# HW2

Make sure you check the syllabus for the due date. Please use the notations adopted in class, even if the problem is stated in the book using a different notation.

We are not looking for very long answers (if you find yourself writing more than 1-2 page(s) of typed text per problem, you are probably on the wrong track). Try to be concise; also keep in mind that good ideas and explanations matter more than exact details.

### PROBLEM 1 [50 points]

Consider the following neural network (left graph), with 8 input units (for data with 8 features), 3 hidden units and 8 output units, and assume the nonlinear functions are all sigmoid.

a)The 8 training data inputs are identical with the outputs, as shown in the right side table. Implement this network and the backpropagation algorithm to compute all the network weights; you should initialize the weights with nontrivial values (i.e not values that already minimize the erorr).

HINT: on the trained network, you should obtain values for the hidden units somehow similar with the ones shown in the table (up to symmetry). Feel free to make changes to the algorithm that suit your implementation, but briefly document them.

b) Since the outputs and inputs are identical for each datapoint, one can view this network as an encoder-decoder mechanism. (this is one of the uses of neural networks). In this context, explain the purpose of the training algorithm. (I expect a rather nontechnical --but documented-- answer).

c) Can this encoder-decoder scheme work with 1 hidden unit? Can it work with 2 hidden units? What if there are more than 8 inputs and outputs? Justify your answers mathematically.

Submit a concise report.

### PROBLEM 2 [50]

In this assignment, you will create a Naive Bayes classifier for detecting e-mail spam, and you will test your classifier on a publicly available spam dataset. You will experiment with three methods for modeling the distribution of features, and you will test your classifier using 10-fold cross-validation.
• Step 1: Download the Spambase dataset available from the UCI Machine Learning Repository.

The Spambase data set consists of 4,601 e-mails, of which 1,813 are spam (39.4%). The data set archive contains a processed version of the e-mails wherein 57 real-valued features have been extracted and the spam/non-spam label has been assigned. You should work with this processed version of the data. The data set archive contains a description of the features extracted as well as some simple statistics over those features.

• Partition the data into 10 folds.

To estimate the generalization (testing) error of your classifier, you will perform cross-validation. In k-fold cross-validation, one would ordinarily partition the data set randomly into k groups of roughly equal size and perform k experiments (the "folds") wherein a model is trained on k-1 of the groups and tested on the remaining group, where each group is used for testing exactly once. (A common value of k is 10.) The generalization error of the classifier is estimated by the average of the performance across all k folds.

While one should perform cross-validation with random partitions, for consistency and comparability of your results, you should partition the data into 10 groups as follows: Consider the 4,601 data points in the order they appear in the processed data file. Each group will consist of every 10th data point in file order, i.e., Group 1 will consist of points {1,11,21,...}, Group 2 will consist of points {2,12,22,...}, ..., and Group 10 will consist of ponts {10,20,30,...}. Finally, Fold k will consist of testing on Group k a model obtained by training on the remaining k-1 groups.

• Step 2: Create three Naive Bayes classifiers by modeling the features in three different ways.

The 57 features are real-valued, and one can model the feature distributions in simple and complex ways. You will explore the effect of this modeling on overall performance.

1. Model the features as simple Bernoulli (Boolean) random variables. Consider a threshold such as the overall mean value of the feature (available in the Spambase documentation), and simply compute the fraction of the time that the feature value is above or below the overall mean value for each class. In other words, for feature fi with overall mean value mui, estimate

• Pr[fi <= mui | spam]
• Pr[fi > mui | spam]
• Pr[fi <= mui | non-spam]
• Pr[fi > mui | non-spam]

and use these estimated values in your Naive Bayes predictor, as appropriate. (I would suggest using Laplace or Laplace-like smoothing to avoid any issues with zero probabilities, and feel free to experiment with various theshold values, if you like.)

2. Model the features as Gaussian random variables, estimating the class conditional mean and variance as appropriate. (Again, I would suggest smoothing your estimate of the variance, to avoid any zero variance issues.)

3. Model the feature distribution via a histogram. Bucket the data into a number of bins and estimate the class conditional probabilities of each bin. For example, for any feature, consider the following values: min-value, mean-value, max-value, and the class conditional (spam/non-spam) mean values, one of which will presumably be lower than the overall mean, and one of which will be higher than the overall mean. Order these values numerically, and create four feature value bins as follows

• [min-value, low-mean-value]
• (low-mean-value, overall-mean-value]
• (overall-mean-value, high-mean-value]
• (high-mean-value, max-value]

and estimate the class conditional probabilities for each of these bins, in a manner similar to that used in the Bernoulli model above. (Again, use Laplace smoothing to avoid any issues with zero probabilities, and feel free to experiment with the number of bins and various theshold values, if you like.)

• Step 3: Evaluate your results.

