Time to Update the P-Value Dichotomy to a Trichotomy

In executing a classical hypothesis test, a small $p$-value allows us to reject the null hypothesis and declare that the alternative hypothesis is true.

This classical decision requires a leap of faith: if the $p$-value is small, either something unusual occurred or the null hypothesis must be false.

These days we should add a third possibility. That we searched over several models and methods to find a small $p$-value. We need to update the $p$-value oath of decision making to state: Either something unusual happened, we searched to find a small $p$-value or the null hypothesis is false.

Note that being Bayesian doesn't necessarily avoid this problem. Suppose a regression model $Y = X\beta+ \mbox{error}$. Apologies for not defining notation, except that $\beta$ is a $p$-vector with elements $\beta_k$. One way to define a one-sided Bayesian $p$-value is the posterior probability that $\beta_k$ is less than zero. If this probability $P(\beta_k \lt 0 | Y)$ is near 0 or near 1, then we declare "significance". Basically the Bayesian $p$-value tells us how much certainty we have about the sign of $\beta_k$. The usual classical $p$-value is approximately twice the smaller of $P(\beta_k \lt 0 | Y)$ and $P(\beta_k \gt 0 | Y)$. How close the approximation is depends on the relative strength of the prior information to the information in the data, the observed Fisher information. The Bayesian $p$-value is subject to the same maximization by search over models as the classical $p$-value.

Bayesians have an alternative to merely searching over models however. We can do a mixture model (George and McCulloch 1993, JASA; Kuo and Mallick 1998, Sankhyā B) and incorporate all the models that we've searched over into a single model to calculate the $p$-value.

Why Be Bayesian? Let Me Count the Ways

In answer to an old friend's question.

  1. Bayesians have more fun.
    1. Our conferences are in better places too.
  2. It's the model not the estimator.
  3. Life's too short to be a frequentist: In an infinite number of replications ...
  4. Software works better.
    1. Rather surprisingly, Bayesian software is a lot more general than frequentist software.
  5. Small sample inference comes standard with most Bayesian model fitting these days.
    1. But if you like your inference asymptotic, that's available, just not high on anyone's priority list.
    2. We can handle the no-data problem, all the way up to very large problems.
    3. Don't need a large enough sample to allow for a bootstrap.
  6. Hierarchical random effects models are better fit with Bayesian models and software.
    1. If a variance component is small, the natural Bayes model doesn't allow zero as an estimate, while the natural maximum likelihood algorithms do allow zero. If you get a zero estimate, then you're going to get poor estimates of standard errors of fixed effects. [More discussion omitted.]
    2. Can handle problems where there are more parameters than data.
  7. Logistic regression models fit better with Bayes
    1. If there's perfect separation on a particular variable, the maximum likelihood estimate of the coefficient is plus or minus infinity which isn't a good estimate.
    2. Bayesian modeling offers (doesn't guarantee it, there's no insurance against stupidity) the opportunity to do the estimation correctly.
    3. Same thing if you're trying to estimate a very tiny (or very large) probability. Suppose you observe 20 out of 20 successes on something that you know doesn't have 100% successes.
    4. To rephrase a bit: In small samples or with rare events, Bayesian estimates shrink towards sensible point estimates, (if your prior is sensible) thus avoiding the large variance of point estimates.
  8. Variance bias trade-off is working in your favor.
  9. Frequentists keep reinventing Bayesian methods
    1. Shrinkage estimates
    2. Empirical Bayes
    3. Lasso
    4. Penalized likelihood
    5. Ridge regression
    6. James-Stein estimators
    7. Regularization
    8. Pittman estimation
    9. Integrated likelihood
    10. In other words, it's just not possible to analyze complex data structures without Bayesian ideas.
  10. Your answers are admissible if you're Bayesian but usually not if you're a frequentist.
    1. Admissibility means never having to say you're sorry.
    2. Alternatively, admissibility means that someone else can't prove that they can do a better job than you.
    3. And if you're a frequentist, someone is clogging our journals with proofs that the latest idiocy is admissible or not.
    4. Unless they are clogging it with yet more ways to estimate the smoothing parameter for a nonparametric estimator.
  11. Bayesian models are generalizations of classical models. That's what the prior buys you: more models
  12. Can handle discrete, categorical, ordered categorical, trees, densities, matrices, missing data and other odd parameter types.
  13. Data and parameters are treated on an equal playing field.
  14. I would argue that cross-validation works because it approximates Bayesian model selection tools.
  15. Bayesian Hypothesis Testing
    1. Treats the null and alternative hypotheses on equal terms
    2. Can handle two or more than two hypotheses
    3. Can handle hypotheses that are
      1. Disjoint
      2. Nested
      3. Overlapping but neither disjoint nor nested
    4. Gives you the probability the alternative hypothesis is true.
    5. Classical inference can only handle the nested null hypothesis problem.
    6. We're all probably misusing p-values anyway.
  16. Provides a language for talking about modeling and uncertainty that is missing in classical statistics.
    1. And thus provides a language for developing new models for new data sets or scientific problems.
    2. Provides a language for thinking about shrinkage estimators and why we want to use them and how to specify the shrinkage.
    3. Bayesian statistics permits discussion of the sampling density of the data given the unknown parameters.
    4. Unfortunately this is all that frequentist statistics allows you to talk about.
    5. Additionally: Bayesians can discuss the distribution of the data unconditional on the parameters.
    6. Bayesian statistics also allows you to discuss the distribution of the parameters.
    7. You may discuss the distribution of the parameters given the data. This is called the posterior, and is the conclusion of a Bayesian analysis.
    8. You can talk about problems that classical statistics can't handle: The probability of nuclear war for example.
  17. Novel computing tools -- but you can often use your old tools as well.
  18. Bayesian methods allow pooling of information from diverse data sources.
    1. Data can come from books, journal articles, older lab data, previous studies, people, experts, the horse's mouth, rats a** or it may have been collected in the traditional form of data.
    2. It isn't automatic, but there is language to think about how to do this pooling.
  19. Less work.
    1. Bayesian inference is via laws of probability, not by some ad hoc procedure that you need to invent for every problem or validate every time you use it.
    2. Don't need to figure out an estimator.
    3. Once you have a model and data set, the conclusion is a computing problem, not a research problem.
    4. Don't need to prove a theorem to show that your posterior is sensible. It is sensible if your assumptions are sensible.
    5. Don't need to publish a bunch of papers to figure out sensible answers given a novel problem
    6. For example, estimating a series of means $mu_1, mu_2, \ldots$ that you know are ordered $mu_j \le mu_{j+1}$ is a computing problem in Bayesian inference, but was the source of numerous papers in the frequentist literature. Finding a (good) frequentist estimator and finding standard errors and confidence intervals took lots of papers to figure out.
  20. Yes, you can still use SAS.
    1. Or R or Stata.
  21. Can incorporate utility functions, if you have one.
  22. Odd bits of other information can be incorporated into the analysis, for example
    1. That a particular parameter, usually allowed to be positive or negative, must be positive.
    2. That a particular parameter is probably positive, but not guaranteed to be positive.
    3. That a given regression coefficient should be close to zero.
    4. That group one's mean is larger than group two's mean.
    5. That the data comes from a distribution that is not a Poisson, Binomial, Exponential or Normal. For example, the data may be better modeled by a t, gamma.
    6. That a collection of parameters come from a distribution that is skewed, or has long tails.
    7. Bayesian nonparametrics can allow you to model an unknown density as a non-parametric mixture of normals (or other density). The uncertainty in estimating this distribution is incorporated in making inferences about group means and regression coefficients.
  23. Bayesian modeling is about the science.
    1. You can calculate the probability that your hypothesis is true.
    2. Bayesian modeling asks if this model describes the data, mother nature, the data generating process correctly, or sufficiently correctly.
    3. Classical inference is all about the statistician and the algorithm, not the science.
    4. In repeated samples, how often (or how accurately) does this algorithm/method/model/inference scheme give the right answer?
    5. Classical inference is more about the robustness (in repeated sampling) of the procedure. In that way, it provides robustness results for Bayesian methods.
  24. Bayesian methods have had notable successes, to wit:
    1. Covariate selection in regression problems
    2. Model selection
    3. Model mixing
    4. And mixture models
    5. Missing data
    6. Multi-level and hierarchical models
    7. Phylogeny

