Andrew Ng Naive Bayes Generative Learning Algorithms

Summary

This video delves into generative learning algorithms, contrasting them with discriminative ones like logistic regression. It highlights the simplicity, quick implementation, and scalability of generative algorithms, especially Naive Bayes, which is ideal for a 'quick and dirty' approach to machine learning problems. The script explains the concepts of discriminative learning, which tries to find a boundary to separate classes, and generative learning, which models each class individually. It also covers the mathematical foundation of generative algorithms, including Bayes' rule, and provides a numerical example to illustrate how these models make predictions.

Takeaways

• 📚 The video discusses different types of learning algorithms beyond linear and logistic regression, focusing on generative learning algorithms.
• 🔍 Generative learning algorithms are advantageous when there are few training examples and are simple, quick to implement, and efficient, even for large datasets.
• 🛠️ The philosophy of implementing a 'quick and dirty' solution and then iterating to improve is highlighted as a practical approach in machine learning.
• 🤖 Most learning algorithms are categorized into discriminative learning algorithms and generative learning algorithms, with the latter being the focus of the video.
• 🔍 Discriminative learning algorithms, such as logistic regression, try to find a boundary to separate different classes, while generative learning algorithms model each class separately.
• 📈 The video provides an intuitive example of how a generative learning algorithm might classify new data points by comparing them to models of 'benign' and 'malignant' tumors.
• 🧠 Discriminative learning algorithms estimate the probability of Y given X directly, whereas generative learning algorithms learn the probability of X given Y and the class prior (P of Y).
• 📝 Bayes' rule is integral to generative learning algorithms, allowing the computation of the posterior probability P(Y=1|X) for classification.
• 📉 The video explains the process of calculating the posterior probability using the terms learned from the generative model.
• 📝 Modeling P(X|Y) and P(Y) are key decisions in building a generative model, which will be further explored in the development of the naive Bayes algorithm in subsequent videos.
• 🔑 The naive Bayes algorithm is introduced as an example of a generative learning algorithm that simplifies the modeling of P(X|Y) and P(Y).

Q & A

What is the main topic discussed in this video script?

The main topic discussed in this video script is generative learning algorithms, with a focus on their advantages and how they differ from discriminative learning algorithms.

Why might generative learning algorithms be preferred when there are very few training examples?

Generative learning algorithms may be preferred when there are very few training examples because they can work better with limited data and are simple, quick to implement, and efficient, which allows them to scale easily even to massive datasets.

What is the philosophy of implementing something 'quick and dirty' and then iterating to improve it?

The philosophy of implementing something 'quick and dirty' is about starting with a simple solution and then refining it through iteration. This approach can be particularly useful in machine learning when you want to quickly get something working and then improve upon it based on feedback and performance.

What are the two main categories of learning algorithms discussed in the script?

The two main categories of learning algorithms discussed in the script are discriminative learning algorithms and generative learning algorithms.

How does a discriminative learning algorithm like logistic regression work?

A discriminative learning algorithm like logistic regression works by trying to find a decision boundary that separates different classes. It does this by fitting a model that estimates the probability of the target variable given the input features.

What is the difference between discriminative and generative learning algorithms in terms of what they model?

Discriminative learning algorithms model the probability of the target variable given the input features (P(Y|X)), while generative learning algorithms model the input features given the target variable (P(X|Y)) and also learn the prior probabilities of the target variable (P(Y)).

How does a generative learning algorithm make a classification prediction?

A generative learning algorithm makes a classification prediction by building models for each class and then comparing a new example to these models to determine which class it looks more like, based on the computed probabilities.

What is Bayes' rule and how is it used in the context of generative learning algorithms?

Bayes' rule is a fundamental theorem in probability that describes the probability of an event based on prior knowledge of conditions that might be related to the event. In the context of generative learning algorithms, Bayes' rule is used to compute the posterior probability P(Y|X), which is used for making predictions.

What are the key terms a generative learning algorithm needs to model?

The key terms a generative learning algorithm needs to model are P(X|Y), which is the probability of the input features given the target variable, and P(Y), which is the prior probability of the target variable.

Can you provide an example of how a generative model computes P(Y=1|X) for a new test example?

Sure, given a new test example X, a generative model computes P(Y=1|X) by using the values of P(X|Y=1), P(Y=1), P(X|Y=0), and P(Y=0). It applies Bayes' rule to calculate the posterior probability, which is then used for classification.

What is the naive Bayes algorithm and how does it relate to the discussion in the script?

The naive Bayes algorithm is a specific type of generative learning algorithm that makes a strong (naive) assumption of feature independence given the target variable. The script discusses the naive Bayes algorithm as an example of how to model P(X|Y) and P(Y) in a generative learning context.

