Which side is correct depends on what’s out there waiting to be discovered, which of course we don’t know. We can only guess. Timid research is rational if you believe there are only marginal improvements that are likely to be discovered.

Sample size increases quickly as the size of the effect you’re trying to find decreases. To establish small differences in effect, you need very large trials.

If you think there are only small improvements on the status quo available to explore, you’ll explore each of the possibilities very carefully. On the other hand, if you think there’s a miracle drug in the pipeline waiting to be discovered, you’ll be willing to risk falsely rejecting small improvements along the way in order to get to the big improvement.

Suppose there are 500 drugs waiting to be tested. All of these are only 10% effective except for one that is 100% effective. You could quickly find the winner by giving each candidate to one patient. For every drug whose patient responded, repeat the process until only one drug is left. One strike and you’re out. You’re likely to find the winner in three rounds, treating fewer than 600 patients. But if all the drugs are 10% effective except one that’s 11% effective, you’d need hundreds of trials with thousands of patients each.

The best research strategy depends on what you believe is out there to be found. People who know nothing about cancer often believe we could find a cure soon if we just spend a little more money on research. Experts are more sanguine, except when they’re asking for money.

]]>You could read this aloud as “the mean of the mean is the mean.” More explicitly, it says that the expected value of the average of some number of samples from some distribution is equal to the expected value of the distribution itself. The shorter reading is confusing since “mean” refers to three different things in the same sentence. In reverse order, these are:

- The mean of the distribution, defined by an integral.
- The sample mean, calculated by averaging samples from the distribution.
- The mean of the sample mean as a random variable.

The hypothesis of this theorem is that the underlying distribution **has** a mean. Lets see where things break down if the distribution does not have a mean.

It’s tempting to say that the Cauchy distribution has mean 0. Or some might want to say that the mean is infinite. But if we take any value to be the mean of a Cauchy distribution — 0, ∞, 42, etc. — then the theorem above would be false. The mean of *n* samples from a Cauchy has the same distribution as the original Cauchy! The variability does not decrease with *n*, as it would with samples from a normal, for example. The sample mean doesn’t converge to any value as *n* increases. It just keeps wandering around with the same distribution, no matter how large the sample. That’s because the mean of the Cauchy distribution simply doesn’t exist.

Statistics is in many ways much more useful for most students than calculus. The problem is, to teach it

wellis extraordinarily difficult. It’s very easy to teach a horrible statistics class where you spit back the definitions of mean and median. But you become dangerous because you think you know something about data when in fact it’s kind of subtle.

A little knowledge is a dangerous thing, more so for statistics than calculus.

This reminds me of a quote by Stephen Senn:

Statistics: A subject which most statisticians find difficult but in which nearly all physicians are expert.

**Related**: Elementary statistics book recommendation

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The terminology used throughout this document

enormously overloadsthe symbol p(). That is, we are using, in each line of this discussion, the function p() to mean something different; its meaning is set by the letters used in its arguments. That is a nomenclatural abomination. I apologize, and encourage my readers to do things that aren’t so ambiguous (like maybe add informative subscripts), but it is so standard in our business that I won’t change (for now).

I found this terribly confusing when I started doing statistics. The meaning is not explicit in the notation but implicit in the conventions surrounding its use, conventions that were foreign to me since I was trained in mathematics and came to statistics later. When I would use letters like *f* and *g* for functions collaborators would say “I don’t know what you’re talking about.” Neither did I understand what they were talking about since they used one letter for everything.

This morning I thought about what Eric said when I saw a little snow. Last Tuesday was predicted to see ice and schools all over the Houston area closed. As it turned out, there was only a tiny amount of ice and the streets were clear. This morning there actually is snow and ice in the area, though not much, and the schools are all open. (There’s snow out in Cypress where I live, but I don’t think there is in Houston proper.)

