Nandram Discusses Bayesian Statistics at DLSU

De La Salle University once again exposed its mathematics and statistics department to a wider world today at the Ariston Estrada lecture room, with a lecture by Dr. Balgobin Nandram, Professor of Statistics at the Worcester Polytechnic Institute in Massachusetts.

Undergraduates taking up Mathematics and Statistics, along with graduates and faculty, attended his workshop on Bayesian Statistics and Small Area Estimation. After a short introduction by faculty member Shirley Ocampo, Dr. Nandram proceeded to his lecture, showing slides in his own handwriting, scanned fresh off a notebook. “It is an honor,” Ocampo said. “This visit from Dr. Nandram is timely,” as it coincides with the university offering MS Statistical Science for the first time, and the opening of the new Statistics Laboratory.

He introduced the basics of Bayesian Statistics, then followed with four simple examples, starting with a Normal Mean Data Model, then a Beta-Binary, an example on Non-response and Poisson-Gamma. His example on Non-response made use of data from his class back in Worcester, with the survey question “Are you from Masachusetts?” Out of his 103 students, he had 80 responses, and 60 of them said ‘yes’. He then continued to illustrate the use of Bayesian statistics in taking consideration the absence of the 23 other responses. He compared the Pattern Mixture Model with the Selection Model.

His workshop ended with the discussion on the Hierarchical Model, and will continue to expound further on this with the Gibbs Sampler during the second part of the workshop, at the Ariston Estrada Lecture Hall, room L126, this coming Friday, June 15, 2012, 2:30pm, exactly on the closing of De La Salle University’s Centennial Celebration.

The notes from his lecture can be requested from the De La Salle Mathematics Department, at room J201. Dr. Nandram will also discuss his most recent paper on June 22, 2012.

To know more about the esteemed Dr. Balgobin Nandram, visit his page at Worchester Polytechnic Institute.

8/23/2010: World Contingencies

What is the probability that a person aged x will not die within the next t years?

What is the probability that a person aged x will survive up to age x+t but die within the following u years after?

If there’s anything that I’d know the probability of, that’d be that all men die. P[ X = die] = 1. No exceptions, no excuses.

Even my mother died when I was born. I lived on as Kately Barton, blind girl from birth. There was no excuse for it. She just died because she couldn’t live. That was the story.

A lot of people who see the things I study, about death and life contingencies, tell me that math is an inhumane and detached science, that it is a study that is completely objective if not oblivious to the world. We try to quantify life, death and all the variables and factors that affect them to try to predict how much a life is worth monetarily.

They feel threatened, as if math was a study of measuring the world in numbers.

And they call the study of arts and letters—of history and literature, of sociology and psychology—the humanities.

But you can’t really call math inhumane, in that sense, because all that arts and letters is trying to do is capture the world in a form of language, bounded by a system of logic. And math is a language and a system of logic. It’s a philosophy.

I was told once by a teacher in high school that math is a science of symbols and patterns. Not of numbers, but of their behaviors and their effects in systems of logic.

Math is a behavioral science.

Symbols. Patterns. Like languages, like literature—like history and politics, the world and the ways of thinking. Like people.

Like the people who call my situation inhumane: to be blind and incomplete and incapable of my maximum potential as a human being. But being blind, or being sick, or being incapable of something doesn’t hinder anyone at all from expressing or experiencing love or happiness, despair or sadness, and the range of other human emotions in between. In fact, I don’t think I’m sick or incapable—I feel fine, just different. I see the world differently. I’m like math. I’m human, but with a different language.

Statistics studies the behavior of things that happen and don’t happen, things that might happen and might not. Math studies the behavior of logic. It isn’t inhumane. In fact, it goes deeper into what the mind itself usually can’t grasp. It sees the world in different ways, like how I do.

And if math is inhumane, then every study of that tries to capture the world by some language or system of logic is inhumane too.

Fact of the matter is, the world can’t be captured at all.

L’Hôpital’s Rule

Calculus is a behavioral science. Granted, it doesn’t deal with the behaviors of humans. But it does make an enormous fuss about behaviors of functions. Functions do have their own behaviors and characteristics.

Johann Bernoulli

One of the basic things we learn in calculus is the limiting process. Basically, it tells us what value the function is going nearest to, when your x is going towards a certain value. Usually, you get a numerical value for a limit, and sometimes you get an infinity, which just simply means your function is growing too big or too small, all too fast for your limits to know where your function’s going.

This is a good example on why you can’t always say that math is an exact science: some things are just too big, too small or too undefined to understand or compute for. And math isn’t just about computations, it’s about behavior.

But what if your limit is a 0/0 or an ∞/∞ ? What would that mean?

To find out, let’s take a page out of our good French Geometrician friend from the 1700’s Guillame de l’Hôpital’s book Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes. (Analyses of Curves Too Infinitely Small to Understand.) In this paper he published in 1696, he included a good number of lessons he learned from his Swiss teacher Johann Bernoulli (the first, not the second or the third) who, like a lot of his Bernoulli family members, was a well known genius in mathematics. Johann, specifically, was best when it came to discovering things about infinitesimal calculus.
So what did they tell us?

Definition: Indeterminate Form.

Let f(x) and g(x) be functions of x.

If  or ∞ and  , then  is of the indeterminate form 0/0 or ∞/∞.

Theorem: L’Hôpital’s Rule.

