Since the comments to my previous post are getting too long, I’m reblogging it here to make more room. I say that the issue raised by J. Berger and Sellke (1987) and Casella and R. Berger (1987) concerns evaluating the evidence in relation to a given hypothesis (using error probabilities). Given the information that *this* hypothesis H* was randomly selected from an urn with 99% true hypothesis, we wouldn’t say this gives a great deal of evidence for the truth of H*, nor suppose that H* had thereby been well-tested. (H* might concern the existence of a standard model-like Higgs.) I think the issues about “science-wise error rates” and long-run performance in dichotomous, diagnostic screening should be taken up separately, but commentators can continue on this, if they wish (perhaps see this related post). Continue reading

# Bayesian/frequentist

## Continued:”P-values overstate the evidence against the null”: legit or fallacious?

## “P-values overstate the evidence against the null”: legit or fallacious? (revised)

** 0. July 20, 2014: **Some of the comments to this post reveal that using the word “fallacy” in my original title might have encouraged running together the current issue with the fallacy of transposing the conditional. Please see a newly added Section 7.

**1. What you should ask…**

Discussions of P-values in the Higgs discovery invariably recapitulate many of the familiar criticisms of P-values (some “howlers”, some not). When you hear the familiar refrain, “We all know that P-values overstate the evidence against the null hypothesis”, denying the P-value aptly measures evidence, what you should ask is:

“What do you mean by overstating the evidence against a hypothesis?”

## Higgs Discovery two years on (1: “Is particle physics bad science?”)

July 4, 2014 was the two year anniversary of the Higgs boson discovery. As the world was celebrating the “5 sigma!” announcement, and we were reading about the statistical aspects of this major accomplishment, I was aghast to be emailed a letter, purportedly instigated by Bayesian Dennis Lindley, through Tony O’Hagan (to the ISBA). Lindley, according to this letter, wanted to know:

“Arethe particle physics community completely wedded to frequentist analysis? If so, has anyone tried to explain what bad science that is?”

Fairly sure it was a joke, I posted it on my “Rejected Posts” blog for a bit until it checked out [1]. (See O’Hagan’s “Digest and Discussion”) Continue reading

## Big Bayes Stories? (draft ii)

* “Wonderful examples, but let’s not close our eyes,” * is David J. Hand’s apt title for his discussion of the recent special issue (Feb 2014) of Statistical Science called “Big Bayes Stories” (edited by Sharon McGrayne, Kerrie Mengersen and Christian Robert.) For your Saturday night/ weekend reading, here are excerpts from Hand, another discussant (Welsh), scattered remarks of mine, along with links to papers and background. I begin with David Hand:

[The papers in this collection] give examples of problems which are well-suited to being tackled using such methods, but one must not lose sight of the merits of having multiple different strategies and tools in one’s inferential armory.(Hand [1])_

…. But I have to ask, is the emphasis on ‘Bayesian’ necessary? That is, do we need further demonstrations aimed at promoting the merits of Bayesian methods? … The examples in this special issue were selected, firstly by the authors, who decided what to write about, and then, secondly, by the editors, in deciding the extent to which the articles conformed to their desiderata of being Bayesian success stories: that they ‘present actual data processing stories where a non-Bayesian solution would have failed or produced sub-optimal results.’ In a way I think this is unfortunate. I am certainly convinced of the power of Bayesian inference for tackling many problems, but the generality and power of the method is not really demonstrated by a collection specifically selected on the grounds that this approach works and others fail. To take just one example, choosing problems which would be difficult to attack using the Neyman-Pearson hypothesis testing strategy would not be a convincing demonstration of a weakness of that approach if those problems lay outside the class that that approach was designed to attack.

Hand goes on to make a philosophical assumption that might well be questioned by Bayesians: Continue reading

## Deconstructing Andrew Gelman: “A Bayesian wants everybody else to be a non-Bayesian.”

At the start of our seminar, I said that “on weekends this spring (in connection with Phil 6334, but not limited to seminar participants) I will post some of my ‘deconstructions‘ of articles”. I began with Andrew Gelman‘s note “Ethics and the statistical use of prior information”[i], but never posted my deconstruction of it. So since it’s Saturday night, and the seminar is just ending, here it is, along with related links to Stat and ESP research (including me, Jack Good, Persi Diaconis and Pat Suppes). Please share comments especially in relation to current day ESP research. Continue reading

