Statistical moments —

Inspired by this blog from paypal team. Moment is a physics concept(or atleast I encountered it first in physics, but it looks it has been generalized in math to apply to other fields).

If you followed that wikipedia math link above, you’ll know the formula for moment is
\mu_n = \int\limits_{-\infty}^{+\infty} (x-c)^n f(x)\ dx\
where x — the value of the variable
n — order of the moment (aka nth moment, we’ll get to that shortly)
c — center or value around which to calculate the moment.

However if you look at a few other pages and links they ignore that part c.. and of course use the summation symbol.**

The reason they don’t put up ‘c’ there is they assume moment around the value 0. As we’ll see below this is well and good in some cases, but not always.

The other part n- order of the moment is an interesting concept. It’s just raising the value to nth power. To begin with if n is even the the negative sign caused by differences goes away. So it’s all a summary and becomes a monotonically increasing function.

I usually would argue that the ‘c’ would be the measure of central tendency like mean/median/mode and a sign of fat-tailed/thin-tailed distributions is that the moments will be different if you choose a different c and the different moments change wildly.

The statsblog I linked above mentions something different.

Higher-order terms(above the 4th) are difficult to estimate and equally difficult to describe in layman’s terms. You’re unlikely to come across any of them in elementary stats. For example, the 5th order is a measure of the relative importance of tails versus center (mode, shoulders) in causing skew. For example, a high 5th means there is a heavy tail with little mode movement and a low 5th means there is more change in the shoulders.

Hmm.. wonder how or why? I can’t figure out how it can be an indication of fat-tails(referred by the phrase importance of tails in the quote above) with the formula they are using. i.e: when the formula doesn’t mention anything about ‘c’.

** — That would be the notation for comparing discrete variables as opposed to continuous variables, but given that most of real-world application of statistics uses sampling at discrete intervals, it’s understandable to use this notation instead of the integral sign.

Types of Statistical Regression Models

Inspired by this and this

Types of Regression:
Attributes of a regression model:
There are 4 key attributes:

1. No. of Independent variables
2. Type of Dependent variables
3. Shape/Curve of the Regression line
4. No. of Dependent variables

Disclaimer: This is by no means exhaustive. Just an attempt to write around key techniques

  1. Linear —

* – Shape of regression==> straight line.
* – (no. of variables is assumed 1 independent vs 1 dependent)
* – Type of output/dependent variable(Numerical)
* – Type of input/independent variable(Numerical)
* – More variables can be used by collapsing them into one(some formula like weighted sum)
* – Or just by adding more variables to the line eqn(and modifying the convergence algorithm): Y = a + b * x1 + c * x2 + …. + xn

2. Logarithmic —

* – Shape of regression ==> exponential/logarithmic curve
* – (no. of variables is assumed 1 independent vs 1 dependent)
* – Type of output/dependent variable(Numerical)
* – Type of input/independent variable(Numerical)
* – More variables can be used by collapsing them into one(some formula like weighted sum)

3. Logistic —

* – Shape ==> S-shaped logistic curve
* – (no. of variables is assumed 1 independent vs 1 dependent)
* – Type of output/dependent variable(Binary-aka 2 categories)
* – Type of input/independent variable(Numerical)
* – More categories in output variable can be used by collapsing them into one(some formula dividing the output curve)

4. Polynomial —

* – Shape ==> Some complex polynomial function of order >= 2
* – (no. of variables is assumed 1 independent vs 1 dependent)
* – Type of output/dependent variable(Numerical)
* – Type of input/independent variable(Numerical)

5. Stepwise —

* – no. of variables is assumed multiple(n)- independent vs 1 dependent
* – Multiple steps, with each of them selecting specific variables based on variance, R-square, t-tests etc..
* – Forward selection , selects predictors/independent variables
* – Backward elimination , starts with all input variables and eliminates them by above mentioned methods.

6. Ridge

* – no. of variables is assumed multiple(n)- independent vs 1 dependent
* – adds a shrinkage (lambda) parameter (to regression estimates) to solve multicollinearity between independent variables
* – also called as l2 regularization.. it’s a regularization method

7. Lasso

* – Least Absolute Shrinkage and Selection Operator
* – also called as l1 regularization.. it’s a regularization method.
* – It shrinks coefficients to zero (exactly zero), which certainly helps in feature selection.
* – If group of predictors are highly correlated, lasso picks only one of them and shrinks the others to zero.