**Related posts**:

Interview with Eric Floehr

Accuracy versus perceived accuracy

History of weather prediction

The problem with big data is that it’s difficult to analyze it when the data is stored in many different ways. How do you analyze data that is distributed across relational database management systems (RDBMS), XML flat-file databases, text-based log files, and binary format storage systems?

If data are in disparate file formats, that’s a pain. And from an IT perspective that may be as far as the difficulty goes.** But why would data be in multiple formats**? Because it’s different kinds of data! That’s the bigger difficulty.

It’s conceivable, for example, that a scientific study would collect the exact same kinds of data at two locations, under as similar conditions as possible, but one site put their data in a relational database and the other put it in XML files. More likely the differences go deeper. Maybe you have lab results for patients stored in a relational database and their phone records stored in flat files. How do you meaningfully combine lab results and phone records in a single analysis? That’s a much harder problem than converting storage formats.

]]>

However, a more fundamental point has been lost. At the core of Ioannidis’ paper is the assertion that **the proportion of true hypotheses under investigation matters**. In terms of Bayes’ theorem, the *posterior* probability of a result being correct depends on the *prior* probability of the result being correct. This prior probability is vitally important, and it varies from field to field.

In a field where it is hard to come up with good hypotheses to investigate, most researchers will be testing false hypotheses, and most of their positive results will be coincidences. In another field where people have a good idea what ought to be true before doing an experiment, most researchers will be testing true hypotheses and most positive results will be correct.

For example, it’s very difficult to come up with a better cancer treatment. Drugs that kill cancer in a petri dish or in animal models usually don’t work in humans. One reason is that these drugs may cause too much collateral damage to healthy tissue. Another reason is that treating human tumors is more complex than treating artificially induced tumors in lab animals. Of all cancer treatments that appear to be an improvement in early trials, very few end up receiving regulatory approval and changing clinical practice.

A greater proportion of physics hypotheses are correct because physics has powerful theories to guide the selection of experiments. Experimental physics often succeeds because it has good support from theoretical physics. Cancer research is more empirical because there is little reliable predictive theory. This means that a published result in physics is more likely to be true than a published result in oncology.

Whether “most” published results are false depends on context. The proportion of false results varies across fields. It is high in some areas and low in others.

]]>By this classification, Hector Cuesta’s new book Practical Data Anaysis is located toward the “hacking skills” corner of the diagram. No single book can cover everything, and this one emphasizes practical software knowledge more than mathematical theory or details of a particular problem domain.

The biggest strength of the book may be that it brings together in one place information on tools that are used together but whose documentation is scattered. The book is great source for sample code. The source code is available on GitHub, though it’s more understandable in the context of the book.

Much of the book uses Python and related modules and tools including:

- NumPy
- mlpy
- PIL
- twython
- Pandas
- NLTK
- IPython
- Wakari

It also uses D3.js (with JSON, CSS, HTML, …), MongoDB (with MapReduce, Mongo Shell, PyMongo, …), and miscellaneous other tools and APIs.

There’s a lot of material here in 360 pages, making it a useful reference.

]]>… a non-informative prior is a placeholder: you can use the non-informative prior to get the analysis started, then if your posterior distribution is less informative than you would like, or if it does not make sense, you can go back and add prior information. …

At first this may sound like tweaking your analysis until you get the conclusion you want. It’s like the old joke about consultants: the client asks what 2+2 equals and the consultant counters by asking the client what he wants it to equal. But that’s not what Andrew is recommending.

A prior distribution cannot strictly be non-informative, but there are common intuitive notions of what it means to be non-informative. It may be helpful to substitute “convenient” or “innocuous” for “non-informative.” My take on Andrew’s advice is something like this.

Start with a prior distribution that’s easy to use and that nobody is going to give you grief for using. Maybe the prior doesn’t make much difference. But if your convenient/innocuous prior leads to too vague a conclusion, go back and use a more realistic prior, one that requires more effort or risks more criticism.

