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NaiveBayes

Naive Bayes Model

The Naive Bayes classifier is an example of the generative approach: we will model {$P(\mathbf{x},y)$}. Consider the toy transportation data below:

x: Inputs/Features/Attributes			y: Class
Distance(miles)	Raining	Flat Tire	Mode
1 mile	no	no	bike
2 miles	yes	no	walk
1 mile	no	yes	bus
1 mile	yes	no	walk
2 miles	yes	no	bus
1 mile	no	no	car
1 mile	yes	yes	bike
10 miles	yes	no	bike
10 miles	no	no	car
4 miles	no	no	bike

We will decompose {$P(\mathbf{x},y)$} into class prior and class model:

{$P(\mathbf{x},y) = \underbrace{P(y)}_{\rm class prior} \underbrace{P(\mathbf{x}\mid y)}_{\rm class model}$}

and estimate them separately as {$\hat{P}(y)$} and {$\hat{P}(\mathbf{x} \mid y)$}. (Class prior should not be confused with parameter prior. They are very similar concepts, but not the same things.)

We will then use our estimates to output a classifier using Bayes rule:

{$ \begin{align*} h(\mathbf{x}) & = \arg\max_y \;\;\hat{P}(y\mid \mathbf{x}) \\ & = \arg\max_y \;\;\;\; \frac{\hat{P}(y)\hat{P}(\mathbf{x} \mid y)}{\sum_{y'}\hat{P}(y')\hat{P}(\mathbf{x} \mid y')} \\ & = \arg\max_y \;\;\;\; \hat{P}(y)\hat{P}(\mathbf{x} \mid y) \end{align*} $}

To estimate our model using MLE, we can separately estimate the two parts of the model:

{$ \begin{align*} \log P(D) & = \sum_i \log P(\mathbf{x}_i,y_i) \\ & = \sum_i \log P(y_i) + \log P(\mathbf{x}_i \mid y_i) \\ & = \log P(D_Y) + \log P(D_X | D_Y) \end{align*} $}

Estimating {$P(y)$}

How do we estimate {$P(y)$}? This is very much like the biased coin, except instead of two outcomes, we have 4 (walk, bike, bus, car). We need 4 parameters to represent this multinomial distribution (3 really, since they must sum to 1): {$(\theta_{\rm walk},\theta_{\rm bike},\theta_{\rm bus},\theta_{\rm car})$}. The MLE estimate (deriving it is a good exercise) is {$\hat{\theta}_y = \frac{1}{n}\sum_i \textbf{1}(y=y_i)$}.

y	parameter {$\theta_{y}$}	MLE {$\hat{\theta}_{y}$}
walk	{$\theta_{\rm walk}$}	0.2
bike	{$\theta_{\rm bike}$}	0.4
bus	{$\theta_{\rm bus}$}	0.2
car	{$\theta_{\rm car}$}	0.2

Estimating {$P(\textbf{x}\mid y)$}

Estimating class models is much more difficult, since the joint distribution of m dimensions of {$\textbf{x}$} can be very complex. Suppose that all the features are binary, like Raining or Flat Tire above. If we have m features, there are {$K*2^m$} possible values of {$(\textbf{x},y)$} and we cannot store or estimate such a distribution explicitly, like we did for {$P(y)$}. The key (naive) assumption of the model is conditional independence of the features given the class. Recall that {$X_k$} is conditionally independent of {$X_j$} given Y if:

{$P(X_j=x_j \mid X_k=x_k, Y=y)= P(X_j=x_j \mid Y=y), \; \forall x_j,x_k,y$}

or equivalently,

{$ \begin{align*} P(X_j=x_j,X_k=x_k\mid Y=y) & = \\ & P(X_j=x_j\mid Y=y)P(X_k=x_k\mid Y=y),\forall x_j,x_k,y \end{align*} $}

More generally, the Naive Bayes assumption is that:

{$\hat{P}(\mathbf{X}\mid Y) = \prod_j \hat{P}(X_j\mid Y)$}

Hence the Naive Bayes classifier is simply:

{$\arg\max_y \hat{P}(Y=y \mid \mathbf{X}) = \arg\max_y \hat{P}(Y=y)\prod_j \hat{P}(X_j\mid Y=y)$}

