Naive Bayes

by Chiyuan Zhang

This notebook illustrates multiclass learning using Naive Bayes in Shogun. A semi-formal introduction to Logistic Regression is provided at the end.

\[ P\left( Y=k | X = x \right) = \frac{P(X=x|Y=k)P(Y=k)}{P(X=x)} \]

The prediction is then made by

\[ y = \operatorname*{argmax}_{k\in\{1,\ldots,K\}}\; P(Y=k|X=x) \]

Since \(P(X=x)\) is a constant factor for all \(P(Y=k|X=x)\), \(k=1,\ldots,K\), there is no need to compute it.

In SHOGUN, CGaussianNaiveBayes implements the Naive Bayes algorithm. It is prefixed with "Gaussian" because the probability model for \(P(X=x|Y=k)\) for each \(k\) is taken to be a multi-variate Gaussian distribution. Furthermore, each dimension of the feature vector \(X\) is assumed to be independent. The Naive independence assumption enables us the learn the model by estimating the parameters for each feature dimension independently, thus the whole learning algorithm runs very quickly. And this is also the reason for its name. However, this assumption can be very restrictive. In this demo, we show a simple 2D example. There are 3 linearly separable classes. The scattered points are training samples with colors indicating their labels. The filled area indicate the hypothesis learned by the CGaussianNaiveBayes. The training samples are actually generated from three Gaussian distributions. But since the covariance for those Gaussian distributions are not diagonal (i.e. there are rotations), the GNB algorithm cannot handle them properly.

We first init the models for generating samples for this demo:

In []:
import numpy as np
import pylab as pl

np.random.seed(0)

n_train = 300

models = [{'mu': [8, 0], 'sigma':
              np.array([[np.cos(-np.pi/4),-np.sin(-np.pi/4)],
                  [np.sin(-np.pi/4), np.cos(-np.pi/4)]]).dot(np.diag([1,4]))},
          {'mu': [0, 0], 'sigma':
              np.array([[np.cos(-np.pi/4),-np.sin(-np.pi/4)],
                  [np.sin(-np.pi/4), np.cos(-np.pi/4)]]).dot(np.diag([1,4]))},
          {'mu': [-8,0], 'sigma':
              np.array([[np.cos(-np.pi/4),-np.sin(-np.pi/4)],
                  [np.sin(-np.pi/4), np.cos(-np.pi/4)]]).dot(np.diag([1,4]))}]

A helper function is defined to generate samples:

In []:
    def gen_samples(n_samples):
        X_all = np.zeros((2, 0))
        Y_all = np.zeros(0)
    
        for i, model in enumerate(models):
            Y = np.zeros(n_samples) + i+1
            X = np.array(model['sigma']).dot(np.random.randn(2, n_samples)) + np.array(model['mu']).reshape((2,1))
    
            X_all = np.hstack((X_all, X))
            Y_all = np.hstack((Y_all, Y))
    
        return (X_all, Y_all)

Then we train the GNB model with SHOGUN:

In []:
from modshogun import GaussianNaiveBayes
from modshogun import RealFeatures
from modshogun import MulticlassLabels

X_train, Y_train = gen_samples(n_train)

machine = GaussianNaiveBayes()
machine.set_features(RealFeatures(X_train))
machine.set_labels(MulticlassLabels(Y_train))

machine.train()
Out[]:
True

Run classification over the whole area to generate color regions:

In []:
delta = 0.1
x = np.arange(-20, 20, delta)
y = np.arange(-20, 20, delta)
X,Y = np.meshgrid(x,y)
Z = machine.apply_multiclass(RealFeatures(np.vstack((X.flatten(), Y.flatten())))).get_labels()

Plot figure:

In []:
pl.figure(figsize=(8,5))
pl.contourf(X, Y, Z.reshape(X.shape), np.arange(0, len(models)+1))
pl.scatter(X_train[0,:],X_train[1,:], c=Y_train)
pl.axis('off')
pl.tight_layout()

Although the independent assumption is usually considered to be too optimistic in reality, Naive Bayes sometimes works very well in some applications. For example, in email spam filtering, Naive Bayes (More specifically, the discrete Naive Bayes is generally used in this scenario. The main difference with Gaussian Naive Bayes is that a tabular instead of a parametric Gaussian distribution is used to describe the likelihood \(P(X=x|K=k)\)) is a very popular and widely used method.

This algorithm is closely related to the Gaussian Mixture Model (GMM) learning algorithm. However, while GMM is an unsupervised learning algorithm, Gaussian Naive Bayes is supervised learning. It uses the training labels to directly estimate the Gaussian parameters for each class, thus avoids the iterative Expectation Maximization procedures in GMM.

The merit of GNB is that both training and predicting are very fast, and it has no hyper-parameters.

Brief Introduction to Logistic Regression

Although named logistic regression, it is actually a classification algorithm. Similar to Naive Bayes, logistic regression computes the posterior \(P(Y=k|X=x)\) and makes prediction by

\[ y = \operatorname*{argmax}_{k\in\{1,\ldots,K\}}\; P(Y=k|X=x) \]

However, Naive Bayes is a generative model, in which the distribution of the input variable \(X\) is also modeled (by a Gaussian distribution in this case). But logistic regression is a discriminative model, which doesn't care about the distribution of \(X\), and models the posterior directly. Actually, the two algorithms are a generative-discriminative pair.

To be specific, logistic regression uses linear functions in \(X\) to model the posterior probabilities:

The training of a logistic regression model is carried out via maximum likelihood estimation of the parameters \(\boldsymbol\beta = \{\beta_{10},\beta_1^T,\ldots,\beta_{(K-1)0},\beta_{K-1}^T\}\). There is no closed form solution for the estimated parameters.

There is no independent implementation of logistic regression in SHOGUN, but the CLibLinear becomes a logistic regression model when constructed with the argument L2R_LR. This model also include a regularization term of the \(\ell_2\)-norm of \(\boldsymbol\beta\). If sparsity in \(\boldsymbol\beta\) is needed, one can also use L1R_LR, which replaces the \(\ell_2\)-norm regularizer with a \(\ell_1\)-norm regularizer.

Unfortunately, while the original LibLinear implementation of Logistic Regression support multiclass case, due to interface incompatability, one cannot use LibLinear as a multiclass-machine in SHOGUN directly so far. An easy work-around is to use multiclass-to-binary reduction instead. Please see the Multiclass Reduction tutorial for details.