It’s odd that realistic priors can be more controversial than unrealistic priors, but that’s been my experience. It’s OK to be unrealistic as long as you’re conventional.

]]>Statistics seems to be a difficult subject for mathematicians, perhaps because its elusive and wide-ranging character mitigates against the traditional theorem-proof method of presentation. It may come as some comfort then that statistics is also a difficult subject for statisticians.

**Related posts**:

Although such derivations are attractive, they don’t apply that often, and they’re suspect when they do apply. There’s often some effect that keeps the prerequisite conditions from being satisfied in practice, so the derivation doesn’t lead to the right result.

The Poisson may be the best example of this. It’s easy to argue that certain count data have a Poisson distribution, and yet empirically the Poisson doesn’t fit so well because, for example, you have a mixture of two populations with different rates rather than one homogeneous population. (Averages of Poisson distributions have a Poisson distribution. Mixtures of Poisson distributions don’t.)

The best scenario is when a theoretical derivation agrees with empirical analysis. Theory suggests the distribution should be X, and our analysis confirms that. Hurray! The theoretical and empirical strengthen each other’s claims.

Theoretical derivations can be useful even when they disagree with empirical analysis. The theoretical distribution forms a sort of baseline, and you can focus on how the data deviate from that baseline.

**Related posts**:

What distribution does my data have?

How do you justify that distribution?

In God we trust, all others bring data. — William Edwards Deming

The footnote to the quote is better than the quote:

On the Web, this quote has been widely attributed to both Deming and Robert W. Hayden; however Professor Hayden told us that he can claim no credit for this quote,and ironically

we could find no “data” confirming that Deming actually said this.

Emphasis added.

The fact that so many people attributed the quote to Deming is evidence that Deming in fact said it. It’s not conclusive: popular attributions can certainly be wrong. But it is evidence.

Another piece of evidence for the authenticity of the quote is the slightly awkward phrasing “all others bring data.” The quote is often stated in the form “all others must bring data.” The latter is better, which lends credibility to the former: a plausible explanation for why the more awkward version survives would be that it is what someone, maybe Deming, actually said.

The inconclusive evidence in support of Deming being the source of the quote is actually representative of the kind of data people are likely to bring someone like Deming.

]]>

Perl has the slogan “There’s more than one way to do it,” abbreviated TMTOWTDI and pronouced “tim toady.” Perl prides itself on variety.

Python takes the opposite approach. The Zen of Python says “There should be one — and preferably only one — obvious way to do it.” Python prides itself on consistency.

Frequentist statistics has a variety of approaches and criteria for various problems. Bayesian critics call this “adhockery.”

Bayesian statistics has one way to do everything: write down a likelihood function and prior distribution, then add data and compute a posterior distribution. This is sometimes called “turning the Bayesian crank.”

]]>But the title was actually “Statistics for People Who (*Think They*) Hate Statistics” which is far less interesting.

Our suffering sinuses are now frequently relieved by

antihistamines. Our suffering philosophy — whether implicit or explicit — of data analysis, or of statistics, or of science and technology needs to be far more frequently relieved byantihubrisines.To the Greeks

hubrismeant the kind of pride that would be punished by the gods. To statisticians, hubris should mean the kind of pride that fosters an inflated idea of one’s powers and thereby keeps one from being more than marginally helpful to others.

Tukey then lists several antihubrisines. The first is this:

]]>The data may not contain the answer. The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data.

You can also compute the higher sample moments in one pass. I’ve extended the previous code to compute skewness and kurtosis in one pass as well.

The new code also lets you split your data, say to process it in parallel on different threads, and then combine the statistics, in the spirit of map-reduce.

Lastly, I’ve posted analogous code for simple linear regression.

]]>

Suppose the probability of a correctly analyzing an email or phone call is not 100% but 99.99%. In other words, there’s one chance in 10,000 of an innocent message being incriminating. Imagine authorities analyzing one message each from 300,000,000 people, roughly the population of the United States. Then around 30,000 innocent people will have some ‘splaining to do. They will have to interrupt their dinner to answer questions from an agent knocking on their door, or maybe they’ll spend a few weeks in custody. If the legal system is 99.99% reliable, then three of them will go to prison.

