Bayesian Learning

Summary

The lecture introduces Bayesian probability and its applications. Key concepts like Bayes' Theorem, hypothesis testing, and probability calculation are explored using a cancer diagnosis example. The lecture delves into Bayesian learning, maximum likelihood hypothesis, and Gaussian distribution, demonstrating their role in finding optimal classifiers. Topics like linear regression, MAP hypothesis, and Gibbs sampling are also discussed, concluding with a focus on Bayes optimal classification and its importance. The session emphasizes understanding Bayesian methods for optimal data classification and decision-making.

Takeaways

• 🔍 Introduction to Bayesian probability, explaining its importance and applications.
• 📊 Detailed explanation of Bayes' Theorem, including its role in updating probabilities based on new evidence.
• 🧠 Use of a medical example to explain probability calculations, such as determining cancer risk based on test results.
• 🧮 Introduction to MAP (Maximum A Posteriori) Hypothesis and its application in Bayesian Learning.
• 🔢 Discussion on calculating posterior probabilities using given values and their implications.
• 📉 Explanation of linear regression and its use in predicting target functions.
• 🎯 Introduction to Maximum Likelihood Hypothesis and Gaussian distribution for hypothesis selection.
• 🧩 Explanation of Bayes Optimal Classifier and its use in classification problems.
• 🧪 Introduction to Gibbs Sampling as a method for hypothesis selection and optimization.
• 🔄 Emphasis on the process of hypothesis testing and validation using training and test data.

Q & A

What is Bayesian probability?

Bayesian probability is a method of probability interpretation that updates the probability estimate as new evidence or information becomes available.

What is Bayes' theorem?

Bayes' theorem describes how to update the probabilities of hypotheses when given evidence. It calculates the probability of a hypothesis based on prior knowledge and the likelihood of current evidence.

How does Bayes' theorem apply to medical testing for cancer?

In the context of medical testing, Bayes' theorem can help calculate the probability that a patient has cancer, considering the prior probability (the general population’s cancer rate) and the test’s accuracy (sensitivity and specificity).

What is Maximum A Posteriori (MAP) hypothesis?

MAP hypothesis refers to the hypothesis that maximizes the posterior probability, which is the likelihood of the hypothesis given the data and prior information.

What is the significance of the likelihood function in Bayesian analysis?

The likelihood function measures how probable the observed data is under various hypotheses. It plays a critical role in updating beliefs about hypotheses based on the data.

What is the concept of Least Squares in linear regression?

Least Squares is a method used to minimize the differences between observed data and the predicted values in linear regression by finding the line that best fits the data points.

What is the relationship between Gaussian distribution and Maximum Likelihood Hypothesis?

When dealing with Gaussian distributions, the Maximum Likelihood Hypothesis is used to maximize the likelihood of data fitting a Gaussian model. It helps in estimating the parameters of the Gaussian distribution.

What is the role of Gibbs sampling in Bayesian classification?

Gibbs sampling is used to approximate the posterior distribution in Bayesian classification. It generates samples from the distribution and allows for estimation when direct calculation is complex.

What is the Bayes Optimal Classifier?

The Bayes Optimal Classifier is a classification model that makes predictions based on the posterior probabilities of all possible hypotheses. It is considered optimal as it minimizes the expected loss.

Why is the Bayes Optimal Classifier considered optimal?

It is considered optimal because it uses the full posterior distribution of hypotheses, making it the most accurate classifier by minimizing classification errors in the long run.

Outlines

00:00

🔍 Introduction to Bayesian Probability and Bayes Theorem

This paragraph introduces the concept of Bayesian probability and Bayes' theorem. It begins with a welcome to the lecture and proceeds to explain how probabilities are applied, particularly in medical cases such as cancer diagnosis. The idea is to determine probabilities based on given data, using Bayes' theorem to calculate the likelihood of events, like a positive test result for cancer, within a population. The focus is on applying Bayes' theorem to real-world problems.

05:04

📈 Linear Regression and Least Squares Line

The second paragraph delves into linear regression and the concept of the least squares line. It discusses the process of learning and estimating functions, with reference to target functions and noise components (epsilon). The text explains how to apply linear regression models and introduces maximum likelihood hypothesis, which is used to estimate the most probable model based on the data.

11:14

🧮 Gaussian Distributions and Function Optimization

This section focuses on Gaussian distributions and how they relate to optimization problems. It highlights the use of argmax (argument of the maximum) in maximizing certain functions. The paragraph explains how parts of the equation remain constant while others contribute to maximizing the outcome. It also mentions studying functions to optimize them for specific events.

16:32

📊 Applying Training Data to Test Examples

Here, the paragraph talks about the application of training data in machine learning to test examples. It emphasizes that while training data is essential, it doesn't necessarily guarantee a direct answer for every test case. The concept of hypothesis spaces (h1, h2, h3) is introduced, and the focus shifts to Bayesian optimal classification, which helps improve predictions using new data.