## You can only become coherent by ‘converting’ non-Bayesianly

“What ever happened to Bayesian foundations?” was one of the final topics of our seminar (Mayo/SpanosPhil6334). In the past 15 years or so, not only have (some? most?) Bayesians come to accept violations of the Likelihood Principle, they have also tended to disown Dutch Book arguments, and the very idea of inductive inference as updating beliefs by Bayesian conditionalization has evanescencd. In one of Thursday’s readings, by Baccus, Kyburg, and Thalos (1990)[1], it is argued that under certain conditions, it is *never* a rational course of action to change belief by Bayesian conditionalization. Here’s a short snippet for your Saturday night reading (the full paper is http://errorstatistics.files.wordpress.com/2014/05/bacchus_kyburg_thalos-against-conditionalization.pdf): Continue reading

## Phil 6334: Foundations of statistics and its consequences: Day #12

We interspersed key issues from the reading for this session (from Howson and Urbach) with portions of my presentation at the Boston Colloquium (Feb, 2014): Revisiting the Foundations of Statistics in the Era of Big Data: Scaling Up to Meet the Challenge. (Slides below)*.

*Someone sent us a recording (mp3)of the panel discussion from that Colloquium (there’s a lot on “big data” and its politics) including: Mayo, Xiao-Li Meng (Harvard), Kent Staley (St. Louis), and Mark van der Laan (Berkeley). *

See if this works: | mp3

*There’s a prelude here to our visitor on April 24: Professor Stanley Young from the National Institute of Statistical Sciences.

## Phil 6334: Notes on Bayesian Inference: Day #11 Slides

A. Spanos Probability/Statistics Lecture Notes 7: An Introduction to Bayesian Inference (4/10/14)

## Phil 6334: Duhem’s Problem, highly probable vs highly probed; Day #9 Slides

April 3, 2014: We interspersed discussion with slides; these cover the main readings of the day (check syllabus): the Duhem’s Probem and the Bayesian Way, and “Highly probable vs Highly Probed”. syllabus four. Slides are below (followers of this blog will be familiar with most of this, e.g., here). We also did further work on misspecification testing.

Monday, April 7, is an optional outing, “a seminar class trip”

you might say, here at Thebes at which time we will analyze the statistical curves of the mountains, pie charts of pizza, and (seriously) study some experiments on the problem of replication in “the Hamlet Effect in social psychology”. If you’re around please bop in!

Mayo’s slides on Duhem’s Problem and more from April 3 (Day#9):

## Who is allowed to cheat? I.J. Good and that after dinner comedy hour….

It was from my Virginia Tech colleague I.J. Good (in statistics), who died five years ago (April 5, 2009), at 93, that I learned most of what I call “howlers” on this blog. His favorites were based on the “paradoxes” of stopping rules. (I had posted this last year here.)

“In conversation I have emphasized to other statisticians, starting in 1950, that, in virtue of the ‘law of the iterated logarithm,’ by optional stopping an arbitrarily high sigmage, and therefore an arbitrarily small tail-area probability, can be attained even when the null hypothesis is true. In other words if a Fisherian is prepared to use optional stopping (which usually he is not) he can be sure of rejecting a true null hypothesis provided that he is prepared to go on sampling for a long time.

The way I usually express this ‘paradox’ is that a Fisherian [but not a Bayesian] can cheat by pretending he has a plane to catch like a gambler who leaves the table when he is ahead” (Good 1983, 135)[*]

## Phil 6334: Day #3: Feb 6, 2014

**Day #3: Spanos lecture notes 2, and reading/resources from Feb 6 seminar **

** 6334 Day 3 slides: Spanos-lecture-2 **

___

Crupi & Tentori (2010). Irrelevant Conjunction: Statement and Solution of a New Paradox, *Phil Sci*, 77, 1–13.

Hawthorne & Fitelson (2004). Re-Solving Irrelevant Conjunction with Probabilistic Independence, *Phil Sci* *71*: 505–514.

Skryms (1975) *Choice and Chance* 2nd ed. Chapter V and Carnap (pp. 206-211), Dickerson Pub. Co.

Mayo posts on the tacking paradox: Oct. 25, 2013: “Bayesian Confirmation Philosophy and the Tacking Paradox (iv)*” & Oct 25.