Outlines

00:00

📚 Introduction to Generative Learning Algorithms

This paragraph introduces the concept of generative learning algorithms, contrasting them with discriminative learning algorithms like logistic regression. It highlights the advantages of generative algorithms, such as their efficiency, simplicity, and quick implementation, which make them ideal for situations with limited training examples. The paragraph also touches on the philosophy of implementing a 'quick and dirty' solution and iterating for improvement, using the naive Bayes algorithm as an example of a simple yet effective generative model. The distinction between discriminative and generative learning is further clarified by explaining how discriminative algorithms, like logistic regression, aim to find a boundary to separate data, whereas generative algorithms build models for each class separately and then make predictions by comparing new examples to these models.

05:01

🔍 Deep Dive into Generative Learning: Naive Bayes

The second paragraph delves deeper into the workings of generative learning algorithms, focusing on the naive Bayes model. It explains the process of learning the probability of features given the class (P(X|Y)) and the class prior probability (P(Y)). The paragraph illustrates how these probabilities are used in conjunction with Bayes' rule to compute the posterior probability (P(Y|X)) for classification. The explanation includes a step-by-step guide on how to calculate this probability using the learned model parameters. The importance of modeling P(X|Y) and P(Y) accurately is emphasized, as these are the core components of a generative model. The paragraph concludes by setting the stage for further exploration of the naive Bayes algorithm in subsequent videos.

10:02

📉 Numerical Example of a Generative Model in Action

The final paragraph provides a numerical example to demonstrate how a generative model, specifically the naive Bayes algorithm, can be applied to make predictions. It outlines the process of using the model's learned probabilities for P(X|Y) and P(Y) to calculate the posterior probability P(Y=1|X) for a new test example. The example includes a step-by-step calculation that simplifies the computation by taking advantage of equal prior probabilities for the classes. The paragraph concludes by emphasizing the importance of choosing the right modeling approach for P(X|Y) and P(Y), which will be further discussed in upcoming videos.

Keywords

💡Learning Algorithm

A learning algorithm is a set of rules that allow a computer program to improve its performance on a specific task through experience. In the context of the video, different types of learning algorithms such as linear regression and logistic regression are discussed, emphasizing the introduction of generative learning algorithms which are efficient and quick to implement, suitable for scenarios with limited training examples.

💡Generative Learning Algorithms

Generative learning algorithms are a category of machine learning algorithms that model the joint probability distribution of the input variables and the output variables. They are highlighted in the video for their simplicity, quick implementation, and efficiency, making them ideal for 'quick and dirty' solutions that can be iterated upon. They are contrasted with discriminative learning algorithms, which try to find a boundary to separate different classes.

💡Naive Bayes

Naive Bayes is a specific type of generative learning algorithm that is based on applying Bayes' theorem with strong (naive) independence assumptions between the features. The video mentions Naive Bayes as a good candidate for a quick implementation due to its simplicity and scalability, even with massive datasets.

💡Discriminative Learning Algorithms

Discriminative learning algorithms are those that try to learn how to classify examples by distinguishing between different classes. In the video, logistic regression is given as an example of a discriminative learning algorithm, which attempts to find a decision boundary to separate data points of different classes, such as malignant and benign tumors.

💡Logistic Regression

Logistic regression is a statistical method for analyzing a dataset in which there are one or more independent variables that determine an outcome. The video describes how logistic regression can be used as a discriminative learning algorithm, using gradient descent to find the best decision boundary that separates classes.

💡Gradient Descent

Gradient descent is an optimization algorithm used to find the values of model parameters that minimize a loss function. In the context of the video, gradient descent is used to iteratively update the parameters of a logistic regression model to find the optimal decision boundary.

💡Bayes' Rule

Bayes' Rule is a fundamental theorem in probability theory and statistics that describes how to update the probabilities of hypotheses when given evidence. The video explains how Bayes' Rule is used in generative learning algorithms to compute the posterior probability of a class given an example, which is essential for making classification predictions.

💡Conditional Probability

Conditional probability is the probability of an event given the occurrence of another event. The video discusses how conditional probabilities, such as P(X given Y), are learned in generative models to understand the likelihood of features given a class label, which is crucial for classification.

💡Class Prior

The class prior, denoted as P(Y) in the video, refers to the a priori probability of a class label occurring. It is an important component in generative models, representing the base rate of a class without considering any specific features, such as the default probability of a tumor being malignant or benign.

💡Feature Space

Feature space is the set of all possible combinations of features that can describe a given phenomenon. In the video, the feature space is used to illustrate where benign and malignant tumors tend to lie, with the generative model building separate distributions for each class within this space.