22:02

⚖️ Bayesian Optimal Classifier and Gibbs Sampling

The fifth paragraph explains the Bayesian optimal classifier and its importance in classification tasks. It touches on how the classifier works optimally by incorporating probabilities. The discussion extends to the use of Gibbs sampling for hypothesis selection and classification, noting that the Gibbs classifier provides a more efficient method by reducing computational effort.

27:24

🎓 Conclusion and Closing Remarks

The final paragraph brings the lecture to a close by summarizing the key points covered. It reinforces the importance of the discussed topics, including Bayesian learning and classifiers. The lecture ends with a thank you message, marking the conclusion of the session.

Keywords

💡Bayesian Probability

Bayesian Probability is a method of statistical inference in which probabilities are updated based on new evidence. In the video, it is introduced as a core concept used for making predictions based on prior knowledge. For instance, the video discusses using Bayesian methods to assess the probability of a patient having cancer based on given data.

💡Bayes' Theorem

Bayes' Theorem is a mathematical formula used to calculate conditional probabilities. It helps in updating the probability estimate for a hypothesis as more evidence or information becomes available. In the video, Bayes' Theorem is applied to calculate the likelihood of a cancer diagnosis based on test results and the prevalence of cancer in the population.

💡Hypothesis

A hypothesis is a proposed explanation for a phenomenon, which can be tested to determine its validity. The video frequently refers to hypotheses, particularly in the context of Bayesian learning, where various hypotheses are tested to find the one with the highest likelihood based on the available data.

💡Maximum Likelihood Hypothesis

The Maximum Likelihood Hypothesis is the hypothesis that has the highest probability of being true, given the data. In the video, this concept is illustrated through the process of identifying the most probable hypothesis in a Gaussian distribution and maximizing it to find the 'arg max' function.

💡Gaussian Distribution

A Gaussian distribution, also known as a normal distribution, is a bell-shaped curve that describes how values are distributed around a mean. In the video, Gaussian distribution is used to model certain data, and it plays a role in calculating the Maximum Likelihood Hypothesis.

💡Training Data

Training data refers to the dataset used to train models in machine learning, allowing the model to make predictions based on learned patterns. The video discusses how training data is applied to specific hypotheses and how it informs the Bayesian learning process.

💡Bayes Optimal Classifier

The Bayes Optimal Classifier is a probabilistic model that makes predictions by averaging over all possible hypotheses, weighted by their posterior probabilities. In the video, this concept is introduced to explain the classification process using Bayesian methods, showing how the optimal classification is derived from multiple hypotheses.

💡MAP Hypothesis

The MAP (Maximum A Posteriori) Hypothesis is the most probable hypothesis after taking into account both prior probability and likelihood. It differs from the Maximum Likelihood Hypothesis by also considering the prior knowledge. The video touches on MAP learning when discussing how hypotheses are chosen in Bayesian inference.

💡Gibbs Sampling

Gibbs Sampling is a technique for approximating complex distributions by generating samples from the joint probability distribution of the variables. The video briefly mentions using Gibbs Sampling in the classification process to reduce the number of hypotheses and focus on the most probable ones.

💡Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In the video, linear regression is referred to as part of a discussion on learning algorithms and how target functions are generated using training data.

Highlights

Introduction to Bayesian probability and its application in analyzing data.

Overview of Bayes' theorem and its use in calculating probabilities.

Example of applying Bayes' theorem to medical diagnosis, particularly cancer probability.

Discussion on how to use given values in problems based on Bayes' theorem.

Introduction to MAP (Maximum A Posteriori) hypothesis and its application.

Explanation of how to calculate probabilities using specific events in a hypothesis.

Introduction to linear regression and its connection to Bayesian learning.

Explanation of the Maximum Likelihood (ML) hypothesis with Gaussian distribution.

Clarification on Argmax and how it helps in finding the optimal hypothesis.

Introduction to the concept of Bayesian optimal classification and its application.

Example of using Gibbs sampling in Bayesian learning models.

How Bayesian classification provides optimal classification with given data.

Use of Bayesian learning to classify new data and derive results.

Understanding the hypothesis space (h1, h2, h3) in Bayesian models.

Final remarks and wrap-up of Bayesian learning applications in classification.

Transcripts

ಶುಭೋದಯ. ಇಂದಿನ ಉಪನ್ಯಾಸಕ್ಕೆ ಸ್ವಾಗತ.

ಇಂದು ನಾವು ಮಾಡ್ಯೂಲ್ ಗಳಿಗೆ ಹೇಗೆ ಬಳಸಲಾಗುತ್ತದೆ

ಎಂಬುದನ್ನು ನಾವು ನೋಡುತ್ತೇವೆ.

