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Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

Probability Theory Basics in Machine Learning

Sarvagya
Sarvagya Last Updated : 20 Nov, 2023
6 min read
This article was published as a part of the Data Science Blogathon.
Probability Theory Basics

Machine Learning is a very interesting branch of Artificial Intelligence where the machine is made to learn from the data with the help of models created by us, and then identify patterns and thus make predictions based on that. If you’re considering taking your future ahead in the field of Data Science, then one of the most important things you should be knowing is “Probability Theory‘. It plays a very crucial role in Data Science and you should be a master of this theory. Have you ever came across the word ‘Probability’? Don’t Worry! We’ll discuss this, and at the end of this lecture, you will be confident about this theory. We’ll discuss all the important things that one should know without wasting any time.

Table of contents

Introduction to Basics of Probability Theory

Probability simply talks about how likely is the event to occur, and its value always lies between 0 and 1 (inclusive of 0 and 1). For example: consider that you have two bags, named A and B, each containing 10 red balls and 10 black balls. If you randomly pick up the ball from any bag (without looking in the bag), you surely don’t know which ball you’re going to pick up. So here is the need of probability where we find how likely you’re going to pick up either a black or a red ball. Note that we’ll be denoting probability as P from now on. P(X) means the probability for an event X to occur.

P(Red ball)= P(Bag A). P(Red ball | Bag A) + P(Bag B). P(Red ball | Bag B), this equation finds the probability of the red ball. Here I have introduced the concept of conditional probability ( which finds probability when we’re provided with the condition). P(Bag A) = 1/2 because we’ve 2 bags out which we’ve to select Bag A. P(Red ball | Bag A) should read as “probability of drawing a red ball given the bag A” here “given” word specifies the condition which is Bag A in this case, so it is 10 red balls out of 20 balls i.e. 10/20. So let’s solve:

P(Red Ball)= 1/2. 10/20 + 1/2. 10/20 = 1/2

Similarly, you can try to find the probability of drawing a black ball? Also, find the probability of drawing two consecutive red balls from the bag after transferring one black ball from bag A to Bag B?

conditional probability

Now if you look at the image above,  you must be thinking what is it? I haven’t introduced the “intersection” in Set Theory. I have already discussed the concept above, there is nothing new in the image given above.  Here we’re finding the probability for an event A to occur given that event B has already occurred. The numerator of the right-hand side of the equation is the probability for both events to occur, divided by the probability for an event B to occur. The numerator has an inverted shape symbol between A and B which we call “Intersection” in set theory.

Basic concepts of probability theory

There are a few key concepts that are important to understand in probability theory. These include:

  • Sample space: The sample space is the collection of all potential outcomes of an experiment. For example, the sample space of flipping a coin is {heads, tails}.
  • Event: An event is a collection of outcomes within the sample space. For example, the event of flipping a head is {heads}.
  • Probability: The probability of an event is a number between 0 and 1 that represents the likelihood of the event occurring. A chance of 0 means that the event is impossible, and a probability of 1 means that the event is specific.

Introduction to Bayes Theorem 

Till now we’ve just discussed the basics of probability theory. Before we proceed let me discuss ‘Random Variable‘ which is a variable whose possible values are numerical outcomes of a random phenomenon. In the above case, Bag is a random variable that can take possible values as Bag A and Bag B. Ball is also a random variable that can take values red and black.

Now imagine the above situation when I say find the probability that ball is drawn from a bag A given that the ball is red in color? Notice that in this question we’re already given the color of the ball and we’ve to find the probability that the red ball is drawn is from bag A. Instead in other questions we used to find the probability of drawing a red ball from bag A. In the case where we’ve to find the probability of event and object is given, this kind of probability is called posterior probability. In the case where we’ve to find the probability of an object given for that event, this kind of probability is known as the prior probability. So the answer to my question asked at the beginning of the paragraph is:

P(Bag A | red ball)= [P(red ball | Bag A). P(Bag A) ] / [P(red ball | Bag A). P(Bag A) + P(red ball | Bag B). P(Bag B) ]

The above equation is what we call “Bayes Theorem”, one of the most significant and important theorems given by the Reverend Thomas Bayes.

Now, I want you to take a look at what we studied so far was about discrete events. In machine learning, we usually need ‘continuous’ events instead of a discrete variable. What we do now? Look at the image given below and try to understand between the two types of events that I just talked about!

Bayes Theorem 

What is Probability Density?

As I told you, we were considering the probabilities of discrete events but our requirement is for continuous events. To achieve this, I would like to introduce the concept of “Probability Density” here at this point. Let the probability of real-valued x within the interval (x, x+ dx) be given by the integral of p(x).dx where dx–> 0, then p(x) is probability density over x. The probability for x lies in the interval [a,b] is given by the integral from a to b of p(x).dx. This is shown in the image given below.

Probability Theory - Probability Density

If you look at the image, I have also specified the two conditions that the probability density should satisfy.

Great! You learned something which is very difficult in a much easier way. Isn’t it?

How does it relate to Machine learning??

So we learned the basics of probability theory, but still, confused about how we relate this to machine learning? Isn’t it? We use probability when we’ve to make predictions. When we have the model in ML and data, we can use it to make predictions based on the trained model. Consider a case where we’ve got a dataset for different temperatures over a region for different dates. Here we can make predictions on how many water bottles are required to be stored in that region with the help of a model.

I have tried to cover as much as I can in this article but there is a lot to learn in probability theory. It is just the basics. So far we just discussed the probability definition, conditional probability, Bayes Theorem, and Probability Density.  Good Luck!!

Conclusion

In summary, we’ve explored the foundational aspects of Probability Theory, including key concepts and an introduction to Bayes’ Theorem. The discussion on Probability Density shed light on its importance in understanding uncertainty. Notably, the relevance of machine learning emphasizes its crucial role in data modeling, enhancing our comprehension of probabilistic reasoning across diverse applications

FAQs

Q1. What is the importance of probability theory?

Probability theory is like a special math tool that helps us deal with things that might happen but aren’t for sure. It lets us figure out how likely different outcomes are, which is useful for making decisions, understanding data, and even building machines that learn from experience.

Q2. What is an example of a probability theory?

Probability theory is used in many areas, including flipping coins, rolling dice, predicting exam scores, estimating weather, modeling disease outbreaks, designing gambling games, assessing engineering risks, and conducting statistical sampling.

The media shown in this article are not owned by Analytics Vidhya and is used at the Author’s discretion. 

basics of probabilityblogathonprobabilityprobability theorem
Sarvagya
Sarvagya

Hi, I'm Sarvagya Agrawal, Software Engineer, with a strong passion for utilizing technology to drive positive change in society. I believe that technology is not just a skill, but an art form that can be leveraged to transform the world.
My primary focus lies in machine learning and web development, with strong programming skills in Python. I have worked on innovative projects, including developing an AI model to calculate cardiovascular risk factors from OCTA scans using computer vision algorithms and creating an AI-based web application for calculating financial risk based on an individual's spending trends.

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