On Statistics: Bayesian Statistics

"Another reason for the applied statistician to care about Bayesian inference is that consumers of statistical answers, at least interval estimates, commonly interpret them as probability statements about the possible values of parameters. Consequently, the answers statisticians provide to consumers should be capable of being interpreted as approximate Bayesian statements." (Donald B Rubin, "Bayesianly justifiable and relevant frequency calculations for the applied statistician", Annals of Statistics 12(4), 1984)

"The practicing Bayesian is well advised to become friends with as many numerical analysts as possible." (James Berger, "Statistical Decision Theory and Bayesian Analysis", 1985)

"Subjective probability, also known as Bayesian statistics, pushes Bayes' theorem further by applying it to statements of the type described as 'unscientific' in the frequency definition. The probability of a theory (e.g. that it will rain tomorrow or that parity is not violated) is considered to be a subjective 'degree of belief - it can perhaps be measured by seeing what odds the person concerned will offer as a bet. Subsequent experimental evidence then modifies the initial degree of belief, making it stronger or weaker according to whether the results agree or disagree with the predictions of the theory in question." (Roger J Barlow, "Statistics: A guide to the use of statistical methods in the physical sciences", 1989)

"In the design of experiments, one has to use some informal prior knowledge. How does one construct blocks in a block design problem for instance? It is stupid to think that use is not made of a prior. But knowing that this prior is utterly casual, it seems ludicrous to go through a lot of integration, etc., to obtain 'exact' posterior probabilities resulting from this prior. So, I believe the situation with respect to Bayesian inference and with respect to inference, in general, has not made progress. Well, Bayesian statistics has led to a great deal of theoretical research. But I don't see any real utilizations in applications, you know. Now no one, as far as I know, has examined the question of whether the inferences that are obtained are, in fact, realized in the predictions that they are used to make." (Oscar Kempthorne, "A conversation with Oscar Kempthorne", Statistical Science, 1995)

"Bayesian computations give you a straightforward answer you can understand and use. It says there is an X% probability that your hypothesis is true-not that there is some convoluted chance that if you assume the null hypothesis is true, you’ll get a similar or more extreme result if you repeated your experiment thousands of times. How does one interpret THAT!" (Steven Goodman, "Bayes offers a new way to make sense of numbers", Science 19, 1999)

"Bayesian methods are complicated enough, that giving researchers user-friendly software could be like handing a loaded gun to a toddler; if the data is crap, you won’t get anything out of it regardless of your political bent." (Brad Carlin, "Bayes offers a new way to make sense of numbers", Science 19, 1999)

"I sometimes think that the only real difference between Bayesian and non-Bayesian hierarchical modelling is whether random effects are labeled with Greek or Roman letters." (Peter Diggle, "Comment on Bayesian analysis of agricultural field experiments", Journal of Royal Statistical Society B vol. 61, 1999)

"I believe that there are many classes of problems where Bayesian analyses are reasonable, mainly classes with which I have little acquaintance." (John Tukey, "The life and professional contributions of John W. Tukey, The Annals of Statistics", Vol 30, 2001)

"Bayesian statistics give us an objective way of combining the observed evidence with our prior knowledge (or subjective belief) to obtain a revised belief and hence a revised prediction of the outcome of the coin’s next toss. [...] This is perhaps the most important role of Bayes’s rule in statistics: we can estimate the conditional probability directly in one direction, for which our judgment is more reliable, and use mathematics to derive the conditional probability in the other direction, for which our judgment is rather hazy. The equation also plays this role in Bayesian networks; we tell the computer the forward  probabilities, and the computer tells us the inverse probabilities when needed." (Judea Pearl & Dana Mackenzie, "The Book of Why: The new science of cause and effect", 2018)

"We thus echo the classical Bayesian literature in concluding that ‘noninformative prior information’ is a contradiction in terms. The flat prior carries information just like any other; it represents the assumption that the effect is likely to be large. This is often not true. Indeed, the signal-to-noise ratio s is often very low and then it is necessary to shrink the unbiased estimate. Failure to do so by inappropriately using the flat prior causes overestimation of effects and subsequent failure to replicate them." (Erik van Zwet & Andrew Gelman, "A proposal for informative default priors scaled by the standard error of estimates", The American Statistician 76, 2022)