8. ElasticNet

* – Hybrid of Ridge and Lasso method, uses l1 and l2 prior as regularizer
* – It encourages group effect in case of highly correlated variables
* – There are no limitations on the number of selected variables
* – It can suffer with double shrinkage

9. Multi-variate —

* – no. of variables is assumed multiple(n)- independent vs 1 dependent

  1. Multi-variable —

* – no. of variables is assumed multiple(n)- independent vs multiple(n) dependent

Update: These are hardly exhaustive. A closer read and reflection will point out that adding multiple predictors, changing the function, or shape of the relationship, or changing the algorithm to find the right(aka converged) coefficients are easy ways to add combinatoric explosions.

Chi-Square — goodness of fit Test

Pre-Script: This was inspired/triggered by this post.

For a long time, I’ve in the past taken a  “religiously blind”TM stance in the frequentists vs Bayesians automatically. (as evident in this post for example) For the most part it was justified in the sense that I didn’t understand the magic tables and the comparisons and how the conclusions were made. But I was also over-zealous and assumed the Bayesian methods were better by default. After realizing it I wrote a blog post (around the resources I found on the topic). This process convinced me that while the standard objections about frequentist statistical methods being used in blind faith by most scientists, may be true, there’s enough power they provide in many situations where Bayesian method would become computationally unwieldy. i.e: in cases where a sampling theory approach would still allow me to make conclusions with rigourous methods based uncertainty estimates, where Bayesian methods would fail.

So without further ado, here’s a summary of my attempt at understanding the Chi-Square test. Okay, first cut Wikipedia: . Ah.. Ok.. abort mission .. that route’s a no-go.. Clearly the Wikipedia Defn:

A chi-squared test, also referred to as a


test (or chi-square test), is any statistical hypothesis test wherein the sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true. Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent.

Has too many assumptions. It’s time to go back and read what’s Chi-squared Distribution first, and may be not just Wikipedia, but also that statistics textbook, that I’ve been ignoring for some time now.

Ok the definition of Chi-squared distribution looks straight forward, except for the independent standard normal part. I know what independent means, but have a more vague idea of standard and normal variables More down the rabbit-hole.
Ok that wikipedia link points here. So it basically assumes the k-number of variables are : a, Independent of each other, b, Are drawn from a population that follows the Standard Normal Distribution.
That sounds fairly rare in practice, but can be created by choosing and combining variables wisely(aka feature engineering in ML jargon). So ok. let’s go beyond that.
The distribution definition is simple. Sum of squares. according to wikipedia, but my text book says, it’s something like (X - \frac{\mu(X)}{\sigma(X)})^2 .

The textbook talks about Karl Pearson’s Chi-Square Test so I’ll pick that one to delve deeper.
According to the textbook, Karl Pearson proved* that the Sum of squares of ( \frac{(Observed - Expected)}{Expected})^2 follows a Chi-Squared Distribution.

The default Null Hypothesis or H0 in a Chi-Square test is that the difference between observed and theoretical/expected(according to your theory) values have no difference.
So clearly that magic comparison values are really just some p-percentage of significance you need on the ideal Chi-square distribution and seeing if your calculated value is less or more.
Conclusion comes from whether calculated value is less .If it’s less it means the Null Hypothesis is true by chance at the given significance level. Or to write it in Bayesian Terms P(Observations | H0) == P(chance/random coincidence)**
If it’s more well here’s what I think it means. P(Observations | H0) != P(chance |random coincidence) and we are p%*** confident about this assertion.

P.S.1: At this point the textbook goes into conditions where a Chi-Squared test is meaningful, I’ll save that for later.
P.S.2: Also that number k is called degrees of freedom, And I really need to figure out what it means in this context. I know what it means in the field of complexity theory and dynamical systems, but in this context I’ll have to look at the proof or atleast math areas the proof draws upon to find out. #TODO for some time. another post.

* — According to the Book the Chi-Squared test does not assume any thing about the distribution of Observed and Expected values and is therefore called non-parametric or distribution-free test. I have a difficult time imagining an approach to a proof that broad, but then I’m not much of a mathematician, for now I’ll take this at face value.

** — I almost put 0.5 here before realizing that’s only to for a coin-toss with a fair coin.

*** — The interpretation what this p-value actually means seems to be thorny issue. So I’ll reserve it for a different post.

Continuity of a function.

Most of us, would have studied(likely in high school) about the idea of functions being continuous.

As the wikipedia section states, we end up with 3 conditions for a function over an interval [a,b].

  • The function should be defined at a constant value c
  • The limit has to exist.
  • The value of the limit must equal to c.


