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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

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
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.
anyone know any internet site where one can find nanotechnology papers?
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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
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.
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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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