Interview Prep
Career
GenAI
Prompt Engg
ChatGPT
LLM
Langchain
RAG
AI Agents
Machine Learning
Deep Learning
GenAI Tools
LLMOps
Python
NLP
SQL
AIML Projects

Reading list

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

An Intuitive Introduction to Bayesian Decision Theory

Chirag Goyal
Chirag Goyal Last Updated : 24 May, 2021
6 min read

This article was published as a part of the Data Science Blogathon

Introduction

Bayesian decision theory refers to the statistical approach based on tradeoff quantification among various classification decisions based on the concept of Probability(Bayes Theorem) and the costs associated with the decision.

It is basically a classification technique that involves the use of the Bayes Theorem which is used to find the conditional probabilities.

In Statistical pattern Recognition, we will focus on the statistical properties of patterns that are generally expressed in probability densities (pdf’s and pmf’s), and this will command most of our attention in this article and try to develop the fundamentals of the Bayesian decision theory.

Prerequisites 

Random Variable

A random variable is a function that maps a possible set of outcomes to some values like while tossing a coin and getting head H as 1 and Tail T as 0 where 0 and 1 are random variables.

Bayes Theorem

The conditional probability of A given B, represented by P(A | B) is the chance of occurrence of A given that B has occurred.

P(A | B) = P(A,B)/P(B) or

By Using the Chain rule, this can also be written as:

P(A,B) = P(A|B)P(B)=P(B|A)P(A)

P(A | B) = P(B|A)P(A)/P(B)    ——-  (1)

Where, P(B) = P(B,A) + P(B,A’) = P(B|A)P(A) + P(B|A’)P(A’)

Here, equation (1) is known as the Bayes Theorem of probability

Our aim is to explore each of the components included in this theorem. Let’s explore step by step:

(a) Prior or State of Nature:

  • Prior probabilities represent how likely is each Class is going to occur.
  • Priors are known before the training process.
  • The state of nature is a random variable P(wi).
  • If there are only two classes, then the sum of the priors is P(w1) + P(w2)=1, if the classes are exhaustive.

(b) Class Conditional Probabilities:

  • It represents the probability of how likely a feature x occurs given that it belongs to the particular class. It is denoted by,   P(X|A) where x is a particular feature
  • It is the probability of how likely the feature x occurs given that it belongs to the class wi.
  • Sometimes, it is also known as the Likelihood.
  • It is the quantity that we have to evaluate while training the data. During the training process, we have input(features) X labeled to corresponding class w and we figure out the likelihood of occurrence of that set of features given the class label.

(c) Evidence:

  • It is the probability of occurrence of a particular feature i.e. P(X).
  • It can be calculated using the chain rule as,  P(X) = Σin P(X | wi) P(wi)
  • As we need the likelihood of class conditional probability is also figure out evidence values during training.

(d) Posterior Probabilities:

  • It is the probability of occurrence of Class A when certain Features are given
  • It is what we aim at computing in the test phase in which we have testing input or features (the given entity) and have to find how likely trained model can predict features belonging to the particular class wi.

 

For a better understanding of the above theory, we consider an example

Problem Description

Suppose we have a classification problem statement where we have to classify among the object-1 and object-2 with the given set of features X = [x1, x2, …, xn]T.

 Objective

The main objective of designing a such classifier is to suggest actions when presented with unseen features, i.e, object not yet seen i.e, not in training data.

In this example let w denotes the state of nature with w = w1 for object-1 and w = w2 for object-2. Here, we need to know that in reality, the state of nature is so unpredictable that we generally consider that was variable that is described probabilistically.

 Priors

  • Generally, we assume that there is some prior value P(w1) that the next object is object-1 and P(w2) that the next object is object-2. If we have no other object as in this problem then the sum of their prior is 1 i.e. the priors are exhaustive.
  • The prior probabilities reflect the prior knowledge of how likely we will get object-1 and object-2. It is domain-dependent as the prior may change based on the time of year they are being caught.

It sounds somewhat strange and when judging multiple objects (as in a more realistic scenario) makes this decision rule stupid as we always make the same decision based on the largest prior even though we know that any other type of objective also might appear governed by the leftover prior probabilities (as priors are exhaustive in nature).

Consider the following different scenarios:

  • If P(ω1)>>> P(ω2), our decision in favor of ω1 will be correct most of the time we predict.
  • But if P(ω1)= P(ω2), half probable of our prediction of being right. In general, the probability of error is the minimum of P(ω1) and P(ω2), and later in this article, we will see that under these conditions no other decision rule can yield a larger probability of being correct.

Feature Extraction process (Extract feature from the images)

A suggested set of features- Length, width, shapes of an object, etc.

In our example, we use the width x, which is more discriminatory to improve the decision rule of our classifier. The different objects will yield different variable-width readings and we usually see this variability in probabilistic terms and also we consider x to be a continuous random variable whose distribution depends on the type of object wj, and is expressed as p(x|ωj) (probability distribution function pdf as a continuous variable) and known as the class-conditional probability density function. Therefore,

The pdf p(x|ω1) is the probability density function for feature x given that the state of nature is ω1 and the same interpretation for p(x|w2).

Introduction to Bayesian Decision Theory | by Rayhaan Rasheed | Towards Data Science

Fig. Picture Showing pdf for both classes

Image Source: Google Images

Suppose that we are well aware of both the prior probabilities P(ωj) and the conditional densities p(x|ωj). Now, we can arrive at the Bayes formula for finding posterior probabilities:

Naïve Bayes Algorithm: Everything you need to know - KDnuggets

Fig. Formula of Bayes Theorem

Image Source: Google Images

Bayes’ formula gives us intuition that by observing the measurement of x we can convert the prior P(ωj) to the posteriors, denoted by P(ωj|x) which is the probability of ωj given that feature value x has been measured.

p(x|ωj) is known as the likelihood of ωj with respect to x.