ಆದ್ದರಿಂದ, ಬಯೆಸಿಯನ್ ಪ್ರೊಬಾಬಿಲಿಟಿ ಆಗಿದೆ,

ಅದನ್ನು ನಾವು ಈಗ ಪರಿಚಯಿಸುತ್ತೇವೆ.

ಆದ್ದರಿಂದ, ಬೇಯೆಸ್ ತೇರಂ ಇದೆ.

ಹೈಪೋಥೀಸಿಸ್ ಆಗಿದೆ.

ಪ್ರೊಬಾಬಿಲಿಟಿ ಅನ್ನು ನೋಡಬಹುದು.

ರೋಗಿಗೆ ಕ್ಯಾನ್ಸರ್ ಫಲಿತಾಂಶವನ್ನು ನೀಡುತ್ತದೆ.

ಇದಲ್ಲದೆ, ಇಡೀ ಜನಸಂಖ್ಯೆಯ 0.008 ಜನರಿಗೆ ಈ ಕ್ಯಾನ್ಸರ್

ಅನ್ನು 0.97 ಎಂದು ನೀಡಲಾಗಿದೆ.

ಆದ್ದರಿಂದ, ಇವುಗಳು ಸಮಸ್ಯೆಯಲ್ಲಿ ನಿಮಗೆ

ಒದಗಿಸಲಾದ ಮೌಲ್ಯಗಳಾಗಿವೆ. ಈಗ, ಇದರ ಆಧಾರದ ಮೇಲೆ

ನಾವು ಬೇಯ್ಸ್ ಥಿಯರಮ್ ಇದೆ.

ಪರೀಕ್ಷೆಯು ಪಾಸಿಟಿವ್ ಮತ್ತು ಈಗ, ನೀವು ಇಲ್ಲಿ

ಮೌಲ್ಯಗಳನ್ನು ಹಾಕಬಹುದು.

ಜೊತೆಗೆ ನೀಡಲಾದ ಕ್ಯಾನ್ಸರ್ ಗಳಿಗೆ ಸಾಮಾನ್ಯವಾಗಿದೆ.

ಎರಡನೆಯದಾಗಿ, ಕ್ಯಾನ್ಸರ್ ಅನ್ನು ಲೆಕ್ಕಾಚಾರ

ಮಾಡಬಹುದು.

ಆದ್ದರಿಂದ, ಈ ಪ್ಲಸ್ ದ ಅನ್ವಯವಾಗಿದೆ.

ಈಗ, ಬೇಯ್ಸ್ ಲರ್ನಿಂಗ್ ಅನ್ನು ಕಂಡುಹಿಡಿಯಬಹುದು.

ಆದ್ದರಿಂದ, ಮ್ಯಾಪ್ ಹೈಪೋಥೀಸಿಸ್ ಅನ್ನು

ನೀವು ಕಂಡುಹಿಡಿಯಲು ಬಯಸುತ್ತೀರಿ.

ಈಗ, P D ಪರ್ಟಿಕುಲಾರ್ ಹೈಪೋಥೀಸಿಸ್ ಅನ್ನು

ಆರಿಸಿಕೊಳ್ಳುತ್ತೇವೆ.

ಈಗ, ಈವೆಂಟ್‌ ಅನ್ನು ಕಂಡುಹಿಡಿಯಲು.

ಈಗ, ಲಿಸ್ಟ್ ಸ್ಕ್ವಯರ್ಡ್ ಲೈನ್ ಅನ್ನು ಕಲಿಯಬೇಕು

ಎಂದು ಭಾವಿಸೋಣ.

ನಾವು ಈಗಾಗಲೇ ಲೀನರ್ ರಿಗ್ರೆಷನ್ f ಇದೆ.

F ಟಾರ್ಗೆಟ್ ಪಂಕ್ಷನ್ i ಎಂದು ಉತ್ಪಾದಿಸಲಾಗುತ್ತದೆ.

ಎಪ್ಸಿಲಾನ್ ಆಗಿ ರಚಿಸಲಾಗಿದೆ ಎಂದು ಹೇಳಬಹುದು.

ಆದ್ದರಿಂದ, ಇದು ನಮ್ಮ x ಮತ್ತು ಇದು ನಮ್ಮ d

ಆಗಿದೆ, ಇದು ಟ್ರೂ ಫಂಕ್ಷನ್ ಅನ್ನು ಬಳಸುತ್ತೇವೆ

ಎಂದು ಹೇಳೋಣ.

ಹಾಗಾದರೆ, h

m l ಎಂದರೇನು? H m

l ಎಂಬುದು ಗರಿಷ್ಟ ಲೈಕ್ಲೀಹುಡ್ ಹೈಪೋಥೀಸಿಸ್ ಉದಾಹರಣೆಗಳ

ಮೇಲೆ

ಏಕೆಂದರೆ ಅವು ಗಾಸಿಯನ್ ಡಿಸ್ಟ್ರಿಬ್ಯೂಷನ್

ಅನ್ನು ಇಲ್ಲಿ ಬರೆಯಬಹುದು.