💡Modeling

In the context of machine learning, modeling refers to the process of creating a mathematical representation of a system or process. The video discusses how generative learning algorithms model P(X given Y) and P(Y) to make predictions, with the naive Bayes algorithm being one approach to this modeling.

Highlights

Introduction to generative learning algorithms as an alternative to linear and logistic regression.

Generative learning algorithms are advantageous when there are few training examples and are simple to implement and run.

Generative algorithms are efficient and can easily scale to massive datasets.

Naive Bayes as a quick and dirty implementation for initial machine learning solutions.

Philosophy of implementing a simple model first and iterating for improvement.

Discriminative learning algorithms versus generative learning algorithms.

Discriminative algorithms like logistic regression search for a boundary to separate classes.

Generative learning focuses on building models for each class separately.

Explanation of how a generative model classifies new examples based on comparison to built models.

Formal definition of discriminative learning algorithms learning P(Y|X) directly.

Formal definition of generative learning algorithms learning P(X|Y) and P(Y).

Use of Bayes' rule to compute P(Y=1|X) for classification predictions.

Calculation of P(X) using the joint distribution of X and Y.

The importance of modeling P(X|Y) and P(Y) in a generative learning algorithm.

A numerical example demonstrating the computation of P(Y=1|X) using a generative model.

The key decision in building a generative model is how to model P(X|Y) and P(Y).

Introduction to the development of the Naive Bayes algorithm in upcoming videos.