Now suppose false positives are really rare, one in a million. If you analyze 100 messages from each person rather than just one, you’re approximately back to the scenario above.

Scientists call indiscriminately looking through large amounts of data “a fishing expedition” or “data dredging.” One way to mitigate the problem of massive false positives from data dredging is to demand a hypothesis: before you look through the data, say what you’re hoping to prove and why you think it’s plausible.

The legal analog of a plausible hypothesis is a **search warrant**. In statistical terms, “probable cause” is a judge’s estimation that the prior probability of a hypothesis is moderately high. Requiring scientists to have a hypothesis and requiring law enforcement to have a search warrant both dramatically reduce the number of false positives.

**Related**:

You commit three felonies a day

You do too have something to hide

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Coontz used the word “outliers”.

There are no such things. There can be mismeasured data, i.e. incorrect data, say when you tried to measure air temperature but your thermometer fell into boiling water. Or there can be errors in recording the data; transposition and such forth. But excluding mistakes, and the numbers you meant to measure are the numbers you meant to measure, there are no outliers.There are only measurements which do not accord with your theory about the thing of interest.

Emphasis added.

I have a slight quibble with this description of outliers. Some people use the term to mean legitimate extreme values, and some use the term to mean values that “didn’t really happen” in some sense. I assume Matt is criticizing the latter. For example, Michael Jordan’s athletic ability is an outlier in the former sense. He’s only an outlier in the latter sense if someone decides he “doesn’t count” in some context.

A few weeks ago I said this about outliers:

When you reject a data point as an outlier, you’re saying that the point is unlikely to occur again, despite the fact that you’ve already seen it. This puts you in the curious position of believing that some values you have not seen are more likely than one of the values you have in fact seen.

Sometimes you have to exclude a data point because you believe it is far more likely to be a mistake than an accurate measurement. You may also decide that an extreme value is legitimate, but that you wish to exclude from your model. The latter should be done with fear and trembling, or at least with an explicit disclaimer.

**Related post**: The cult of average

This afternoon I was reading a paper that used a normal approximation to a binomial when *n* was around 10 and *p* around 0.001. The relative error was enormous. The paper used the approximation to find an analytical expression for something else and the error propagated.

**A common rule of thumb** is that the normal approximation works well when *np* > 5 and *n*(1-*p*) > 5. This says that the closer *p* is to 0 or 1, the larger *n* needs to be. In this case *p* was very small, but *n* was not large enough to compensate since *np* was on the order of 0.01, far less than 5.

**Another rule of thumb** is that normal approximations in general hold well near the center of the distribution but not in the tails. In particular the *relative* error in the tails can be unbounded. This paper was looking out toward the tails, and relative error mattered.

For more details, see these notes on the normal approximation to the binomial.

]]>Suppose *X* is a mixture of *n* random variables *X*_{i} with weights *w*_{i}, non-negative numbers adding to 1. Then the *j*th central moment of *X* is given by

where μ_{i} is the mean of *X*_{i}.

In my particular application, I’m interested in a mixture of normals and so the code below computes the moments for a mixture of normals. It could easily be modified for other distributions.

from scipy.misc import factorialk, comb def mixture_central_moment(mixture, moment): '''Compute the higher moments of a mixture of normal rvs. mixture is a list of (mu, sigma, weight) triples. moment is the central moment to compute.''' mix_mean = sum( [w*m for (m, s, w) in mixture] ) mixture_moment = 0.0 for triple in mixture: mu, sigma, weight = triple for k in range(moment+1): prod = comb(moment, k) * (mu-mix_mean)**(moment-k) prod *= weight*normal_central_moment(sigma, k) mixture_moment += prod return mixture_moment def normal_central_moment(sigma, moment): '''Central moments of a normal distribution''' if moment % 2 == 1: return 0.0 else: # If Z is a std normal and n is even, E(Z^n) == (n-1)!! # So E (sigma Z)^n = sigma^n (n-1)!! return sigma**moment * factorialk(moment-1, 2)

