## Requirement

### Single pixels as features

• Features: The basic feature set consists of a single binary indicator feature for each pixel. Specifically, the feature indicates the status of the pixel. Its value is 1 if the pixel is foreground (no need to distinguish between the two different foreground values), and 0 if it is background. The images are of size 28*28, so there are 784 features in total.

• Training: The goal of the training stage is to estimate every pixel location(i,j) and for every digit class from 0 to 9. The likelihood estimate is defined.

In addition, as discussed in the lecture, you have to smooth the likelihoods to ensure that there are no zero counts. Laplace smoothing is a very simple method that increases the observation count of every value f by some constant k. This corresponds to adding k to the numerator above, and k*V to the denominator (where V is the number of possible values the feature can take on). The higher the value of k, the stronger the smoothing. Experiment with different values of k (say, from 0.1 to 10) and find the one that gives the highest classification accuracy.

You should also estimate the priors `P(class)` by the empirical frequencies of different classes in the training set.

• Testing: You will performmaximum a posteriori (MAP) classification of test digits according to the learned Naive Bayes model. Suppose a test image has feature values. According to this model, the posterior probability (up to scale) of each class given the digit is given.

Note that in order to avoid underflow, it is standard to work with the log of the above quantity.

After you compute the above decision function values for all ten classes for every test image, you will use them for MAP classification.

• Evaluation: Use the true class labels of the test images from the test labels file to check the correctness of the estimated label for each test digit. Report your performance in terms of the classification rate for each digit(percentage of all test images of a given digit correctly classified). Also report your confusion matrix. This is a 10x10 matrix whose entry in row r and column c is the percentage of test images from class r that are classified as class c. In addition, for each digit class, show the test examples from that class that have the highest and the lowest posterior probabilities according to your classifier. You can think of these as the most and least “prototypical” instances of each digit class (and the least “prototypical” one is probably misclassified).

Important: The ground truth labels of test images should be usedonlyto evaluate classification accuracy. They should not be used in any way during the decision process.

Tip: You should be able to achieve at least 70% accuracy on the test set. One “warning sign” that you have a bug in your implementation is if some digit gets 100% or 0% classification accuracy (that is, your system either labels all the test images as the same class, or never wants to label any test images as some particular class).

Odds ratios: When using classifiers in real domains, it is important to be able to inspect what they have learned. One way to inspect a naive Bayes model is to look at the most likely features for a given label. Another tool for understanding the parameters is to look at odds ratios. For each pixel featureFijand pair of classes c1, c2, the odds ratio is defined.

This ratio will be greater than one for features which cause belief in c1to increase over the belief in c2. The features that have the greatest impact on classification are those with both a high probability (because they appear often in the data) and a high odds ratio (because they strongly bias one label versus another).

Take four pairs of digits that have the highest confusion rates according to your confusion matrix, and for each pair, display the maps of feature likelihoods for both classes as well as the odds ratio for the two classes. For example, the figure below shows the log likelihood maps for 1 (left), 8 (center), and the log odds ratio for 1 over 8 (right).

If you cannot do a graphical display like the one above, you can display the maps in ASCII format using some coding scheme of your choice. For example, for the odds ratio map, you can use ‘+’ to denote features with positive log odds, ‘ ‘ for features with log odds close to 1, and ‘-‘ for features with negative log odds.

### Pixel groups as features

Instead of each feature corresponding to a single pixel, we can form features from groups of adjacent pixels. We can view this as a relaxation of the Naive Bayes assumption that allows us to have a more accurate model of the dependencies between the individual random variables. Specifically, consider a 2*2 square of pixels with top left coordinate i,j and define a feature Gi,jthat corresponds to the ordered tuple of the four pixel values.

We can generalize the above examples of 2*2 features to define features corresponding to n*m disjoint or overlapping pixel patches. An n*m feature will have 2n*mdistinct values, and as many entries in the conditional probability table for each class. Laplace smoothing applies to these features analogously as to the single pixel features.

In this part, you should build Naive Bayes classifiers for feature sets of n*m disjoint/overlapping pixel patches and report the following:

• Test set accuracies for disjoint patches of size 2*2, 2*4, 4*2, 4*4.
• Test set accuracies for overlapping patches of size 2*2, 2*4, 4*2, 4*4, 2*3, 3*2, 3*3.
• Discussion of the trends you have observed for the different feature sets (including single pixels), in particular, why certain features work better than others for this task.
• Brief discussion of running time for training and testing for the different feature sets (which ones are faster and why, and how does the running time scale with feature set size).

Tip: You should be able to achieve over 80% accuracy with your best feature set.

## Extra Credit

• Experiment with yet more features to improve the accuracy of the Naive Bayes model. For example, instead of using binary pixel values, implement ternary features.
• Apply your Naive Bayes classifier with various features to thisface data. It is in a similar format to that of the digit data, and contains training and test images and binary labels, where 0 corresponds to ‘non-face’ and 1 corresponds to ‘face’. The images themselves are higher-resolution than the digit images, and each pixel value is either ‘#’, corresponding to an edge being found at that location, or ‘ ‘, corresponding to a non-edge pixel.