Multivariate Normal, Multivariate - mariginal and conditional, Correlation Coefficient

You have to write your own code, but it is ok to discuss or look up specific algorithmic/language code details. Before starting coding, make sure to understand :

- how
2-dim Gaussians work

- Naive
Bayes independence of features, thus the product of
probabilities

- languages
like Python or Matlab or R can help immensely in dealing with
multidimensional data, including the means and covariances in
2-dim.

- E and M
steps for EM algorithm. You can look into existing
EM code for help.

Since you have 57 real value features, each of the 2gaussians (for + class and for - class) will have a mean vector with 57 components, and a they will have

- either a common (shared) covariance matrix size 57x57. This
covariance is estimated from all training data (both classes)

- or two separate covariance 57x57 matrices (estimated
separately for each class)

Looking at the training and testing performance, does it appear that the gaussian assumption (normal distributed data) holds for this particular dataset?

Create a set of *Naive Bayes classifiers* for detecting e-mail spam and test the classifiers on the Spambase dataset via 10-fold cross validation.

Create three distinct Naive Bayes classifiers by varying the likelihood distribution. In particular, try:

Modeling the likelihood using a Bernoulli distribution

Modeling the likelihood using a Gaussian distribution

Modeling the likelihood non-parameterically / histogram (optional)

For real-valued features, a Bernoulli likelihood may be obtained by thresholding against a scalar-valued statistic, such as the sample mean \(\mu\). Similarly, a Gaussian likelihood function can be obtained by estimating the [class conditional] expected value \(\mu\) and variance \(\sigma\). For the non-parameteric setting, model the feature distribution either with a kernel density estimate (KDE) or a histogram.

*Bernoulli example:* Consider some threshold \(\mu \in \mathbb{R}\). To compute the conditional probability of a feature \(f_i\), compute the fraction by which \(f_i\) is above or below \(\mu\) over all the data within its class. In other words, for feature \(f_i\) with expected value \(\mu_i\), estimate: \[ P(f_i \leq \mu_i \mid \text{spam} ) \text{ and } P(f_i > \mu_i \mid \text{spam} )\] \[ P(f_i \leq \mu_i \mid \text{non-spam} ) \text{ and } P(f_i > \mu_i \mid \text{non-spam} )\] and use these estimated values in your Naive Bayes predictor.

For all the models mentioned, you may want to consider using some kind of additive smoothing technique as a regularizer.

Evaluate the performance of the classifiers using the following three performance summaries.

*Error tables:* Create a table with one row per fold showing the false positive, false negative, and overall error rates of the classifiers. Add one final row per table yielding the average error rates across all folds.

*ROC Curves:* Graph the Receiver Operating Characteristic (ROC) curve for each of your classifiers on *Fold 1* and calculate their area-under-the-curve (AUC) statistics. You should draw all three curves on the same plot so that you can compare them.

For this problem, the *false positive rate* is the fraction of non-spam testing examples that are misclassified as spam, while the *false negative rate* is the fraction of spam testing examples that are misclassified as non-spam.

**Context:** In many situations, false positive (Type I) and false negative (Type II) errors incur different costs. In spam filtering, for example, a false positive 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).

When using Naive Bayes, one can easily make such trade-offs. For example, in the usual Bayes formulation, with \(\mathbf{x}\) the data vector and \(y\) the class variable, one would predict “spam” if:

\[ P(y = \text{spam} | \mathbf{x}) > P(y = \text{non-spam} | \mathbf{x}) \]

or equivalently, in a log-odds formulation,

\[ \ln(P(y = \text{spam} | \mathbf{x}) / P(y = \text{non-spam} | \mathbf{x})) > 0 \]

However, note that one could choose to classify an e-mail as spam for any threshold \(\tau\), i.e.:

\[ \ln(P(y = \text{spam} | \mathbf{x}) / P(y = \text{non-spam} | \mathbf{x})) > \tau \]

Larger values of \(\tau\) reduce the number of spam classifications, reducing false positives at the expense of more false negatives. Negative values of \(\tau\) have the converse effect.

Most users are willing to accept some false negative examples so long as very few legitimate e-mails are misclassified. Given your classifiers and their ROC curves, what value of \(\tau\) would you choose to deploy in a real email spam filter?

Annotate on the right the sections of the code with your explanation of what the code does. Submit as pdf.

mean_1 [3,3]); cov_1 = [[1,0],[0,3]]; n1=2000 points

mean_2 =[7,4]; cov_2 = [[1,0.5],[0.5,1]]; ; n2=4000 points

You should obtain a result visually like this (you dont necessarily have to plot it)

B) Same problem for 2-dim data on file 3gaussian.txt , generated using a mixture of three Gaussians. Verify your findings against the true parameters used generate the data below.

mean_1 = [3,3] ; cov_1 = [[1,0],[0,3]]; n1=2000

mean_2 = [7,4] ; cov_2 = [[1,0.5],[0.5,1]] ; n2=3000

mean_3 = [5,7] ; cov_3 = [[1,0.2],[0.2,1]] ); n3=5000

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?

A.Generate mixture data (coin flips). Pick a p,r,pi as in the EM example discussed in class (or in notes). Say p=.75, r=.4, pi=.8, but you should try this for several sets of values. Generate the outcome of the coin experiment by first picking a coin (pi probability for first coin, 1-pi probability for the second coin), then flip that coin K times (use K=10) with probability of head (p if first coin is picked, r if the second coin is picked) and finally write down a 1 if the head is seen, 0 for the tail. Repeat this M=1000 times or more; so your outcome is a stream of M sequences of K flips : (1001110001; 0001110001; 1010100101 etc)

B.Infer parameters from data. Now using the stream of 1 and 0 observed, recover p,r,pi using the EM algorithm; K is known in advance. Report in a table the parameter values found by comparison with the ones used to generate data; try several sets of (p,r,pi). Here are some useful notes, and other readings (1 , 2 , 3 , 4) for the coin mixture.

C(more coins, optional). Repeat A) and B) with T coins instead of two. You will need more mixture parameters.

Testing: for each testing point , apply the Naive Bayes classifier as before: take the log-odd of product of probabilities over features mixtures (and the prior), separately for positive side and negative side; use the overall probability ratio as an output score, and compute the AUC for testing set. Do this for a 10-fold cross validation setup. Is the overall 10-fold average AUC better than before, when each feature model was a single Gaussian?

a) Somebody tosses a fair coin and if the result is heads, you
get nothing, otherwise you get $5. How much would you be pay to
play this game? What if the win is $500 instead of $5?

b) Suppose you play instead the following game: At the beginning
of each game you pay an entry fee of $100. A coin is tossed until
a head appears, counting n = the number of tosses it took to see
the first head. Your reward is 2^{n} (that is: if a head
appears first at the 4th toss, you get $16). Would you be willing
to play this game (why)?

c) Lets assume you answered "yes" at part b (if you did not, you
need to fix your math on expected values). What is the probability
that you make a profit in one game? How about in two games?

d) [difficult] After
about how many games (estimate) the probability of making a profit
overall is bigger than 50% ?

a) DHS CH2, Pb 45

The standard method for calculating the probability of a sequence in a given HMM is to use the forward probabilities αi(t).