# Is there a Bayesian equivalent to a Wald-Wolfowitz runs test?

I have a sequence of observations and I would like to determine if the observations in the sequence are mutually independent. Wald-Wolfowitz is a non-parametric test that can be used to check for randomness in the sequence. I think I have a general idea of how to formulate the problem, but don't want to sprain my brain if it's been done before. [My literature searches and Google searches didn't turn up anything, but I may not be asking the question properly.] Thanks in advance.

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Check Diaconis and Rolles, Bayesian analysis for reversible Markov chains -- it may be what you're looking for. –  Cyan Mar 22 '12 at 1:53
Interesting. I'll have a look. Thanks! –  Aengus Mar 24 '12 at 16:13
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## 1 Answer

In Chapter 6 of Bayesian Data Analysis by Gelman, Carlin, Rubin, and Stern, they give an example of checking the assumption of independence is binomial trials (it comes on page 163 of the second edition).

Specifically, they consider the case where a sequence of observations $y_{1},\cdots,y_{n}$ are modeled as iid Bernoulli with a uniform prior on the probability of success, $\theta$.

They argue that observing autocorrelation in the outcome sequence is evidence for non-independence. The way they quantify this is to set up a test statistic, $T$, equal to the number of switches between zero and one in the observed set.

They simulate draws from the posterior distribution of their model and compute the test statistic for the drawn simulations, $T_{sim,k}$, where $k$ represents the number of times they repeat the whole process.

They plot the histogram of this test statistic $T_{sim,k}$ and overlay the actual value observed in their data, $T$, which reveals that $T$ is far lower than they'd expect under the independence assumption.

Finally, they reduce it down to a Bayesian p-value of the form $P(T_{sim}\leq T(y) | y)$, and they compute this p-value by simulating and checking the condition $T_{sim}\leq T(y)$.

If your data are not discrete, you could consider computing an autocorrelation function and then testing the hypothesis that some of the coefficients for small lags should be close to zero if the data are independent.

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I think this is pretty close to what I'm looking for. Thanks! Embarrassingly, I have that example tagged in the first edition. –  Aengus May 1 '12 at 18:50
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