Once we have code for central moments, it’s simple to add code for computing skewness and kurtosis.

def mixture_skew(mixture): variance = mixture_central_moment(mixture, 2) third = mixture_central_moment(mixture, 3) return third / variance**(1.5) def mixture_kurtosis(mixture): variance = mixture_central_moment(mixture, 2) fourth = mixture_central_moment(mixture, 4) return fourth / variance**2 - 3.0

Here’s an example of how the code might be used.

# Test on a mixture of 30% Normal(-2, 1) and 70% Normal(1, 3) mixture = [(-2, 1, 0.3), (1, 3, 0.7)] print "Skewness = ", mixture_skew(mixture) print "Kurtosis = ", mixture_kurtosis(mixture)

**Related post**: General formula for normal moments

Isn’t it ironic that almost all known results in asymptotic statistics don’t scale well with data?

There are several things people could mean when they say that complex models don’t scale well.

First, they may mean that the *implementation* of complex models doesn’t scale. The computational effort required to fit the model increases disproportionately with the amount of data.

Second, they could mean that complex models aren’t necessary. A complex model might do even better than a simple model, but simple models work well enough given lots of data.

A third possibility, less charitable than the first two, is that the complex models are a bad fit, and this becomes apparent given enough data. The data calls the model’s bluff. If a statistical model performs poorly with lots of data, it must have performed poorly with a small amount of data too, but you couldn’t tell. It’s simple over-fitting.

I believe that’s what Giuseppe had in mind in his remark above. When I replied that the problem is modeling error, he said “Yes, big time.” The results of asymptotic statistics scale beautifully *when the model is correct*. But giving a poorly fitting model more data isn’t going to make it perform better.

The wrong conclusion would be to say that complex models work well for small data. I think the conclusion is that you *can’t tell* that complex models are *not* working well with small data. It’s a researcher’s paradise. You can fit a sequence of ever more complex models, getting a publication out of each. Evaluate your model using simulations based on your assumptions and you can avoid the accountability of the real world.

If the robustness of simple models is important with huge data sets, it’s even more important with small data sets.

Model complexity should increase with data, not decrease. I don’t mean that it should necessarily increase, but that it could. With more data, you have the ability to test the fit of more complex models. When people say that simple models scale better, they may mean that they haven’t been able to do better, that the data has exposed the problems with other things they’ve tried.

**Related posts**:

Floating point error is the least of my worries

Robustness of equal weights

Occam’s razor and Bayes’ theorem

One can do just as well by selecting a set of scores that have some validity for predicting the outcome and adjusting the values to make them comparable (by using standard scores or ranks). A formula that combines these predictors with equal weights is likely to be just as accurate in predicting new cases as the multiple-regression model that was optimal in the original sample. More recent research went further: formulas that assign equal weights to all the predictors are often superior, because they are not affected by accidents of sampling.

If the data really do come from an approximately linear system, and you’ve identified the correct variables, then linear regression is optimal in some sense. If a simple-minded approach works nearly as well, one of these assumptions is wrong.

- Maybe the system isn’t approximately linear. In that case it would not be surprising that the best fit of an inappropriate model doesn’t work better than a crude fit.
- Maybe the linear regression model is missing important predictors or has some extraneous predictors that are adding noise.
- Maybe the system is linear, you’ve identified the right variables, but the application of your model is robust to errors in the coefficients.

Regarding the first point, it can be hard to detect nonlinearities when you have several regression variables. It is especially hard to find nonlinearities when you assume that they must not exist.