Now this is a perfectly useful notion for most of the functions we encounter in high school. But there are functions that would satisfy these three conditions, but still won’t be helpful for us to move forward. And I just discovered while reading up for my self-educating on statistics. One moment there I was trying to understand what’s the beta distribution, or for that matter what sense does it make to talk about a probability density function, I mean understand what’s probability, but how can it have density and that sort of thing.. I lose focus a few seconds, and find myself tumbling down a click-hole to find a curiouser idea about 3 levels/types(ordered by strictness) of continuity of a function namely


The last one being what we studied and what I described above.
Now let’s get Climb up one more step on the ladder of abstraction and see what’s uniform continuity.
Ah we have five more types of continuous there namely

Ok I won’t act vogonic and try to understand or explain all of those . I just put them out there to tease feel free to click your way in.

To quote first line from the uniform continuous wiki:

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves. For instance, any isometry (distance-preserving map) between metric spaces is uniformly continuous.

So what does this mean and how does it differ from the ordinary continuity? Well they say it up there that the maximum distance between f(x) and f(y) cannot depend on x and y themselves. i.e: the dist.function: df(f(x), f(y)) has neither x or y in it’s expression/input/right hand side.

The more formal definition can be quoted like this:

Given metric spaces (X, d1) and (Y, d2), a function f : X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d1(x, y) < δ, we have that d2(f(x), f(y)) < ε.

Now why would this be relevant or useful and why is it higher/stricter than ordinary continuity. Well note that it doesn’t say anything about an interval. The notion of ordinary continuity is always defined on an interval in the input space and clearly is confined to that. i.e: it is a property that is local to the given interval in input/domain space and may or may not apply on other different intervals.

On the other hand if you can say the function is uniformly continuous you’re effectively saying the function is continuous at all intervals.

Now how do we find a more general definition(i.e: absolute continuity?) Well look at the 3 conditions we defined at the start of this blog post. The first two can be collapsed to say the function must be differentiable over the given interval [a,b]. The third is the distance/measure concept we used in uniform continuous definition to remove the bounds on the interval and say everywhere. So obviously for the absolute continuous definition we do the next step and say the function must be uniformly continuous and differentiable everywhere.(aka uniformly differentiable).

Ok all of this is great, except where the hell is this useful. I mean are there function that belong to different continuous classes, so that these definitions/properties and theorems built on these can be used to differentiate and reason about functions. Turns out there are . I’ll start with something i glimpsed on my way down the click-hole, the cantor distribution. . It’s the exception that causes as to create a new class of continuous. It’s neither discrete nor continuous.

It’s distribution therefore has no point mass or probability mass function or probability density function.* Therefore throws a lot of the reasoning/theorem systems for a loop.

For the other example i.e: something ordinarily continuous but not uniformly continuous see here. . It’s a proof by contradiction approach.

* — Ok, I confess, the last point about point mass and probability mass/density function still escapes me. I’ll revisit that later, perhaps this time with the help of that excellent norvig’s ipython notebook on probability.

September 6 – OMG! GMO! — Little facts about science

Today’s factismal: We’ve been using genetically-modified organisms to save lives for 38 years. If you keep up with the news, you are aware that there is a lot of arguing going on over the use of genetically-modified organisms, also known as GMOs to the acronym-lovers out there. On the one hand, there are those who […]

via September 6 – OMG! GMO! — Little facts about science

Indian startups advice — 3.0

As I tend to be brash and stupid and offer advice/criticisms about running startups before(here and here and here.), I’ll do it again.

I shall dispense this advice now.

1. Treat your early employees more like partners than wage slave.*

2. This follows from the previous one. After every hire(and fire) re-consider your selection process.

3. Remember the Charlie Munger advice on trust here (or quoted below)**.

4. The best problem solvers, prefer to focus on solving the problem(s) and go right on to the next problem. They would much rather leave the performance reviews, raises (promised at the time of joining) etc.. to others. So if you do promise any review and raise based on that, follow through, Don’t delay with “we’ll do this in a formal setting in two weeks” dodge and then fail to follow through. You won’t build the best possible team with that approach.

5. Find the product/market fit..(Meh. I’m not qualified to say much about this without hands-on finding one).

5. Build a monopoly niche. Don’t compete on price, use your skills and knowledge to build a big manic monopoly, that would be the biggest barrier of entry to any competitors.
By the way the last two are just me re-gurgitating what I think makes sense from what I have read around. Only currently experimenting with implementing them.

** – “The highest form that civilization can reach is a seamless web of deserved trust — not much procedure, just totally reliable people correctly trusting one another. … In your own life what you want is a seamless web of deserved trust. And if your proposed marriage contract has forty-seven pages, I suggest you not enter.”
Source: Wesco Financial annual meeting, 2008 (quoted in Stanford Business School paper)

* — Note how I didn’t say anything about politeness or good salary or on time salary etc. That’s because all of those can be wrong ones to emphasize. My whole reason for this point is that they should have skin-in-the-game. Everything else can be worked around. Just don’t do this.

