In our previous Machine Learning From Scratch series, we have covered Linear Regression and Logistic Regression. Today, we will be covering all details about Naive Bayes Algorithm from scratch.
Naive Bayes is a classification algorithm based on the “Bayes Theorem”. So let’s get introduced to the Bayes Theorem first.
Bayes Theorem is used to find the probability of an event occurring given the probability of another event that has already occurred. Here B is the evidence and A is the hypothesis. Here P(A) is known as prior, P(A/B) is posterior, and P(B/A) is the likelihood.
NAIVE BAYES ALGORITHM
The name Naive is used because the presence of one independent feature doesn’t affect (influence or change the value of) other features. The most important assumption that Naive Bayes makes is that all the features are independent of each other. Being less prone to overfitting, Naive Bayes algorithm works on Bayes theorem to predict unknown data sets.
We are given a table that contains a dataset about age, income, student, credit-rating, buying a computer, and their respective features. From the above dataset, we need to find whether a youth student with medium income having a fair credit rating buys a computer or not.
I.e. B = (Youth, Medium, Yes, Fair)
In the above dataset, we can apply the Bayesian theorem.
- A = ( Yes / No ) under buying computer
- B = ( Youth, Medium, Student, Fair)
So, P(A/B) means the probability of buying a computer given that conditions are “Youth age”, “Medium Income”, “Student”, and “fair credit-rating”.
Before starting, we assume that all the given features are independent of each other.
Step 1: Calculate probabilities of buying a computer from above dataset
Step 2: Calculate probabilities under Credit-rating buying a computer from the above dataset
Let’s understand how we calculated the above probabilities. From the table we can see that there are 8 fair credit ratings among which 6 buy computers and 2 don’t buy. Similarly, 6 have excellent credit ratings among which 3 buy computers and 3 don’t. As a whole 9 (6+3) buy computers and 5 (2+5) don’t.
P(fair / Yes) means the probability of credit rating being fair when someone buys a computer. Hence, P(fair / Yes) = P( fair buying computer) / P ( total number of buying computer) i.e. 6/9.
Step 3: Calculate probabilities under Student buying a computer from the above dataset
Step 4: Calculate probabilities under Income level buying a computer from the above dataset
P( High / Yes ) = 2/9 P( Mid / Yes ) = 4/9 P( Low / Yes ) = 3/9
P( High / No ) = 2/5 P( Mid / No ) = 2/5 P( Low / No ) = 1/5
Step 5: Calculate probabilities under age level buying a computer from the above dataset
P( Youth / Yes ) = 2/9 P( Mid / Yes ) = 4/9 P( Senior / Yes ) = 3/9
P( Youth / No ) = 3/5 P( Mid / No ) = 0 P( Senior / No ) = ⅖
B = ( Youth, Medium, Student, Fair)
P(Yes) * P(B / Yes ) = P(Yes) * P( Youth / Yes ) * P( Mid / Yes) * P( S Yes / Yes) * P( Fair / Yes)
= 9/14 * 2/9 * 4/9 * 6/9 * 6/9
P(No) * P(B / No ) = P(No) * P( Youth / No ) * P( Mid / No) * P( S Yes / No) * P( Fair / No)
= 5/14 * 3/5 * 2/5 * 1/5 * 2/5
P( B ) = P (Youth) * P( Mid) * P(Student Yes) * P( Fair)
= 5/14 * 6/14 * 7/14 * 8/14
P(Yes / B) = P(Yes) * P(B / Yes ) / P(B)
= 0.02821 / 0.04373
P(No / B) = P(No) * P(B / No ) / P(B)
= 0.0068 / 0.04373
Here, P(Yes / B) is greater than P(No / B) i.e posterior Yes is greater than posterior No. So the class B ( Youth, Mid, yes, fair) buys a computer.
NAIVE BAYES FROM SCRATCH
Classify whether a given person is a male or a female based on the measured features. The features include height, weight, and foot size.
Now, defining a dataframe which consists if above provided data.
Creating another data frame containing the feature value of height as 6 feet, weight as 130 lbs and foot size as 8 inches. using Naive Bayes from scratch we are trying to find whether the gender is male or female.