Transcripts

in this video we're going to talk about

the different type of learning algorithm

than the one so far like linear

regression and logistic regression in

particular we want to talk about

generative learning algorithms

generative learning algorithms may work

better if we have very few training

examples and there are a second

advantage is that they're very simple

and very quick to implement and also

very quick to run and because they're so

efficient they often scale very easily

even to massive datasets

moreover turns out sometimes if you're

facing some machine learning problem you

just want to get something to work on

well the thing the best thing to do

sometimes isn't to overthink or to

over-design the algorithm but rather to

implement something quick and dirty and

then to iterate and to improve the

algorithm from there I'll say more about

this later of philosophy of implementing

something quick and dirty and iterating

and improving but because the algorithm

we're going to talk about naive Bayes is

so simple to implement and because it

runs so quickly scales even well to

massive datasets this generative

learning algorithm naive Bayes is often

a good candidate for a quick and dirty

implementation of something it turns out

most learning algorithms categorize into

two classes the first of them is what we

call discriminative learning algorithms

and the second is generative learning

algorithms and discriminative learning

algorithms is what we've seen so far

such as logistic regression which we may

fit with gradient descent right then

I'll say more what general I'll say

later what generative learning hours are

so here's the intuition given a training

set

what a discriminative learning algorithm

does by logistic regression is maybe try

to search for a straight line to

separate the two concepts right in

particular it turns out if you say

initialize logistic regressions

parameters randomly maybe initially you

end

with a vision bounty like that well

that's not such a good one so when you

start to do gradient descent with my

discretion

maybe one after one innovation you get

that design boundary of the two

iterations you get that not there bunch

of iterations you get that decision

boundary weave separates the data and so

there's a sense that logistic regression

is trying to search for a straight line

to separate the two courses and so you

know where where once again if these are

what malignant tumors and these are

benign humans we're basically looking at

all of our data and trying to find a

straight line that separates the

malignant humans from the benign tumors

that's what the discriminative learning

algorithm does in contrast this is what

a generative learning algorithm does

which is rather than looking at both

classes and trying to find something

separate them

we'll say well let's just focus on the

benign tumors to start right these are

the benign tumors and you know looking

only at these blue circles let's build a

model for what benign tumors look like

right so let's say you know build a

model and say well the most benign

tumors tend to lie in this region of

space and then we'll turn our attention

and then having built in a model of what

benign tumors look like will then turn

our attention to the malignant tumors

and now we just focus our attention on

the malignant tumors and only look at

the malignant tumors and ask you know

try to build a model of what the

malignant tumors look like it looks

looks like most of malignant tumors live

in that region of space now let me get

rid of the training set now if a new

patient comes in and you know if you

plot their features x1 x2 if it lies

there say you say oh look this black dot

it looks looks like as well though it

looks like the benign tumors I've seen

so classified this is benign whereas in

your patient whose features live there

you say oh look this looks like this

line in my red oval and a red circle

this looks like the malignant tumors

I've seen before so I'm going to

classify that example

malignant in other words well what

genitive album does is it builds a model

of each of the two classes and then mix

classification predictions based on

looking at your example and comparing it

to your two models to see whether it

looks like more like your benign or like

your malignant tumour model let's

formalize this notion of discriminative

and generative learning algorithms what

a discriminative learning algorithm does

is it learns P of Y given X directly and

this is you know the probability of Y

given X and so the justic regression

right uses the sigmoid of logistic

function to estimate this directly and I

should say there are also some

discriminative learning algorithms that

try to learn you know hypotheses that

outputs a value 0 or 1 directly or so

for example if you were to take logistic

regression and threshold this output 0.5

so the hypothesis outputs either 0 1

that would oppose a classification

prediction 0 1 directly that would be

another this ruins an album but the

intuition is yeah we so I've tried to

estimate where Y is directly as a

function of X we try to estimate the

probability of Y directly as a function

of X in contrast what a generative

learning algorithm does is instead

learns P of X given Y and it turns out

that it also learns P of Y if with the

first term is this is more important

maybe interesting one just say what that

means in our earlier example Y the class

label indicates whether a tumor is

malignant or benign and X are the

features right and so what this

algorithm is is doing is learning well

what are the features like conditioned

on a tumor being

lignin say so say what what what do

malignant tumors look like what are the

features like condition on y equals one

or is asking what are the features like

conditions on y equals zero so in other

words what do benign breast cancer

tumors look like okay

and then the other term on P of Y this

is called the class prior and this just

the you know a priori oh is the prior

probability of y equals 0 or y equals 1

in other words think of it as if a

patient walks into your office and you

don't know any features about them you

haven't measured anything yet you have

no idea right what are the odds did the

next patient to walk that your office

won't be will have a malignant versus

benign tumor or for a different example

if you're doing your spam classification

you know this is well not having seen

the next email you get tomorrow yet what

are the odds that the next piece of

email you get will be spam or non-spam

that's what the cost prior is now

modeling these two terms P of X given Y

and P of Y are the two ketones we'll

need to model in a generative learning

algorithm so suppose that we have come

up with a way to model P of x given Y

and P of Y it turns out that given the

new example X we can then compute P of y

equals 1 given X as follows in

particular this is what we need to

compute if we want to make a prediction

on a new test example X right so this

thing on the left is equal to this one

of a P of X and this is owned by Bayes

rule depending on how much probability

remember this rule here is called Bayes

rule where so P of y equals when you're

like this P of X given Y times P of Y

divided by P of X in case this doesn't

look familiar if you multiply both sides

by P of X then

no you know this becomes so the rule of

conditional probability but hopefully

maybe remember what Bayes rule is and

then let's see and howdy Callum so these

two terms in the numerator P of x given

Y and P of Y and we can get directly

from our model right because our model

tries to your estimate P of X in the

line P of Y and the denominator well

turns out you can compute that too so by

the definition of them how you

marginalize probability distributions P

of X is sum of Y of the Joint

Distribution between x and y and the

Joint Distribution and this is therefore

equal to we can also write this out as P

of x given y equals 0 oops usually P of

y equals 1 times P of y equals 1 plus P

of x given y equals 0 times P of y

equals 0 ok and so you know this is a

through some manipulations of

probability you can show that this is

equal to this which is therefore equal

to that and the reason I did this of

course is because each of these terms P

of x given Y and also these terms P of Y

but both of these are the terms that our

model gives us and so you can you know

put plug these into these four terms as

well in order to compute P of X and

using that you can then compute P of y

equals 1 given X and therefore make a

prediction on make a classification

prediction on what Y will be given X

finally to wrap up this video let's work

through I'd like to ask you to work

through a numerical example supposed to

train a generative model and so if we've

trained a generative model what that

really means is what build models of P

of X given Y and P of Y and suppose we

now going to new test example X let's

say our model tells us you know these

things P of X given

zero y equals one and so on P of Y are

equal to these quantities one is P of y

equals one conditions on X hopefully you

got that this is a right answer and just

very quickly set through the calculation

P of y equals 1 given X by the rule we

worked on just now it is equal to this

times P of 1 plus 1 divided by P of x

given y equals 1 times P of y equals 1

plus P of x given y equals 0 times P of

y equals 0 but because in our example P

of y equals 1 equals P of y equals 0

equals 0.5 you know we can cancel out

these terms in the numerator and

denominator because all three of those

one thing is I just 0.5 and so this

leaves us with the numerator P of x

given y equals 1 is 0.03 divided by and

then the other terms are go point O 3

plus this term plus 0.01 which is just

0.75 and this shows how using these

quantities learn from a generative model

we can then compute P of Y given X and

of course when deciding in building the

generative model the key decision is how

do you model P of X given Y and how do

you model P of Y and in the next few

videos we'll develop the naive Bayes

algorithm which is one way of modeling

these two terms