The bottom line: More tools. Faster progress.

Comments

Student name: Changhee Lee
Department: Electrical engineering

Mathematics Departments and the Talented Mr. Teacher

Today we have a guest post from a colleague named Mathprof. The pseudonym perhaps is needed as Mathprof's colleagues might not be pleased to read all mathprof's comments. I did some very minor editing, but otherwise the content is Mathprof's.

I asked Mathprof about Alexander Coward. You can read some background here: http://alexandercoward.com/BlowingTheWhistleOnUCBerkeleyMathematics.html

Mathprof's response to me:

Nationwide, mathematics departments, higher education or otherwise, mirror his [Alexander Coward's] interpretation and description of events. He perfectly described my department. Just today, some of my students alerted me that my peers have been complaining that I try too hard and that my approach is hurting students. In their opinion, it is not proper for the teacher to try to teach - it is the student’s responsibility to try to learn. I am, supposedly, doing my students a disservice as I raise their knowledge base through the use of sound, research based, pedagogical practices by letting my students become accustomed to a style of learning they will likely never experience again.

Most mathematicians at large universities are grounded in pedagogical, epistemological, ontological, and methodological paradigms that uphold and maintain the current mathematics education paradigm. There are deep seated beliefs about what is education, who can and should access it, how it should look, and to what end. Although the socialization in this manner of thinking occurs mainly in colleges, it is often introduced in early education, as mathematics undergraduates fill most of the K12 mathematics teacher positions, and they often bring this paradigm with them. This is how the system perpetuates itself and how it constructs continuity between K-12 and higher education.

What I have learned in the few years of reading and writing extensively on this topic is that the form and function of education are completely at odds with one another. This is particularly true in mathematics. If you ask math teachers what they hope to achieve and then observe their method of approach, it is quickly evident that for the majority of teachers, form rules over function. Strategies whose function is to increase learning are aggressively de-emphasized and de-legitimized simply because they clash with the dominant paradigm view of education. This occurs both in practice and in research. Recreating their learning experience in not a means to an end, but an end onto itself. The demonstration of a superior capacity and knowledge in mathematics is the form that must rule over function. For these teachers, learning is a mere externality. Those that are meant to learn math do - everyone else learns their place. There is no maximization of exposure, customization of approach, or an intellectualization of the process. It is complete madness.

I could go on forever but I will stop. I feel bad for the guy [Alexander Coward]. It sounds like he is doing it the right way. That said, welcome to education. Watch out for gravity - you never know which way it will pull tomorrow.

-- Mathprof