Calculating the total number of males and females and their probabilities i.e priors:
Calculating mean and variance of male and female of the feature height, weight and foot size.
- posterior (male) = P(male)*P(height|male)*P(weight|male)*P(foot size|male) / evidence
- posterior (female) = P(female)*P(height|female)*P(weight|female)*P(foot size|female) / evidence
- Evidence = P(male)*P(height|male)*P(weight|male)*P(foot size|male) + P(female) * P(height|female) * P(weight|female)*P(foot size|female)
The evidence may be ignored since it is a positive constant. (Normal distributions are always positive.)
Calculation of P(height | Male )
- mean of the height of male = 5.855
- variance ( Square of S.D.) of the height of a male is square of 3.5033e-02
- and x i.e. given height is 6 feet
- Substituting the values in the above equation we get P(height | Male ) = 1.5789
- P(weight|male) = 5.9881e-06
- P(foot size|male) = 1.3112e-3
- P(height|female) = 2.2346e-1
- P(weight|female) = 1.6789e-2
- P(foot size|female) = 2.8669e-1
Posterior (male)*evidence = P(male)*P(height|male)*P(weight|male)*P(foot size|male) = 6.1984e-09
Posterior (female)*evidence = P(female)*P(height|female)*P(weight|female)*P(foot size|female)= 5.3778e-04
Since Posterior (female)*evidence > Posterior (male)*evidence, the sample is female.
NAIVE BAYES USING SCIKIT-LEARN
Though we have very small dataset, we are dividing the dataset into train and test do that it can be used in other model prediction. We are importing gnb() from sklearn and we are training the model with out dataset.
Now, our model is ready. Let’s use this model to predict on new data.
We have come to an end of Naive Bayes from Scratch. If you have any queries, feedback, or suggestions then you can leave a comment or mail us on email@example.com. We will see you in the next tutorial. Stay safe !! Happy Coding !!!
24 Comments on “NAIVE BAYES ALGORITHM FROM SCRATCH”
Merely wanna remark that you have a very decent web site, I love the design it really stands out.
Look forward to exploring your web page for a second time.
I do agree with all the ideas you have presented in your post.
Hello! I just wish to give a huge thumbs up for the good info you have right here on this post. I will be coming back to your weblog for more soon.
Does your website have a contact page? I’m having problems locating it but, I’d like to send you an email. I’ve got some creative ideas for your blog you might be interested in hearing. Either way, great blog and I look forward to seeing it develop over time.
Wow! This can be one particular of the most beneficial blogs We’ve ever arrive across on this subject. Basically Excellent. I am also a specialist in this topic therefore I can understand your effort.
Thank you for sharing excellent informations.
Woah! I’m really enjoying Exceptional Blog!
Nice blog here!
Informative Site Hello guys here are some links that contains information that you may find useful yourselves. Its Worth Checking out.
Keep working ,splendid job!
It’s really a cool and useful piece of info. I’m glad that you shared this helpful info with us. Please keep us up to date like this. Thanks for sharing.
This really answered my downside, thank you!
In step 3, you have 7/9 as P(S Yes/Yes). I think this is wrong and should be 6/9. Unfortunately this means all the results you calculate at the end of the example need to be fixed.
Please let me know if you agree or if I am missing something.
yeah, that was a typing mistake and now it’s corrected. The calculation part is correct. Thank you for pointing it out.
The condition associated with the person seeking medical care is theimportant factor in your choice associated with a long-term care service.
I will right away clutch your rss as I can’t to find your email subscription hyperlink or e-newsletter service.
Do you have any? Kindly allow me recognise so that I may just subscribe.
fantastic submit, very informative. I’m wondering why the other experts of this sector do not notice
this. You must proceed your writing. I am confident, you’ve a huge readers’
Hey there! Someone in my Myspace group shared this site with us so I
came to give it a look. I’m definitely enjoying the information. I’m book-marking and will be tweeting this to my followers!
Wonderful blog and outstanding design. adreamoftrains best
web hosting company