ಆದ್ದರಿಂದ, ಇದು Arg max

h ಎಂದು ತಿರುಗುತ್ತದೆ. ಆದ್ದರಿಂದ, ಈ ಉತ್ಪನ್ನ

ಅನ್ನು ಕಡಿಮೆಗೊಳಿಸಲಾಗುತ್ತದೆ.

ಏಕೆ? ಏಕೆಂದರೆ ಈ ಭಾಗವು ಕಾನ್ಸ್ಟಂಟ್ ಅನ್ನು

ಗರಿಷ್ಠಗೊಳಿಸಿದರೆ ಅದು ಈ ಭಾಗವನ್ನು ಮಾತ್ರ

ಹೆಚ್ಚಿಸುತ್ತದೆ. ಆದ್ದರಿಂದ, ನೀವು ಇದರ

ಪಾಸಿಟಿವ್ ಆಗಿದೆ.

ಆದ್ದರಿಂದ, ಇದರ ಆಧಾರದ ಮೇಲೆ, ನಾವು ಒಂದು ಫಂಕ್ಷನ್

ಎಂದರೇನು ಎಂಬುದರ ಕುರಿತು ಅಧ್ಯಯನ ಮಾಡುತ್ತೇವೆ.

ಈಗ, ಪ್ರಶ್ನೆಯು ನಮಗೆ ಕೆಲವು ಟ್ರೈನಿಂಗ್

ಡೇಟಾ ಅನ್ನು ಪರೀಕ್ಷಾ ಉದಾಹರಣೆಗೆ ಅನ್ವಯಿಸುತ್ತೀರಿ

ಎಂಬುದು ನೇರ ಉತ್ತರವಾಗಿದೆ, ಆದರೆ ಇದು ಅಗತ್ಯವಾಗಿ

ಅಲ್ಲ. ಆದ್ದರಿಂದ, ನಿಮಗೆ ಟ್ರೈನಿಂಗ್ ಡೇಟಾ ಫಾರ್ಮ್

ಅಲ್ಲ.

ಉದಾಹರಣೆಗೆ, h1, h2, h3 ಗಳು ಹೈಪೋಥೀಸಿಸ್ ಸ್ಪೇಸ್

ಅನ್ನು ಹೊಂದಿದೆ.

ನಾವು ಹೊಸ ಡೇಟಾ 0.4 ಆಗಿದೆ.

ಆದ್ದರಿಂದ, ನಾವು ಏನು ಹೊಂದಿದ್ದೇವೆ, ನಾವು

ಬೇಯೆಸ್ ಆಪ್ಟಿಮಲ್ ಕ್ಲಾಸಿಫಿಕೇಶನ್ ಎಂದು

ಕರೆಯಲಾಗುತ್ತದೆ.

ಬೇಯಸ್ ಆಪ್ಟಿಮಲ್ ಕ್ಲಾಸಿಫೈಯರ್ ಅನ್ನು ತ್ವರಿತವಾಗಿ

ನೋಡಬಹುದು.

ಆದ್ದರಿಂದ, ಬೇಯಸ್ ಆಪ್ಟಿಮಲ್ ಕ್ಲಾಸಿಫಿಕೇಶನ್

ಎಂದು ನೋಡುತ್ತೇವೆ - 0.6 ಆಗಿದೆ.

ಆದ್ದರಿಂದ, ಇದನ್ನು ಏಕೆ ಆಪ್ಟಿಮಲ್ ಅನ್ನು

ಬಳಸಬಹುದು.

ಆದ್ದರಿಂದ, ಗಿಬ್ಸ್ ಸ್ಯಾಂಪ್ಲಿಂಗ್ ಮಾಡಲು

ಅದನ್ನು ಬಳಸುತ್ತೇವೆ.

ಆದ್ದರಿಂದ, ನಾವು ವಿತರಣೆಯಿಂದ ಕೇವಲ ಒಂದು ಹೈಪೋಥೀಸಿಸ್

ನ ಎರಡು ಪಟ್ಟು ಕಡಿಮೆಯಾಗಿದೆ.

ಆದ್ದರಿಂದ, ಗಿಬ್ಸ್ ಕ್ಲಾಸಿಫೈರ್ ಅನ್ನು

ನೀಡುತ್ತದೆ. ಇದರೊಂದಿಗೆ ನಾವು ಇಂದಿನ ಉಪನ್ಯಾಸದ

ತೀರ್ಮಾನಕ್ಕೆ ಬರುತ್ತೇವೆ.

ಧನ್ಯವಾದ.