The evidence factor, p(x), works as merely a scale factor that guarantees that the posterior probabilities sum up to one for all the classes.

Bayes’ Decision Rule

The decision rule given the posterior probabilities is as follows

If P(w1|x) > P(w2|x) we would decide that the object belongs to class w1, or else class w2.

Probability of Error

To justify our decision we look at the probability of error, whenever we observe x, we have,

P(error|x)= P(w1|x) if we decide w2, and P(w2|x) if we decide w1

As they are exhaustive and if we choose the correct nature of an object by probability P then the leftover probability (1-P) will show how probable is the decision that it the not the decided object.

We can minimize the probability of error by deciding the one which has a greater posterior and the rest as the probability of error will be minimum as possible. So we finally get,

P(error|x) = min [P(ω1|x),P(ω2|x)]

And our Bayes decision rule as,

Decide ω1 if P(ω1|x) >P(ω2|x); otherwise decide ω2

This type of decision rule highlights the role of the posterior probabilities. With the help Bayes theorem, we can express the rule in terms of conditional and prior probabilities.

The evidence is unimportant as far as the decision is concerned. As we discussed earlier it is working as just a scale factor that states how frequently we will measure the feature with value x; it assures P(ω1|x)+ P(ω2|x) = 1.

So by eliminating the unrequired scale factor in our decision rule we have, the similar decision rule by Bayes theorem as,

Decide ω1 if p(x|ω1)P(ω1) >p(x|ω2)P(ω2); otherwise decide ω2

Now, let’s consider 2 cases:

  • Case-1: If class conditionals are equal i.e, p(x|ω1)= p(x|ω2), then we arrive at our premature decision rule governed by just priors.
  • Case-2: On the other hand, if priors are equal i.e, P(ω1)= P(ω2) then the decision is entirely based on class conditionals p(x|ωj).

This completes our example formulation!

 

Generalization of the preceding ideas for Multiple Features and Classes

Bayes classification: Posterior, likelihood, prior, and evidence

P(wi | X)= P(X | wi) P(wi) / P(X)

Posterior = Likelihood* Prior/Evidence

We now discuss those cases which have multiple features as well as multiple classes,

Let the Multiple Features be X1, X2, … Xn and Multiple Classes be w1, w2, … wn, then:

P(wi | X1, …. Xn) = P(X1,…. , Xn|wi)*P(wi)/P(X1,… Xn)

Where, 

Posterior = P(wi | X1, …. Xn)

Likelihood = P(X1,…. , Xn|wi)

Prior = P(wi)

Evidence = P(X1,… ,Xn)

In cases of the same incoming patterns, we might need to use a drastically different cost function, which will lead to different actions altogether. Generally, different decision tasks may require features and yield boundaries quite different from those useful for our original categorization problem.

So, In the later articles, we will discuss the Cost function, Risk Analysis, and decisive action which will further help to understand the Bayes decision theory in a better way.

End Notes

Thanks for reading!

If you liked this and want to know more, go visit my other articles on Data Science and Machine Learning by clicking on the Link

Please feel free to contact me on Linkedin, Email.

Something not mentioned or want to share your thoughts? Feel free to comment below And I’ll get back to you.

About the author

Chirag Goyal

Currently, I am pursuing my Bachelor of Technology (B.Tech) in Computer Science and Engineering from the Indian Institute of Technology Jodhpur(IITJ). I am very enthusiastic about Machine learning, Deep Learning, and Artificial Intelligence.

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

 

Chirag Goyal
Chirag Goyal

I am a B.Tech. student (Computer Science major) currently in the pre-final year of my undergrad. My interest lies in the field of Data Science and Machine Learning. I have been pursuing this interest and am eager to work more in these directions. I feel proud to share that I am one of the best students in my class who has a desire to learn many new things in my field.

Free Courses

Generative AI 4.7

Generative AI - A Way of Life

Explore Generative AI for beginners: create text and images, use top AI tools, learn practical skills, and ethics.
Large Language Models 4.5

Getting Started with Large Language Models

Master Large Language Models (LLMs) with this course, offering clear guidance in NLP and model training made simple.
Prompt Engineering 4.6

Building LLM Applications using Prompt Engineering

This free course guides you on building LLM apps, mastering prompt engineering, and developing chatbots with enterprise data.
LlamaIndex 4.8

Improving Real World RAG Systems: Key Challenges & Practical Solutions

Explore practical solutions, advanced retrieval strategies, and agentic RAG systems to improve context, relevance, and accuracy in AI-driven applications.
RAG systems using LlamaIndex 4.7

Microsoft Excel: Formulas & Functions

Master MS Excel for data analysis with key formulas, functions, and LookUp tools in this comprehensive course.

Recommended Articles

Responses From Readers

Clear

Rajveer Shekhawat
Rajveer Shekhawat

Good attempt Chirag to explain quite a complex topic. It would make more sense if you include more examples but different applications. Also providing some links to next level of learning (text books, reference books) would be more help. Cheers. Rajveer

123

Write for us

Write, captivate, and earn accolades and rewards for your work
  • Reach a Global Audience
  • Get Expert Feedback
  • Build Your Brand & Audience
  • Cash In on Your Knowledge
  • Join a Thriving Community
  • Level Up Your Data Science Game
imag

We use cookies essential for this site to function well. Please click to help us improve its usefulness with additional cookies. Learn about our use of cookies in our Privacy Policy & Cookies Policy.

Show details