1. Error tables: For each of your three classifiers, create a table with one row per fold showing your false positive, false negative, and overall error rates, and add one final row per table corresponding to the average error rates across all folds. For this problem, the false positive rate is the fraction of non-spam testing examples that are misclassified as spam, the false negative rate is the fraction of spam testing examples that are misclassified as non-spam, and the overall error rate is the fraction of overall examples that are misclassified.

2. ROC Curves: In many situations, there is a different cost associated with false positive (Type I) and false negative (Type II) errors. In spam filtering, for example, a false positive error is a legitimate e-mail that is misclassified as spam (and perhaps automatically redirected to a "spam folder" or, worse, auto-deleted) while a false negative is a spam message that is misclassified as legitimate (and sent to one's inbox). Most users are willing to accept some false negative examples so long as very few legitimate e-mails are misclassified.

When using Naive Bayes, one can easily make such trade-offs. For example, in the usual Bayes formulation, one would predict "spam" if

Pr[spam | data] > Pr[non-spam | data]

or equivalently, in a log-odds formulation,

ln (Pr[spam | data] / Pr[non-spam | data]) > 0.

Note that one could choose to classify an e-mail as spam for any threshold tau

ln (Pr[spam | data] / Pr[non-spam | data]) > tau

where positive values of tau effectively reduce the number of spam classifications, thus decreasing false positives at the expense of increasing false negatives, while negative values of tau have the opposite effect.

One mechanism for visualizing this trade-off is through an ROC curve. (See, for example, the Wikipedia page on ROC curves.) An ROC curve effectively visualizes the true positive (detection) rate vs. the false positive (false alarm) rate for all possible thresholds. (The true positive rate is the fraction of spam messages above threshold; the false positive rate is the fraction of non-spam messages above threshold. Note that the true positive rate is 1 minus the false negative rate.)

One simple way to generate an ROC curve is to sort all m testing points by their log-odds, and for the top k testing points, for all k between 1 and m, calculate the true positive and false positive rates associated with the top k items; plot these points.

Create an ROC curve for each of your classifiers for Fold 1. Preferable draw all three curves on the same plot so that you can compare them.

3. AUC: An ROC curve visualizes the tradeoff between false positive rate and true positive rate (which is 1 minus the false negative rate) at various operating points. One measure for summarizing this data is to compute the area under the ROC curve or AUC. The AUC, as its name suggests, is simply the area under the ROC curve, which for a perfect curve is 1 and for a random predictor is 1/2. This area has an interesting probabilistic interpretation: it is the chance that the predictor will rank a randomly chosen positive example above a randomly chosen negative example.

The AUC can be calculated fairly simply using the trapezoidal rule for estimating the area under a curve: If our m ROC data points are

(x1, y1), (x2, y2), ..., (xm, ym)

in order, where

(x1, y1) = (0,0)

and

(xm, ym) = (1,1)

then the AUC is calculated as follows:

(1/2) sumk=2 to m (xk - xk-1) (yk + yk-1).

Calculate the AUC of your three classifiers.

• Step 5: Submit a concise report (results and issues). Be ready to demo.

### PROBLEM 2.1 [Extra Credit].

Run the Gradient Discriminant Analysis on the spambase data. Use the k-folds from the previous problem (1 for testing, k-1 for training, for each fold)
Since you have 57 real value features, each gaussian will have a mean  vector with 57 components, and a they will have a common (shared) covariance matrix size 57x57. Looking at the result, does it appear that the gaussian assumption (normal distributed data) holds for this particular dataset?

### PROBLEM 3 [20 points]

a) Prove that

b) You are given a coin which you know is either fair or double-headed. You believe
that the a priori odds of it being fair are F to 1; i.e., you believe that the a priori probability of the coin
being fair is F/(F+1) . You now begin to flip the coin in order to learn more. Obviously, if you ever see a tail,
you know immediately that the coin is fair. As a function of F, how many heads in a row would you need to see before becoming convinced that there is a better than even chance that the coin is double-headed?

DHS CH2, Pb 43

### PROBLEM 4 part 2 [Extra Credit]

a) DHS CH2, Pb 45

### PROBLEM 6 [15 points]

Read prof Andrew Ng's lecture on ML practice advice. Write a brief summary (1 page) explaining the quantities in the lecture and the advice.

### PROBLEM 7 [15 points]

b) DHS CH2, Pb 44

### PROBLEM 8 [15 points]

DHS ch6, Pb1
Show that if the transfer function of the hidden units is linear, a three-layer network is equivalent to a two-layer one. Explain why, therefore, that a three-layer network with linear hidden units cannot solve a non-linearly separable problem such as XOR or n-bit parity.

### PROBLEM 9 [extra credit]

For a function f(x1,x2,..., xn) with real values, the "Hessian" is the matrix of partial second derivatives

Consider the log-likelihood function for logistic regression

Show that its Hessian matrix H is negative semidefinite, i.e. for any vector z satisfies

Remark: This fact is sometimes written $\mathrm{H\le 0}$and implies the log-likelihood function is concave.

Hint: