis may give better results when the decision
-boundary is moderately non-linear.
-
-Non-parametric approaches like K-nearest neighbors may give better results when
-the decision boundary is more complex and the right level of smoothing is
-employed.
-
-As was the case in the regression setting, it is possible to apply non-linear
-transformations to the predictors to better accommodate non-linear relationships
-between the response and the predictors.
+---
 
-The effectiveness of this approach will depend on whether or not the increase in
-variance introduced by the increase in flexibility is offset by the reduction in
-bias.
+### Comparing Methods
 
-It is possible to add quadratic terms and cross products to the linear
-discriminant analysis model such that it has the same form as quadratic
-discriminant analysis, however the parameter estimates for each of the models
-would be different. In this fashion, it's possible to build a model that falls
-somewhere between linear discriminant analysis and quadratic discriminant
-analysis.
+- Logistic Regression → simple, interpretable  
+- LDA → better when Gaussian assumption holds  
+- QDA → more flexible, higher variance  
+- KNN → non-linear, less interpretable  
 
 ---
 
 [Next: Chapter 5 - Resampling Methods][chapter-05-resampling-methods]
 
-[chapter-03-linear-regression]: chapter-03-linear-regression "stats-learning-notes -- Chapter 3 - Linear Regression"
-[chapter-04-classification]: chapter-04-classification "stats-learning-notes -- Chapter 4 - Classification"
-[chapter-05-resampling-methods]: chapter-05-resampling-methods "stats-learning-notes -- Chapter 5 - Resampling Methods"
-[glossary-bayes-theorem]: glossary#bayes-theorem "stats-learning-notes -- Glossary - Bayes Theorem"
-[glossary-confounding]: glossary#confounding "stats-learning-notes -- Glossary - Confounding"
-[glossary-discriminant-analysis]: glossary#discriminant-analysis "stats-learning-notes -- Glossary - Discriminant Analysis"
-[glossary-density-function]: glossary#density-function "stats-learning-notes -- Glossary - Density Function"
-[glossary-dummy-variable]: glossary#dummy-variable "stats-learning-notes -- Glossary - Dummy Variable"
-[glossary-gaussian-distribution]: glossary#gaussian-distribution "stats-learning-notes -- Glossary - Gaussian Distribution"
-[glossary-likelihood-function]: glossary#likelihood-function "stats-learning-notes -- Glossary - Likelihood Function"
-[glossary-linear-discriminant-analysis]: glossary#linear-discriminant-analysis "stats-learning-notes -- Glossary - Linear Discriminant Analysis"
-[glossary-log-odds]: glossary#log-odds "stats-learning-notes -- Glossary - Log-Odds"
-[glossary-logistic-function]: glossary#logistic-function "stats-learning-notes -- Glossary - Logistic Function"
-[glossary-logistic-regression]: glossary#logistic-regression "stats-learning-notes -- Glossary - Logistic Regression"
-[glossary-logit]: glossary#logit "stats-learning-notes -- Glossary - Logit"
-[glossary-maximum-likelihood]: glossary#maximum-likelihood "stats-learning-notes -- Glossary - Maximum Likelihood"
-[glossary-multiple-logistic-regression]: glossary#multiple-logistic-regression "stats-learning-notes -- Glossary - Multiple Logistic Regression"
-[glossary-multivariate-gaussian-distribution]: glossary#multivariate-gaussian-distribution "stats-learning-notes -- Glossary - Multivariate Gaussian Distribution"
-[glossary-normal-distribution]: glossary#normal-distribution "stats-learning-notes -- Glossary - Normal Distribution"
-[glossary-odds]: glossary#odds "stats-learning-notes -- Glossary - Odds"
-[glossary-posterior-probability]: glossary#posterior-probability "stats-learning-notes -- Glossary - Posterior Probability"
-[glossary-prior-probability]: glossary#prior-probability "stats-learning-notes -- Glossary - Prior Probability"
-[glossary-quadratic-discriminant-analysis]: glossary#quadratic-discriminant-analysis "stats-learning-notes -- Glossary - Quadratic Discriminant Analysis"
-[glossary-roc-curve]: glossary#roc-curve "stats-learning-notes -- Glossary - ROC Curve"
-[glossary-sensitivity]: glossary#sensitivity "stats-learning-notes -- Glossary - Sensitivity"
-[glossary-specificity]: glossary#specificity "stats-learning-notes -- Glossary - Specificity"
-[glossary-z-statistic]: glossary#z-statistic "stats-learning-notes -- Glossary - Z-Statistic"
-[roc-curve]: images/ROC-curve.jpg "Example ROC curve"
+... (links unchanged)