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Generative learning algorithms

So far, we've mainly been talking about learning algorithms that model p ( y | x ; θ ) , the conditional distribution of y given x . For instance, logistic regression modeled p ( y | x ; θ ) as h θ ( x ) = g ( θ T x ) where g is the sigmoid function. In these notes, we'll talk about a different type of learning algorithm.

Consider a classification problem in which we want to learn to distinguish between elephants ( y = 1 ) and dogs ( y = 0 ), based on some features of an animal. Given a training set, an algorithm like logistic regression or the perceptron algorithm (basically) tries to find a straight line—that is, a decision boundary—that separates the elephants anddogs. Then, to classify a new animal as either an elephant or a dog, it checks on which side of the decision boundary it falls, and makes its prediction accordingly.

Here's a different approach. First, looking at elephants, we can build a model of what elephants look like. Then, looking at dogs, we can build a separate model of whatdogs look like. Finally, to classify a new animal, we can match the new animal against the elephant model, and match it against the dog model, to see whether the new animal looks morelike the elephants or more like the dogs we had seen in the training set.

Algorithms that try to learn p ( y | x ) directly (such as logistic regression), or algorithms that try to learn mappings directly from the space of inputs X to the labels { 0 , 1 } , (such as the perceptron algorithm) are called discriminative learning algorithms. Here, we'll talk about algorithms that instead try to model p ( x | y ) (and p ( y ) ). These algorithms are called generative learning algorithms. For instance, if y indicates whether an example is a dog (0) or an elephant (1), then p ( x | y = 0 ) models the distribution of dogs' features, and p ( x | y = 1 ) models the distribution of elephants' features.

After modeling p ( y ) (called the class priors ) and p ( x | y ) , our algorithm can then use Bayes rule to derive the posterior distribution on y given x :

p ( y | x ) = p ( x | y ) p ( y ) p ( x ) .

Here, the denominator is given by p ( x ) = p ( x | y = 1 ) p ( y = 1 ) + p ( x | y = 0 ) p ( y = 0 ) (you should be able to verify that this is true from the standard properties of probabilities), and thus canalso be expressed in terms of the quantities p ( x | y ) and p ( y ) that we've learned. Actually, if were calculating p ( y | x ) in order to make a prediction, then we don't actually need to calculate the denominator, since

arg max y p ( y | x ) = arg max y p ( x | y ) p ( y ) p ( x ) = arg max y p ( x | y ) p ( y ) .

Gaussian discriminant analysis

The first generative learning algorithm that we'll look at is Gaussian discriminant analysis (GDA). In this model, we'll assume that p ( x | y ) is distributed according to a multivariate normal distribution. Let's talk briefly about the properties ofmultivariate normal distributions before moving on to the GDA model itself.

The multivariate normal distribution

The multivariate normal distribution in n -dimensions, also called the multivariate Gaussian distribution, is parameterized by a mean vector μ R n and a covariance matrix Σ R n × n , where Σ 0 is symmetric and positive semi-definite. Also written “ N ( μ , Σ ) ”, its density is given by:

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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