If  is of the indeterminate forms 0/0 or ∞/∞, then

So if you ever do encounter those two forms 0/0 or ∞/∞, then you know what to do. Just differentiate both the numerator and the denominator, and use the limiting process again. The rule also applies if we replace x→a with x→a+,x→a-,x→+∞ and x→-∞

Example 1.

If we evaluate the limit by plugging in the value x=2 in the equation, then we’d have:

Now, we see that evaluating the limit like any usual limit would give us an indeterminate form 0/0. We use L’Hôpital’s Rule. Let’s differentiate both the numerator and the denominator.

And then we evaluate just like usual.

And now that we have arrived at a known value for the limit, we have an answer.
Here are a couple more examples.

Example 2.

Example 3.

We see that after applying L’Hôpital’s Rule, we get the indeterminate form 0/0 again. It’s just a simple matter of repeating the process, or moving around the trigonometric identities to get an answer.

Don’t give up. Try using the trigonometric identities.

But there are more indeterminate forms.  Getting the answers is simply moving around the function to get an indeterminate form of 0/0 or ∞/∞ before you apply L’Hôpital’s Rule.

Related articles

• Will the real continuous function please stand up? (maa.org)
• What is infinity times infinity minus infinity (wiki.answers.com)
• What does 0^0 equal? Why do mathematicians and high school teachers disagree?(askamathematician.com)
• Integrals of limits of sequences of functions (explainingmaths.wordpress.com)

Functions

There’s always a billion and one ways to think about something. So here’s a fresh perspective on something basic: a function.

A function, according to any English dictionary, but really this definition is just off of the top of my mind right now, is basically what a thing does. Our very trusty friend, Merriam Webster, explains it as such:

func•tion n. \ˈfəŋ(k)-shən\ – 2: the action for which a person or thing is specially fitted or used or for which a thing exists : purpose

In math, it’s something, sorta, kinda like that, but not really.

A function is an equation that has an entire life on its own. Okay, so right now, I’m confusing you more. What I’m trying to say is basically, a function is an equation. It doesn’t matter what letter you use or how many variables you have, but when you put in or assign a value to one of your variables, then the others would have a value too. Usually.

Rene Déscartes (read it like, Day-Cart), a philosopher (“I think, therefore I am”) and mathematician devised what is known as a Cartesian Coordinate Plane. By assigning a value for an x (which usually shows horizontal movement or width, left to right) you get a corresponding value for a y (showing height, or vertical movement, down to up). And these values you have are coordinates, used to know where you put the points in a graph. Connecting these points, you get the graph of your equation. We know what an x is, but in functions, the y is explicitly represented as f(x) which means “function of x”. It means, that for every value of x, you have a certain value of y, which is its function.

The function of x is basically “what x does”. Told you it was sorta, kinda, a bit like the dictionary explanation but sorta not really like it.

If your teacher ever told you something about a vertical line test, then they’re partly right, and also partly wrong.

The vertical line test proves if something is a function. Of x. Just because it doesn’t pass for the vertical line test, it’s not a function anymore. It still is. But it’s a function of y. (If it passes the horizontal line test.) The vertical line test shows that for every value of x, you only have one value for f(x). Same goes for a horizontal line test. For every value of y, there is a one (and only one) corresponding value for f(y).

But for the sake of uniformity, everyone just says that the values of x is in your domain, and f(x) is your co-domain.

And as with all things mathematical, the best way to learn is by example.

Example: f(x) = 3x+5

x	f(x) = 3x +5
-3	-4
-2	-1
-1	2
0	5
1	8
2	11
3	14
4	17

As you can see, for every value of x, there’s only one value for y or f(x). Yup, yup. In functions, we just usually say that y is f(x). y=3x+5 is called “explicit”, because we really do show that y is the function. If we wrote that as 3x-y+5=0, then that’s called “implicit”, because we just imply and not exactly state that y is the function of x. In a sense, it can be written as f(x,y) = 3x-y+5. Now, it’s a function of both x and y. X is not the function of Y, Y is not the function of X. The entire thing is a function of both of them.

What is a domain? It’s not the name of a website. Well it is, but not in math. A domain is the set of all the values of x, or the values of your independent variable.

Oh, and by the way, x and y are called variables, because they vary. They change. It can take any value. The value of your x, however, is independent (in this case, and the case of most functions of x). But since y is implicitly the function of x, then that means your y only has a value if x has a value. What value y takes is dependent on the value of x. X basically dictates whatever y is.

And your co-domain or range is the set of the values of y.

Get it?

If for every value of x in the domain, there is exactly one corresponding value of y in the co-domain, then the function is said to have a one-to-one relationship. The y values can have a lot of corresponding x’s. If for every x, there are a lot of corresponding y’s, then it is not a function.

In set theory, for the sets X and Y, and the function f: X–>Y, there are three correspondence relationships.

Surjective means that for every Y, there exists a corresponding X. Injective means that for every X, there is a corresponding value in Y. If there is a corresponding value for everything, which means it is both Surjective and Injective, we call the function to be Bijective.

Anyway, enough of this. I hope you have been confused.

–

There is an approach to teaching mathematics which uses less examples and applications. Sometimes, one of the best ways to learn math is to know the theorems by heart, and have them down packed upside down and inside out. Granted, it’s not the easier route to take, but learning that way is the way you practice your mind to think in loops and circles and find new ways and ideas and concepts, like theorems and corollaries, that may connect to each other.

This wouldn’t be a nice way to learn things if you’re vying to become an engineer or an economist. But it is, if you want to learn pure mathematics or be a philosopher.