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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

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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 .?
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Stoney Reply
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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.
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Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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Akash Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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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