Home > data science, finance, hedge funds, statistics > Bayesian regressions (part 1)

Bayesian regressions (part 1)

October 6, 2011

I’ve decided to talk about how to set up a linear regression with Bayesian priors because it’s super effective and not as hard as it sounds. Since I’m not a trained statistician, and certainly not a trained Bayesian, I’ll be coming at it from a completely unorthodox point of view. For a more typical “correct” way to look at it see for example this book (which has its own webpage).

The goal of today’s post is to abstractly discuss “bayesian priors” and illustrate their use with an example. In later posts, though, I promise to actually write and share python code illustrating bayesian regression.

The way I plan to be unorthodox is that I’m completely ignoring distributional discussions. My perspective is, I have some time series (the $x_i$‘s) and I want to predict some other time series (the $y$) with them, and let’s see if using a regression will help me- if it doesn’t then I’ll look for some other tool. But what I don’t want to do is spend all day deciding whether things are in fact student-t distributed or normal or something else. I’d like to just think of this as a machine that will be judged on its outputs. Feel free to comment if this is palpably the wrong approach or dangerous in any way.

A “bayesian prior” can be thought of as equivalent to data you’ve already seen before starting on your dataset. Since we think of the signals (the $x_i$‘s) and response ($y$) as already known, we are looking for the most likely coefficients $\beta_i$ that would explain it all. So the form a bayesian prior takes is: some information on what those $\beta_i$‘s look like.

The information you need to know about the $\beta_i$‘s is two-fold. First you need to know their values and second you need to have a covariance matrix to describe their statistical relationship to each other. When I was working as a quant, we almost always had strong convictions about the latter but not the former, although in the literature I’ve been reading lately I see more examples where the values (really the mean values) for the $\beta_i$‘s are chosen but with an “uninformative covariance assumption”.

Let me illustrate with an example. Suppose you are working on the simplest possible model: you are taking a single time series and seeing how earlier values of $x$ predict the next value of $x$. So in a given update of your regression, $y= x_t$ and each $x_i$ is of the form $x_{t-a}$ for some $a>0.$

What is your prior for this? Turns out you already have one (two actually) if you work in finance. Namely, you expect the signal of the most recent data to be stronger than whatever signal is coming from older data (after you decide how many past signals to use by first looking at a lagged correlation plot). This is just a way of saying that the sizes of the coefficients should go down as you go further back in time. You can make a prior for that by working on the diagonal of the covariance matrix.

Moreover, you expect the signals to vary continuously- you (probably) don’t expect the third-from recent variable $x_{t-3}$ to have a positive signal but the second-from recent variable $x_{t-2}$ to have a negative signal (especially if your lagged autocorrelation plot looks like this). This prior is expressed as a dampening of the (symmetrical) covariance matrix along the subdiagonal and superdiagonal.

In my next post I’ll talk about how to combine exponential down-weighting of old data, which is sacrosanct in finance, with bayesian priors. Turns out it’s pretty interesting and you do it differently depending on circumstances. By the way, I haven’t found any references for this particular topic so please comment if you know of any.

1. October 7, 2011 at 3:25 am

I’m having trouble following, although I might have an idea from http://en.wikipedia.org/wiki/Bayesian_linear_regression, if that is really related. Is it something like this for a model where the response depends on the two most recent signals?

y_i = b_0 x_i + b_1 x_{i-1} + e_i

The e_i are iid N(0,sigma^2). The correct coefficients (b_0, b_1) as well as sigma are unknown, but we presume from the start that our particular problem was picked at random from a space of possibilities with known prior distribution. Then as we make observations, we can use Bayes to form a posterior distribution of what the bs and sigma are likely to be given our observations. Having done that, you can deduce statistics about the bs, eg, maybe the distribution center bears a relationship to the least squares solution that fits observations.

There may be an art to picking a prior distribution that results in a tractable posterior distribution, and to make iteration nice, you want them in the same family. The Wikipedia link suggests something like normal in bs times inverse-gamma in sigma is a good choice.

Now the bs are a 2D random variable with prior and posterior covariance matrices in this framework. And the off-diagonals represent autocorrelation over one period in this case, and in larger models, further diagonals represent autocorrelation over greater periods? So if you have a feel for the autocorrelation from the beginning, your prior distribution on the bs should respect that rather than require the observations to force that to develop in posterior distributions? I’m basing this take on a nice OLS write up: http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-timeseries-regression.pdf

My rendition is pretty botched, but maybe there was some use in laying it out. And for next steps you will look at cases where in some sense (b0, b1, b2, …) can be thought of as an exponentially decreasing sequence?

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2. October 7, 2011 at 5:34 am
3. October 7, 2011 at 2:01 pm

The closest I’ve come to finding EWMA-style regression in a Bayesian context are state-space methods, where adding state variance effectively down-weights older data. My reference for this is Bayesian Forecasting and Dynamic Models, by West and Harrison.

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• October 7, 2011 at 2:14 pm

Very cool, thanks! I just bought a copy on Amazon.

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4. November 22, 2013 at 1:17 pm

Came across this, and have been desperately seeking the Part 2 with Code examples! See my question in the Pystatsmodels group on the topic! https://groups.google.com/d/msg/pystatsmodels/ShG-9RyYTYg/PPR2F-lfF_QJ

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• November 22, 2013 at 6:52 pm

Look in my book “Doing Data Science”. My chapter on finance.

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5. November 23, 2013 at 1:11 pm

OK, Bought it :-) The book looks quite useful in general, Skimmed the Finance chapter, looks helpful, but I still have a way to go :-) Before I can build something that does as well as my simplistic projection noted above.. The Bayesian time series examples still seem to me to be pretty “rough”? not sure if that is the right term but continuing to build knowledge, Thanks!

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• November 23, 2013 at 1:12 pm

Tom,

Happy to hear feedback, thanks!

Cathy

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1. October 12, 2011 at 6:43 am