Regarding the last point, depending on the purpose you put your model to, an accurate fit might not be that important. If the regression model is being used as a classifier, for example, maybe you could do about as good a job at classification with a crude fit.

The context of Dawes’ paper, and Kahneman’s commentary on it, is a discussion of clinical judgement versus simple formulas. Neither author is discouraging regression but rather saying that a simple formula can easily outperform clinical judgment in some circumstances.

**Related posts**:

The robustness of simple rules

More theoretical power, less real power

Data calls the model’s bluff

Prob( A | B ) > Prob( A ) ⇒ Prob( B | A ) > Prob( B ).

The proof is trivial: Apply the definition of conditional probability and observe that if Prob( A ∩ B ) / Prob( B ) > Prob( A ), then Prob( A ∩ B ) / Prob( A ) > Prob( B ).

Let A be the event that someone was born in Arkansas and let B be the event that this person has been president of the United States. There are five living current and former US presidents, and one of them, Bill Clinton, was born in Arkansas, a state with about 1% of the US population. Knowing that someone has been president increases your estimation of the probability that this person is from Arkansas. Similarly, knowing that someone is from Arkansas should increase your estimation of the chances that this person has been president.

The chances that an American selected at random has been president are very small, but as small as this probability is, it goes up if you know the person is from Arkansas. In fact, it goes up by the same proportion as the opposite probability. Knowing that someone has been president increases their probability of being from Arkansas by a factor of 20, so knowing that someone is from Arkansas increases the probability that they have been president by a factor of 20 as well. This is because

Prob( A | B ) / Prob( A ) = Prob( B | A ) / Prob( B ).

This isn’t controversial when we’re talking about presidents and where they were born. But it becomes more controversial when we apply the same reasoning, for example, to deciding who should be screened at airports.

When I jokingly said that being an Emacs user makes you a better programmer, it appears a few Vim users got upset. Whether they were serious or not, it does seem that they thought “Hey, what does that say about me? I use Vim. Does that mean I’m a bad programmer?”

Assume for the sake of argument that Emacs users are better programmers, i.e.

Prob( good programmer | Emacs user ) > Prob( good programmer ).

We’re not assuming that Emacs users are necessarily better programmers, only that a larger proportion of Emacs users are good programmers. And we’re not saying anything about causality, only probability.

Does this imply that being a Vim user lowers your chance of being a good programmer? i.e.

Prob( good programmer | Vim user ) < Prob( good programmer )?

No, because being a Vim user is a specific alternative to being an Emacs user, and there are programmers who use neither Emacs nor Vim. What the above statement about Emacs *would* imply is that

Prob( good programmer | not a Emacs user ) < Prob( good programmer ).

That is, if knowing that someone uses Emacs increases the chances that they are a good programmer, then knowing that they are not an Emacs user does indeed lower the chances that they are a good programmer, *if we have no other information*. In general

Prob( A | B ) > Prob( A ) ⇒ Prob( A | not B ) < Prob( A ).

To take a more plausible example, suppose that spending four years at MIT obtaining a computer science degree makes you a better programmer. Then knowing that someone has a CS degree from MIT increases the probability that this person is a good programmer. But if that’s true, it must also be true that **absent any other information**, knowing that someone does not have a CS degree from MIT decreases the probability that this person is a good programmer. If a larger proportion of good programmers come from MIT, then a smaller proportion must not come from MIT.

***

This post uses the ideas of information and conditional probability interchangeably. If you’d like to read more on that perspective, I recommend Probability Theory: The Logic of Science by E. T. Jaynes.

]]>Visualization can surprise you, but it doesn’t scale well.

Modelling scales well, but it can’t surprise you.

Visualization can show you something in your data that you didn’t expect. But some things are hard to see, and visualization is a slow, human process.

Modeling might tell you something slightly unexpected, but your choice of model restricts what you’re going to find once you’ve fit it.

So you iterate. Visualization suggests a model, and then you use your model to factor out some feature of the data. Then you visualize again.

