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Then in parts of notation, I guess, I define this term here to be the likelihood of theta. And the likely of theta is just the probability of the data Y, right? Given X and prioritized by theta. To test the likelihood and probability are often confused. So the likelihood of theta is the same thing as the probability of the data you saw. So likely and probably are, sort of, the same thing. Except that when I use the term likelihood I’m trying to emphasize that I’m taking this thing and viewing it as a function of theta. Okay? So likelihood and for probability, they’re really the same thing except that when I want to view this thing as a function of theta holding X and Y fix are then called likelihood. Okay? So hopefully you hear me say the likelihood of the parameters and the probability of the data, right? Rather than the likelihood of the data or probability of parameters. So try to be consistent in that terminology.

So given that the probability of the data is this and this is also the likelihood of the parameters, how do you estimate the parameters theta? So given a training set, what parameters theta do you want to choose for your model? Well, the principle of maximum likelihood estimation says that, right? You can choose the value of theta that makes the data as probable as possible, right? So choose theta to maximize the likelihood. Or in other words choose the parameters that make the data as probable as possible, right? So this is massive likely your estimation from six to six. So it’s choose the parameters that makes it as likely as probable as possible for me to have seen the data I just did.

So for mathematical convenience, let me define lower case l of theta. This is called the log likelihood function and it’s just log of capital L of theta. So this is log over product over I to find sigma E to that. I won’t bother to write out what’s in the exponent for now. It’s just saying this from the previous board. Log and a product is the same as the sum of over logs, right? So it’s a sum of the logs of – which simplifies to m times one over root two pi sigma plus and then log of explanation cancel each other, right? So if log of E of something is just whatever’s inside the exponent. So, you know what, let me write this on the next board.

Okay. So maximizing the likelihood or maximizing the log likelihood is the same as minimizing that term over there. Well, you get it, right? Because there’s a minus sign. So maximizing this because of the minus sign is the same as minimizing this as a function of theta. And this is, of course, just the same quadratic cos function that we had last time, J of theta, right? So what we’ve just shown is that the ordinary least squares algorithm, that we worked on the previous lecture, is just maximum likelihood assuming this probabilistic model, assuming IID Gaussian errors on our data. Okay?

One thing that we’ll actually leave is that, in the next lecture notice that the value of sigma squared doesn’t matter, right? That somehow no matter what the value of sigma squared is, I mean, sigma squared has to be a positive number. It’s a variance of a Gaussian. So that no matter what sigma squared is since it’s a positive number the value of theta we end up with will be the same, right? So because minimizing this you get the same value of theta no matter what sigma squared is. So it’s as if in this model the value of sigma squared doesn’t really matter. Just remember that for the next lecture. We’ll come back to this again. Any questions about this? Actually, let me clean up another couple of boards and then I’ll see what questions you have.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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
ya I also want to know the raman spectra
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.
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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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