Few other links:

Bayesians vs Frequentists(aka sampling theorists)

This is a long standing debate/argument and like most polarized arguments, both sides have some valid and good reasons for their stand. (There goes the punchline/ TLDR). I’ll try to go a few levels deeper and try to explain the reasons why I think this is kind of a fake argument. (Disclaimer: am just a math enthusiast, and a (willing-to-speculate) novice. Do your own research, if this post helps as a starting point for that, I’d have done my job.)

  • As EY writes in this post about how bayes theorem is  a law that’s more general and should be observed over whatever frequentist tools we have developed?
  • If you read the original post carefully, he doesn’t mention the original/underlying distribution, guesses about it or confidence interval(see calibration game)
  •  He points to a chapter(in the addendum) here.
  • Most of the post otherwise is about using tools vs using a general theory and how the general theory is more powerful and saves a lot of time
  •  My first reaction to the post was but obviously there’s a reason those two cases should be treated different. They both have the same number of samples, but different ways of taking the samples. One sampling method(one who does sample till 60% success) is a biased way of gathering data .
  • As a different blog and some comments point out, if we’re dealing with robots(deterministic algorithmic data-collector) that precisely take data in a rigourous deterministic algorithmic manner the bayesian priors are the same.
  • However in real life, it’s going to be humans, who’ll have a lot more decisions to make about considering a data point or not. (Like for example, what stage of the patient should be when he’s considered a candidate for the experimental drug)
  • The point I however am going to make or am interested in making is related to known-unknowns vs unknown-unknowns debate.
  • My point being even if you have a robot collecting the data, if the underlying nature of the distribution is unknown-unknown(or for that matter depends on a unknown-unknown factor, say location, as some diseases are more widespread in some areas)  mean that they can gather same results, even if they were seeing different local distributions.
  • A contiguous point is that determining the right sample size is a harder problem in a lot of cases to be confident about the representativeness of the sample.
  • To be fair, EY is not ignorant of this problem described above. He even refers to it a bit in his 0 and 1 are not probabilities post here. So the original post might have over-simplified for the sake of rhetoric or simply because he hadn’t read The Red queen.
  • The Red queen details about a bunch of evolutionary theories eventually arguing that the constant race between parasite and host immune system is why we have sex  as a reproductive mechanism and we have two genders/sexes.
  • The medicine/biology example is a lot more complex system than it seems so this mistake is easier to make.
  • Yes in all of the cases above, the bayesian method (which is simpler to use and understand) will work, if the factors(priors) are known before doing the analysis.
  • But my point is that we don’t know all the factors(priors) and may not even be able to list all of them, let alone screen, and find the prior probability of each of them.

With that I’ll wind this post down. But leave you with a couple more posts I found around the topic, that seem to dig into more detail. (here and here)


P.S: Here’s a funny Chuck Norris style facts about Eliezer Yudkowsky.(Which I happened upon when trying to find the original post and was not aware of before composing the post in my head.) And here’s an xkcd comic about frequentists vs bayesians.

UPDATE-1(5-6 hrs after original post conception): I realized my disclaimer doesn’t really inform the bayesian prior to judge my post. So here’s my history/whatever with statistics. I’ve had trouble understanding the logic/reasoning/proof behind standard (frequentist?) statistical tests, and was never a fan of rote just doing the steps. So am still trying to understand the logic behind those tests, but today if I were to bet I’d rather bet on results from the bayesian method than from any conventional methods**.

UPDATE-2(5-6 hrs after original post conception):  A good example might be the counter example. i.e: given the same data(aka in this case frequency of a distribution, nothing else really, i.e: mean, variance, kurtosis or skewness) show that bayesian method gives different results based on how it(data) was collected and frequentist doesn’t. I’m not sure it’s possible though given the number of methods frequentist/standard methods use.

UPDATE-3 (a few weeks after original writing): Here’s another post about the difference in approaches between the two.

UPDATE-4 (A month or so after): I came across this post with mentions more than two buckets, but obviously they are not all disjoint sets(buckets).

UPDATE-5(Further a couple of months after): There’s a slightly different approach to splitting the two cultures from a different perspective here.

UPDATE-6: A discussion in my favourite community can be found here.
** — I might tweak the amount I’d bet based on the results from it .

Elementary Worldly Wisdom

The great lesson in microeconomics is to discriminate between when technology is going to help you and when it’s going to kill you. And most people do not get this straight in their heads. But a fellow like Buffett does.

For example, when we were in the textile business, which is a terrible commodity business, we were making low-end textiles—which are a real commodity product. And one day, the people came to Warren and said, “They’ve invented a new loom that we think will do twice as much work as our old ones