**An update on this issue will appear shortly in a separate blogpost.**

_

**READING FOR NEXT WEEK
**Selection (pp. 35-59) from: Popper (1962).

*Conjectures and Refutations*:

*The Growth of Scientific Knowledge.*Basic Books.

## Objective/subjective, dirty hands and all that: Gelman/ Wasserman blogolog (ii)

Andrew Gelman says that as a philosopher, I should appreciate his blog today in which he records his frustration: “Against aggressive definitions: No, I don’t think it helps to describe Bayes as ‘the analysis of subjective beliefs’…” Gelman writes:

I get frustrated with what might be called “aggressive definitions,” where people use a restrictive definition of something they don’t like. For example, Larry Wasserman writes (as reported by Deborah Mayo):

“I wish people were clearer about what Bayes is/is not and what frequentist inference is/is not. Bayes is the analysis of subjective beliefs but provides no frequency guarantees. Frequentist inference is about making procedures that have frequency guarantees but makes no pretense of representing anyone’s beliefs.”

I’ll accept Larry’s definition of frequentist inference. But as for his definition of Bayesian inference: No no no no no. The probabilities we use in our Bayesian inference are not subjective, or, they’re no more subjective than the logistic regressions and normal distributions and Poisson distributions and so forth that fill up all the textbooks on frequentist inference.

To quickly record some of my own frustrations:*: First, I would disagree with Wasserman’s characterization of frequentist inference, but as is clear from Larry’s comments to (my reaction to him), I think he concurs that he was just giving a broad contrast. Please see Note [1] for a remark from my post: Comments on Wasserman’s “what is Bayesian/frequentist inference?” Also relevant is a Gelman post on the Bayesian name: [2].

Second, Gelman’s “no more subjective than…” evokes remarks I’ve made before. For example, in “What should philosophers of science do…” I wrote:

Arguments given for some very popular slogans (mostly by non-philosophers), are too readily taken on faith as canon by others, and are repeated as gospel. Examples are easily found: all models are false, no models are falsifiable, everything is subjective, or equally subjective and objective, and the only properly epistemological use of probability is to supply posterior probabilities for quantifying actual or rational degrees of belief. Then there is the cluster of “howlers” allegedly committed by frequentist error statistical methods repeated verbatim (discussed on this blog).

I’ve written a lot about objectivity on this blog, e.g., here, here and here (and in real life), but what’s the point if people just rehearse the “everything is a mixture…” line, without making deeply important distinctions? I really think that, next to the “all models are false” slogan, the most confusion has been engendered by the “no methods are objective” slogan. However much we may aim at objective constraints, it is often urged, we can never have “clean hands” free of the influence of beliefs and interests, and we invariably sully methods of inquiry by the entry of background beliefs and personal judgments in their specification and interpretation. Continue reading

## Mascots of Bayesneon statistics (rejected post)

** (see rejected posts)**

## U-Phil: Deconstructions [of J. Berger]: Irony & Bad Faith 3

**Memory Lane: 2 years ago:**

*M**y efficient Errorstat Blogpeople ^{1} have put forward the following 3 reader-contributed interpretive efforts^{2} as a result of the “deconstruction” exercise from December 11, (mine, from the earlier blog, is at the end) of what I consider:*

“….an especially intriguing remark by Jim Berger that I think bears upon the current mindset (Jim is aware of my efforts):

Too often I see people pretending to be subjectivists, and then using “weakly informative” priors that the objective Bayesian community knows are terrible and will give ridiculous answers; subjectivism is then being used as a shield to hide ignorance. . . . In my own more provocative moments, I claim that the only true subjectivists are the objective Bayesians, because they refuse to use subjectivism as a shield against criticism of sloppy pseudo-Bayesian practice. (Berger 2006, 463)” (From blogpost, Dec. 11, 2011)

_________________________________________________

Andrew Gelman:

The statistics literature is big enough that I assume there really is some bad stuff out there that Berger is reacting to, but I think that when he’s talking about weakly informative priors, Berger is not referring to the work in this area that I like, as I think of weakly informative priors as specifically being designed to give answers that are _not_ “ridiculous.”

Keeping things unridiculous is what regularization’s all about, and one challenge of regularization (as compared to pure subjective priors) is that the answer to the question, What is a good regularizing prior?, will depend on the likelihood. There’s a lot of interesting theory and practice relating to weakly informative priors for regularization, a lot out there that goes beyond the idea of noninformativity.

