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Probability Learning: Naive Bayes


This post will describe various simplifications of Bayes' Theorem, that make it more practical and applicable to real world problems: these simplifications are known by the name of Naive Bayes. Also, to clarify everything we will see a very illustrative example of how Naive Bayes can be applied for classification.



By Jaime Zornoza, Universidad Politecnica de Madrid

Figure

 

Hello dear readers. This is post number five of the Probability Learning series. The previous posts are:

I deeply encourage you to read them, as they are fun and full of useful information about probabilistic Machine Learning.

In the previous post, we covered the math Behind Bayes theorem for Machine Learning. This post will describe various simplifications of this theorem, that make it more practical and applicable to real world problems: these simplifications are known by the name of Naive Bayes. Also, to clarify everything we will see a very illustrative example of how Naive Bayes can be applied for classification.

 

Why don’t we always use Bayes?

 
As mentioned in the previous posts, Bayes’ theorem tells us how to gradually update our knowledge on something as we get more evidence or that about that something.

We saw that in Machine Learning this is reflected by updating certain parameter distributions in the evidence of new data. We also saw how Bayes theorem can be used for classification by calculating the probability of a new data point belonging to a certain class and assigning this new point to the class that reports the highest probability. We mentioned that the strength of this was the ability to incorporate previous or prior knowledge into our models.

Let's recover the most basic Bayes’ formula for a second:

Figure

General version of Bayes Formula

 

This formula can be customised to calculate the probability of a data point xbelonging to a certain class ci, like so:

Figure

Bayes formula particularised for class i and the data point x

 

Approaches like this can be used for classification: we calculate the probability of a data point belonging to every possible class and then assign this new point to the class that yields the highest probability. This could be used for both binary and multi-class classification.

The problem for this application of Bayes Theorem comes when we have models with data points that have more than one feature: calculating the likelihood term P(x|ci) is not straightforward. This term accounts for the probability of a data point (represented by its features), given a certain class. This conditional probability calculation, if the features are related in-between them, can be very computationally heavy. Also, if there are a lot of features and we have to calculate the joint probability of all the features, the computation can be quite extensive too.

This is why we don’t always use Bayes, but sometimes have to resort to simpler alternatives.

 

So what is Naive Bayes then?

 
Naive Bayes is a simplification of Bayes’ theorem which is used as a classification algorithm for binary of multi-class problems. It is called naive because it makes a very important but somehow unreal assumption: that all the features of the data points are independent of each other. By doing this it largely simplifies the calculations needed for Bayes’ classification, while maintaining pretty decent results. These kinds of algorithms are often used as a baseline for classification problems.

Let's see an example to clarify what this meansand the differences with Bayes: Imagine you like to go for a short walk every morning in the park next to your house. After doing this for a while, you start to meet a very wise old man, which some days takes the same walk as you do. When you meet him, he explains data science concepts to you in the simplest of terms, breaking down complex matters with elegance and clarity.

Some days, however, you go out for a walk all excited to hear more from the old man, and he is not there. Those days you wish you had never left your home, and feel a bit sad.

Figure

One of your lovely walks in the park

 

To solve this problem, you go on walks every day for a week, and write down the weather conditions for each day, and if the old man was out walking or not. The next table represents the information you gathered. The “Walk” column refers to whether or not the old man went for a walk in the park.

Figure

Table with the information collected during one week

 

Using this information, and something this data science expert once mentioned, the Naive Bayes classification algorithm, you will calculate the probability of the old man going out for a walk every day depending on the weather conditions of that day, and then decide if you think this probability is high enough for you to go out to try to meet this wise genius.

If we model every one of our categorical variables as a 0 when the value of the field is “no” and a 1 when the value of the field is “yes”, the first row of our table, for example, would be:

111001 | 0

where the 0 after the vertical bar indicates the target label.

If we used the normal Bayes algorithm to calculate the posterior probability for each class (walk or not walk) for every possible weather scenario, we would have to calculate the probability of every possible combination of 0s and 1s for each class. In this case, we would have to calculate the probability of two to the power of 6 possible combinations for each class, as we have 6 variables. The general reasoning is the following:

This has various problems: first, we would need a lot of data to be able to calculate the probabilities for every scenario. Then, if we had this data available, the calculations would take considerably longer than in other kinds of approaches, and this time would greatly increase with the number of variables or features. Lastly, if we thought some of these variables were related (like being sunny with the temperature for example), we would have to take this relationship into account when calculating the probabilities, which would lead to a longer calculation time.

How does Naive Bayes fix all this? By assuming each feature variable is independent of the rest: this means we just have to calculate the probability of each separate feature given each class, reducing the needed calculations from 2^n to 2n. Also, it means we don’t care about the possible relationships between our variables, like the sun and the temperature.

Let's describe it step by step so you can all see more clearly what I’m talking about:

1. First, we calculate the prior probabilities of each class, using the table shown above.

Figure

Prior probabilities for each class

 

2. Then, for each feature, we calculate the probabilities of the different categorical values given each class (In our example we only have “yes” and “no” as the possible values for each feature, but this could be different depending on the data). The following example shows this for the feature “Sun”. We would have to do this for each feature.

Figure

Probabilities of sun values given each class

 

3. Now, when we get a new data point as a set of meteorological conditions, we can calculate the probability of each class by multiplying the individual probabilities of each feature given that class and the prior probabilities of each class. Then, we would assign this new data point to the class that yields the highest probability.

Let's see an example. Imagine we observe the following weather conditions.

Figure

New data point

 

First, we would calculate the probability of the old man walking, given these conditions.

Figure

Probabilities needed to calculate the chance of the old man walking

 

If we do the product of all this, we get 0.0217. Now let's do the same but for the other target class: not walking.

Figure

Probabilities needed to calculate the chance of the old man not walking

 

Again, if we do the product, we get 0.00027. Now, if we compare both probabilities (the man walking and the man not walking), we get a higher probability for the option where the man walks, so we put on some trainers, grab a coat just in case (there are clouds) and head out to the park.

Notice how in this example we didn’t have any probabilities equal to zero. This has to do with the concrete data point we observed, and the amount of data that we have. If any of the calculated probabilities were zero, then the whole product would be null, which is not very realistic. To avoid these, techniques by the name of smoothing are used, but we will not cover them on this post.

That is it! Now, when we wake up and want to see the chance that we will find the old man taking a walk, all we have to do is look at the weather conditions and do a quick calculation like in the previous example!

 

Conclusion

 
We have seen how we can use some simplifications of Bayes Theorem for classification problems. In the next post, we will talk about the application of Naive Bayes for Natural Language Processing.

To check it out follow me on Medium, and stay tuned!

That is all, I hope you liked the post. Feel Free to connect with me on LinkedIn or follow me on Twitter at @jaimezorno. Also, you can take a look at my other posts on Data Science and Machine Learning here. Have a good read!

 

Additional Resources

 
In case you are hungry for more information, you can use the following resources:

and as always, contact me with any questions. Have a fantastic day and keep learning.

 
Bio: Jaime Zornoza is an Industrial Engineer with a bachelor specialized in Electronics and a Masters degree specialized in Computer Science.

Original. Reposted with permission.

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