**Related posts**:

So here’s the plan. 365 of you write vignettes about your statistical lives. Get into the nitty gritty—tell me what you do, and why you’re doing it. I’ll collect these and then post them at the Statistics Forum, one a day for a year. I think that could be great, truly a unique resource into what statistics and quantitative research is really like. Also it will be perfect for the Statistics Forum: people will want to tune in every day to see what comes next.

If you’re interested in contributing, please contact Andrew. You can read more about the project here and you can find Andrew’s contact info here.

]]>When asked for book recommendations, people will often recommend the textbook used in a course they had. But I never had an elementary statistics course. I had a PhD in math before I became interested in statistics, so I learned statistics from more advanced books. I’ve looked at a number of elementary books, but I haven’t found one I’m excited about.

Elementary statistics books may do more harm than good. They often brush difficulties under the rug. They avoid mathematical and philosophical details. They don’t define terms carefully, and even say things that are false. And they imply that statistical analysis is a matter of applying a set of rules by rote. (And it is, for many statisticians. But that’s a topic for another time.) If a statistics book doesn’t have fairly steep prerequisites, it will be hard for it not to be misleading.

This leads to another frequently asked question: Do I intend to write my own elementary statistics book? No. I don’t know whether I could write such a book that I’d be proud of. And if I could, it would take more time than I could afford to devote to it at this point in my life.

(I’ll write soon about what “this point in my life” is. If you don’t want to wait, here’s the news in a nutshell.)

]]>I knew nothing about statistics at the time and was surprised to find that there was a bitter rivalry between two schools of statistics. The rivalry is still there, though it’s not as bitter as it once was.

I find it grating when someone asks “Are you a Bayesian?” It implies an inappropriate degree of commitment and exclusivity. Bayesian statistics is just a tool. Statistics itself is just tool, one way of understanding the world.

My car has a manual transmission. I prefer manual transmissions. But if someone asked whether I was a manual transmissionist, I’d look at them like they’re crazy. I don’t have any moral objections to automatic transmissions.

I evaluate a car by how well it works. And for most purposes, I prefer the way a manual transmission works. But when I’m teaching one of my kids to drive, we go out in my wife’s car with an automatic transmission. Similarly, I evaluate a mathematical model (statistical or otherwise) by how it works for a given purpose. Sometimes a Bayesian and a frequentist approach lead to the same conclusions, but the latter is easier to understand or implement. Sometimes a Bayesian method leads to a better result because it can use more information or is easier to interpret. Sometimes it’s a toss up and I use a Bayesian approach because its more familiar, just like my old car.

**Related post**: Bayes isn’t magic

Image from Figure 6 of John’s paper.

]]>Here’s my latest idea: Approximating random inequalities with Edgeworth expansions

An Edgeworth expansion is like a Fourier series, except you use derivatives of the normal density as your basis rather than sine functions. Sometimes the full Edgeworth expansion does not converge and yet the first few terms make a good approximation. The tech report explicitly considers Edgeworth approximations with just two terms, but demonstrates the integration tricks necessary to use more terms. The result is computed in closed form, no numerical integration required, and so may be much faster than other approaches.

One advantage of the Edgeworth approach is that it only depends on the moments of the distributions in the inequality. This means it provides an approximation that’s waiting to be used on new families of distributions. But because it’s not specific to a distribution family, its performance in a particular case needs to be explored. In the case of beta distributions, for example, even a single-term approximation does pretty well.

**More blog posts on random inequalities**:

Introduction

Analytical results

Numerical results

Cauchy distributions

Beta distributions

Gamma distributions

Three or more random variables

Folded normals

A Bayesian view of Amazon Resellers

Fast approximation of beta inequalities

Shifting probability distributions

Denote the normal PDF by

Then the product of two normal PDFs is given by the equation

where

and

Note that the product of two normal random variables is not normal, but the product of their PDFs is proportional to the PDF of another normal.

]]>