To put it another way: We all know that there’s no such thing as a purely noninformative prior: any model conveys some information. But, more and more, I’m coming across applied problems where I wouldn’t want to be noninformative even if I could, problems where some weak prior information regularizes my inferences and keeps them sane and under control. Continue reading

## A. Spanos lecture on “Frequentist Hypothesis Testing”

I attended a lecture by Aris Spanos to his graduate econometrics class here at Va Tech last week[i]. This course, which Spanos teaches every fall, gives a superb illumination of the disparate pieces involved in statistical inference and modeling, and affords clear foundations for how they are linked together. His slides follow the intro section. Some examples with severity assessments are also included.

**Frequentist Hypothesis Testing: A Coherent Approach**

Aris Spanos

1 Inherent difficulties in learning statistical testing

Statistical testing is arguably the most important, but also the most difficult and confusing chapter of statistical inference for several reasons, including the following.

(i) The need to introduce numerous new notions, concepts and procedures before one can paint — even in broad brushes — a coherent picture of hypothesis testing.

(ii) The current textbook discussion of statistical testing is both highly confusing and confused. There are several sources of confusion.

- (a) Testing is conceptually one of the most sophisticated sub-fields of any scientific discipline.
- (b) Inadequate knowledge by textbook writers who often do not have the technical skills to read and understand the original sources, and have to rely on second hand accounts of previous textbook writers that are often misleading or just outright erroneous. In most of these textbooks hypothesis testing is poorly explained as an idiot’s guide to combining off-the-shelf formulae with statistical tables like the Normal, the Student’s t, the chi-square, etc., where the underlying statistical model that gives rise to the testing procedure is hidden in the background.
- (c) The misleading portrayal of Neyman-Pearson testing as essentially decision-theoretic in nature, when in fact the latter has much greater affinity with the Bayesian rather than the frequentist inference.
- (d) A deliberate attempt to distort and cannibalize frequentist testing by certain Bayesian drumbeaters who revel in (unfairly) maligning frequentist inference in their attempts to motivate their preferred view on statistical inference.

(iii) The discussion of frequentist testing is rather incomplete in so far as it has been beleaguered by serious foundational problems since the 1930s. As a result, different applied fields have generated their own secondary literatures attempting to address these problems, but often making things much worse! Indeed, in some fields like psychology it has reached the stage where one has to correct the ‘corrections’ of those chastising the initial correctors!

In an attempt to alleviate problem (i), the discussion that follows uses a sketchy historical development of frequentist testing. To ameliorate problem (ii), the discussion includes ‘red flag’ pointers (¥) designed to highlight important points that shed light on certain erroneous in- terpretations or misleading arguments. The discussion will pay special attention to (iii), addressing some of the key foundational problems.

[i] It is based on Ch. 14 of Spanos (1999) *Probability Theory and Statistical Inference. *Cambridge[ii].

[ii] You can win a free copy of this 700+ page text by creating a simple palindrome! http://errorstatistics.com/palindrome/march-contest/

## The error statistician has a complex, messy, subtle, ingenious, piece-meal approach

A comment today by Stephen Senn leads me to post the last few sentences of my (2010) paper with David Cox, “Frequentist Statistics as a Theory of Inductive Inference”:

A fundamental tenet of the conception of inductive learning most at home with the frequentist philosophy is that inductive inference requires building up incisive arguments and inferences by putting together several different piece-meal results; we have set out considerations to guide these pieces[i]. Although the complexity of the issues makes it more difficult to set out neatly, as, for example, one could by imagining that a single algorithm encompasses the whole of inductive inference, the payoff is an account that approaches the kind of arguments that scientists build up in order to obtain reliable knowledge and understanding of a field.” (273)[ii]

A reread for Saturday night?

[i]The pieces hang together by dint of the rationale growing out of a severity criterion (or something akin but using a different term.)

*[ii]Error and Inference: Recent Exchanges on Experimental Reasoning, Reliability and the Objectivity and Rationality of Science* (D Mayo and A. Spanos eds.), Cambridge: Cambridge University Press: 1-27. This paper appeared in *The Second Erich L. Lehmann Symposium: Optimality*, 2006, Lecture Notes-Monograph Series, Volume 49, Institute of Mathematical Statistics, pp. 247-275.