If the feature {$X_j$} is discrete like Raining, then we need to estimate K distributions for it, one for each class, {$P(X_j | Y = k)$}. We have 4 parameters, ({$\theta_{{\rm R}\mid\rm walk}, \theta_{{\rm R}\mid\rm bike}, \theta_{{\rm R}\mid\rm bus}, \theta_{{\rm R}\mid\rm car}$}), denoting probability of Raining=yes given transportation taken. The MLE estimate (deriving it is also a good exercise) is {$\hat{\theta}_{{\rm R}\mid y} = \frac{\sum_i \textbf{1}(R=yes, y=y_i)}{\sum_i \textbf{1}(y=y_i)}$}. For example, {$P({\rm R}\mid Y)$} is

y	parameter {$\theta_{{\rm R}\mid y}$}	MLE {$\hat{\theta}_{R \mid y}$}
walk	{$\theta_{{\rm R}\mid\rm walk}$}	1
bike	{$\theta_{{\rm R}\mid\rm bike}$}	0.5
bus	{$\theta_{{\rm R}\mid\rm bus}$}	0.5
car	{$\theta_{{\rm R}\mid\rm car}$}	0

For a continuous variable like Distance, there are many possible choices of models, with Gaussian being the simplest. We need to estimate K distributions for each feature, one for each class, {$P(X_j | Y = k)$}. For example, {$P({\rm D}\mid Y)$} is

y	parameters {$\mu_{{\rm D}\mid y}$} and {$\sigma_{{\rm D}\mid y}$}	MLE {$\hat{\mu}_{{\rm D}\mid y}$}	MLE {$\hat{\sigma}_{{\rm D}\mid y}$}
walk	{$\mu_{{\rm D}\mid\rm walk}$},{$\sigma_{{\rm D}\mid\rm walk}$}	1.5	0.5
bike	{$\mu_{{\rm D}\mid\rm bike}$}, {$\sigma_{{\rm D}\mid\rm bike}$}	4	3.7
bus	{$\mu_{{\rm D}\mid\rm bus}$}, {$\sigma_{{\rm D}\mid\rm bus}$}	1.5	0.5
car	{$\mu_{{\rm D}\mid\rm car}$}, {$\sigma_{{\rm D}\mid\rm car}$}	5.5	4.5

Note that for words, instead of using a binary variable (word present or absent), we could also use the number of occurrences of the word in the document, for example by assuming a binomial distribution for each word.

MLE vs. MAP

Note the danger of using MLE estimates. For example, consider the estimate of conditional distribution of Raining=yes: {$\hat{P}({\rm Raining = yes} \mid y={\rm car}) = 0$}. So if we know it's raining, no matter the distance, the probability of taking the car is 0, which is not a good estimate. This is a general problem due to scarcity of data: we never saw an example with car and raining. Using MAP estimation with Beta priors (with {$\alpha,\beta>1$}), estimates will never be zero, since additional "counts" are added.

Examples

2-dimensional, 2-class example

Suppose our data is from two classes (plus and circle) in two dimensions ({$x_1$} and {$x_2$}) and looks like this:

The Naive Bayes classifier will estimate a Gaussian for each class and each dimension. We can visualize the estimated distribution by drawing a contour of the density. The decision boundary, where the probability of each class given the input is equal, is shown in red.

Text classification: bag-of-words representation

In classifying text documents, like news articles or emails or web pages, the input is a very complex, structured object. Fortunately, for simple tasks like deciding about spam vs. not spam, politics vs sports, etc., a very simple representation of the input suffices. The standard way to represent a document is to completely disregard the order of the words in it and just consider their counts. So the email below might be represented as:

The Naive Bayes classifier then learns {$\hat{P}(spam)$}, and {$\hat{P}(word\mid spam)$} and {$\hat{P}(word \mid ham)$} for each word in our dictionary by using MLE/MAP as above. It then predicts prediction 'spam' if:

{$\hat{P}(spam)\prod_{{\rm word} \in {\rm email}} \hat{P}(word\mid spam) > \hat{P}(ham)\prod_{{\rm word} \in {\rm email}} \hat{P}(word\mid ham)$}

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