You can express a Student-t distribution as a continuous mixture of normal distributions. Some properties of the t distribution are easier to prove in this form. Here are notes with details.
I ran across this tidbit reading Bayesian Data Analysis by Gelman et al.
Related post: Beer, Wine, and Statistics (origin of the Student-t distribution)
I was asking about backup software for Windows the other day and a couple people recommended Cobian Backup. It’s simple to use, but also very configurable. And it’s free.
You can have the software simply copy files or you can have it zip the output (.zip or .7z format). In either case, you don’t need the backup software in order to restore your files.
The software has all the features you’d expect. You can perform full, incremental, or differential backups. You can run backups manually or as scheduled tasks. Etc.
Random number generators are challenging to test.
These are some of the questions I address in Chapter 10 of Beautiful Testing.
The book is now in stock at Amazon. It is supposed to be in book stores by Friday. All profits from Beautiful Testing go to Nothing But Nets, a project to distribute anti-malarial bed nets.
Related: Need help with randomization?
I recently ran across this quote from Mithat Gönen of Memorial Sloan-Kettering Cancer Center:
While there are certainly some at other centers, the bulk of applied Bayesian clinical trial design in this country is largely confined to a single zip code.
from “Bayesian clinical trials: no more excuses,” Clinical Trials 2009; 6; 203.
The zip code Gönen alludes to is 77030, the zip code of M. D. Anderson Cancer Center. I can’t say how much activity there is elsewhere, but certainly we design and conduct a lot of Bayesian clinical trials at MDACC.
Update: After over a decade working at MDACC, I left to start my own consulting business. If you’d like help with adaptive clinical trials please let me know.
The open source community has a saying: With enough eyes, all bugs are shallow. When enough people look at a piece of code, someone is going find and fix the bugs.
A related principle is that with enough users, all bugs will be reported. With enough people use the software, someone else is going to run into the problem. Someone will report it. Someone will talk about it in an online forum. Someone will blog about it and post a work-around until the bug is fixed. This principle deserves more attention; it’s not cited as often as the shallow bugs principle.
Ideally, you want to use software with lots of eyes and lots of users. Firefox is an open source product with lots eyes and lots of users. But more often you have to pick eyes or users. You have to choose between open but obscure software and closed but popular software. Open source projects may have more people looking at the source code, and so they have the “many eyes make shallow bugs” maxim working for them. But the user base for many open source projects is tiny compared to their commercial counterparts. The number of users to find and report bugs is small, and the number who document fixes and work-arounds is even smaller.
I’m not ideologically attached to open source or commercial software. I use both. I just want my software to work. And when it doesn’t work, I want to find a solution quickly.
How can we extend the idea of derivative so that more functions are differentiable? Why would we want to do so? How can we make sense of a delta “function” that isn’t really a function? We’ll answer these questions in this post.
Suppose f(x) is a differentiable function of one variable. Suppose φ(x) is an infinitely differentiable function that is zero outside of some finite interval. Functions like φ are called test functions. Integration by parts says that
where the integrals are over the entire real line. (The fact that φ is zero outside a finite interval mean the “uv” term from integration by parts is zero.) Now suppose f(x) is not differentiable. Then the left side of the equation above does not make sense, but the right side does. We use the right hand side to develop the definition of the generalized derivative.
We think of the function f not as a function of real numbers, but as a distribution that operates on tests functions. That is, we associate with f the linear functional on the space of tests functions that maps φ to ∫ f(x) φ(x) dx. Then the distributional derivative of this functional is another linear functional, the distribution that maps test functions φ to -∫ f(x) φ'(x) dx. In summary,
We can use this procedure to define as many derivatives of f as we’d like, as long as f is integrable. So f could be some horribly ill-behaved function, differentiable nowhere in the classical sense, and we could define its 37th derivative by repeatedly applying this idea. (Distributions are also called “generalized functions.” Distributional derivatives are also called “generalized derivatives” or “weak derivatives.”)
By the way, this same procedure is used to make sense of the delta function. The delta function isn’t a function at all. It is the distribution δ that evaluates test functions at zero, i.e. δ maps φ to φ(0). (The delta function often nonsensically defined to be a function that is infinite at zero and zero everywhere else.)
Why would we want to be able to differentiate more functions? When we can differentiate more functions, we can look in a bigger space for solutions to differential equations. Sometimes this allows us to find solutions to equations that do not have classical solutions. Other times this allows us to find classical solutions more easily. We may first prove that a generalized solution exists, and then prove that the generalized solution is in fact a classical solution.
Here’s an analogy that explains how generalized solutions might lead to classical solutions. Suppose you want to find the minimum value of a function for integer arguments. You might first look for a real number that minimizes the function. This lets you, for example, use derivatives in your search for the minimum. If the real minimum you find happens to also be an integer, then you’ve solved your original problem. Distributions and generalized derivatives work much the same way. You might find a classical solution by first looking in a larger space of possible solutions, a space that allows you to use more powerful techniques in your search for a solution.
Here’s an interesting exercise. If you’re writing code in a language like C# or C++ that has catch statements, write a script to report all catch blocks. You might be surprised at what you find. Some questions to ask:
Here’s a PowerShell script that will report all catch statements plus the five lines following the catch statement.
Related post: Finding embarrassing and unhelpful error messages
Here are some notes on upper and lower bounds on the probability P(Z > t) for a standard normal random variable Z. I wrote up these notes to settle a issue that came up in a probability class I’m teaching. It’s surprising that there are simple functions that provide efficient bounds on the normal distribution function.
I sold six technical books to a used book store on the way home today. The store paid me $5 total for four of the books. Two books they didn’t want at all. The books were not that old, but they were practically worthless.
It’s sobering to think how little a technical book is worth a few years after it is printed. It’s a good reminder to focus on things that will last.
I enjoyed listening to Moira Gunn’s interview with Kevin Maney, author of the new book Trade-Off: Why Some Things Catch On and Others Don’t (ISBN 038552594X).
The book was a little disappointing after listening to the interview. I felt I had heard most of what Maney had to say before I read the book.
In a nutshell, the message of the book is that you should either strive for fidelity (exclusivity, quality) or convenience (accessibility, affordability). You can succeed by excelling at fidelity or at convenience. But if you strive for both, you’ll lose to companies that are better at one criteria or the other. Maney gives several interesting examples of companies that have succeeded along the edges of the fidelity/convenience graph but then failed when they started pursuing the diagonal.
Related post: I am not an operating system (how Microsoft and Apple are forced into their respective marketing positions)