## Stephen Senn: Dawid’s Selection Paradox (guest post)

**Stephen Senn**

Head, Methodology and Statistics Group,

Competence Center for Methodology and Statistics (CCMS),

Luxembourg

**“Dawid’s Selection Paradox”**

You can protest, of course, that Dawid’s Selection Paradox is no such thing but then those who believe in the inexorable triumph of logic will deny that *anything* is a paradox. In a challenging paper published nearly 20 years ago (Dawid 1994), Philip Dawid drew attention to a ‘paradox’ of Bayesian inference. To describe it, I can do no better than to cite the abstract of the paper, which is available from Project Euclid, here: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?

When the inference to be made is selected after looking at the data, the classical statistical approach demands — as seems intuitively sensible — that allowance be made for the bias thus introduced. From a Bayesian viewpoint, however, no such adjustment is required, even when the Bayesian inference closely mimics the unadjusted classical one. In this paper we examine more closely this seeming inadequacy of the Bayesian approach. In particular, it is argued that conjugate priors for multivariate problems typically embody an unreasonable determinism property, at variance with the above intuition.

I consider this to be an important paper not only for Bayesians but also for frequentists, yet it has only been cited 14 times as of 15 November 2013 according to Google Scholar. In fact I wrote a paper about it in the *American Statistician* a few years back (Senn 2008) and have also referred to it in a previous blogpost (12 May 2012). That I think it is important and neglected is excuse enough to write about it again.

Philip Dawid is not responsible for my interpretation of his paradox but the way that I understand it can be explained by considering what it means to have a prior distribution. First, as a reminder, if you are going to be 100% Bayesian, which is to say that all of what you will do by way of inference will be to turn a prior into a posterior distribution using the likelihood and the operation of Bayes theorem, then your prior distribution has to satisfy two conditions. First, it must be what you would use to bet now (that is to say at the moment it is established) and second no amount of subsequent data will change your prior *qua* prior. It will, of course, be updated by Bayes theorem to form a posterior distribution once further data are obtained but that is another matter. The relevant time here is your observation time not the time when the data were collected, so that data that were available in principle but only came to your attention after you established your prior distribution count as further data.

Now suppose that you are going to make an inference about a population mean, θ, using a random sample from the population and choose the standard conjugate prior distribution. Then in that case you will use a Normal distribution with known (to you) parameters μ and σ^{2}. If σ^{2} is large compared to the random variation you might expect for the means in your sample, then the prior distribution is fairly *uninformative* and if it is small then fairly *informative* but being uninformative is not in itself a virtue. Being not informative enough runs the risk that your prior distribution is not one you might wish to use to bet now and being too informative that your prior distribution is one you might be tempted to change given further information. In either of these two cases your prior distribution will be wrong. Thus the task is to be neither too informative nor not informative enough. Continue reading

## Highly probable vs highly probed: Bayesian/ error statistical differences

A reader asks: “Can you tell me about disagreements on numbers between a severity assessment within error statistics, and a Bayesian assessment of posterior probabilities?” Sure.

There are differences between Bayesian posterior probabilities and formal error statistical measures, as well as between the latter and a severity (SEV) assessment, which differs from the standard type 1 and 2 error probabilities, p-values, and confidence levels—despite the numerical relationships. Here are some random thoughts that will hopefully be relevant for both types of differences. (Please search this blog for specifics.)

1. The most noteworthy difference is that error statistical inference makes use of outcomes other than the one observed, even after the data are available: there’s no other way to ask things like, how often would you find 1 nominally statistically significant difference in a hunting expedition over *k* or more factors? Or to distinguish optional stopping with sequential trials from fixed sample size experiments. Here’s a quote I came across just yesterday:

“[S]topping ‘when the data looks good’ can be a serious error when combined with frequentist measures of evidence. For instance, if one used the stopping rule [above]…but analyzed the data as if a

fixedsample had been taken, one couldguaranteearbitrarily strong frequentist ‘significance’ againstH_{0}.” (Berger and Wolpert, 1988, 77).

The worry about being guaranteed to erroneously exclude the true parameter value here is an error statistical affliction that the Bayesian is spared (even though I don’t think they can be too happy about it, especially when HPD intervals are assured of excluding the true parameter value.) See this post for an amusing note; Mayo and Kruse (2001) below; and, if interested, search the (strong) likelihood principle, and Birnbaum.

2. *Highly probable vs. highly probed*. SEV doesn’t obey the probability calculus: for any test T and outcome ** x**, the severity for both

*H*and ~

*H*might be horribly low. Moreover, an error statistical analysis is not in the business of probabilifying hypotheses but evaluating and controlling the capabilities of methods to discern inferential flaws (problems with linking statistical and scientific claims, problems of interpreting statistical tests and estimates, and problems of underlying model assumptions). This is the basis for applying what may be called the Severity principle. Continue reading

## Gelman est effectivement une erreur statistician

A reader calls my attention to Andrew Gelman’s blog announcing a talk that he’s giving today in French: “Philosophie et practique de la statistique bayésienne”. He blogs:

I’ll try to update the slides a bit since a few years ago, to add some thoughts I’ve had recently about problems with noninformative priors, even in simple settings.

The location of the talk will not be convenient for most of you, but anyone who comes to the trouble of showing up will have the opportunity to laugh at my accent.

P.S. For those of you who are interested in the topic but can’t make it to the talk, I recommend these two papers on my non-inductive Bayesian philosophy:

[2013] Philosophy and the practice of Bayesian statistics (with discussion). British Journal of Mathematical and Statistical Psychology, 8–18. (Andrew Gelman and Cosma Shalizi) [2013] Rejoinder to discussion. (Andrew Gelman and Cosma Shalizi)

*These papers, especially Gelman (2011), are discussed on this blog (in “U-Phils”). **Comments *by Senn, Wasserman, and Hennig may be found here, and here,with a response here (please use search for more).

*As I say in **my comments *on Gelman and Shalizi, I think Gelman’s position is (or intends to be) inductive– in the sense of being ampliative (going beyond the data)– but simply not probabilist, i.e., not a matter of updating priors. (A blog post is here)[i]. *Here’s a snippet from my comments: Continue reading *

## “When Bayesian Inference Shatters” Owhadi, Scovel, and Sullivan (guest post)

I’m extremely grateful to Drs. Owhadi, Scovel and Sullivan for replying to my request for “a plain Jane” explication of their interesting paper, “When Bayesian Inference Shatters”, and especially for permission to post it. If readers want to ponder the paper awhile and send me comments for guest posts or “U-PHILS*” (by OCT 15), let me know. Feel free to comment as usual in the mean time.

—————————————-

**Houman Owhadi**

Professor of Applied and Computational Mathematics and Control and Dynamical Systems, Computing + Mathematical Sciences,

California Institute of Technology, USA

California Institute of Technology, USA

University of Warwick, UK

**“When Bayesian Inference Shatters: A plain Jane explanation”**

*This is an attempt at a “plain Jane” presentation of the results discussed in the recent arxiv paper “When Bayesian Inference Shatters” located at http://arxiv.org/abs/1308.6306 with the following abstract:*

“With the advent of high-performance computing, Bayesian methods are increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods impact the making of sometimes critical decisions in increasingly complicated contexts, the sensitivity of their posterior conclusions with respect to the underlying models and prior beliefs is becoming a pressing question. We report new results suggesting that, although Bayesian methods are robust when the number of possible outcomes is finite or when only a finite number of marginals of the data-generating distribution are unknown, they are generically brittle when applied to continuous systems with finite information on the data-generating distribution. This brittleness persists beyond the discretization of continuous systems and suggests that Bayesian inference is generically ill-posed in the sense of Hadamard when applied to such systems: if closeness is defined in terms of the total variation metric or the matching of a finite system of moments, then (1) two practitioners who use arbitrarily close models and observe the same (possibly arbitrarily large amount of) data may reach diametrically opposite conclusions; and (2) any given prior and model can be slightly perturbed to achieve any desired posterior conclusions.”

Now, it is already known from classical Robust Bayesian Inference that Bayesian Inference has some robustness if the random outcomes live in a finite space or if the class of priors considered is finite-dimensional (i.e. what you know is infinite and what you do not know is finite). What we have shown is that if the random outcomes live in an approximation of a continuous space (for instance, when they are decimal numbers given to finite precision) and your class of priors is finite co-dimensional (i.e. what you know is finite and what you do not know may be infinite) then, if the data is observed at a fine enough resolution, the range of posterior values is the deterministic range of the quantity of interest, irrespective of the size of the data. Continue reading