$$ However, the link is helpful. Is there a way to smoothly increase the density of points in a volume using the 'Distribute points in volume' node? The final line applies the definition of Jeffreys prior on $\varphi{(\theta)}$. Since that time he has been in private practice concentrating in Criminal Defense and Driver License Restorations. but mostly it's because it's really unclear exactly what's being sought, which is why I wanted to express it as a functional equation in the first place. When we drop the bars, we can cancel $h'^{-1}$ and $h'$, giving, $$ \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi = \int_{a}^{b}p_{\theta}(\theta) d\theta$$, $$ P(a \le \theta \le b) = P(h(a) \le \phi \le h(b))$$, Now, we need to show that a prior chosen as the square root of the Fisher Information admits this property. If we take $\theta(\phi)$ as a function of $\phi$, then, $$ I will add some clarifications to my answer regarding your question about the invariance depending on the likelihood to my answer. To me the term "invariant" would seem to imply something along the lines of for any arbitrary smooth monotonic transformation $h$. The key point is we want the following: If $\phi = h(\theta)$ for a monotone transformation $h$, then: $$P(a \le \theta \le b) = P(h(a) \le \phi \le h(b))$$. It is trivial to define an. Regarding your edit, that's not right. What Jeffreys provides is a prior construction method $M$ which has this property. Having come back to this question and thought about it a bit more, I believe I have finally worked out how to formally express the sense of "invari for any smooth function $\varphi(\theta)$. The use of these "Uninformative priors" is completely problem-dependent and not a general method of forming priors. \end{align*}, $\frac{d \log p(y|\theta(\phi))}{d \theta}$, Understanding the Proof for why Jeffreys' prior is invariant, Moderation strike: Results of negotiations, Our Design Vision for Stack Overflow and the Stack Exchange network, Parametrisation invariance/covariance of the Jeffreys prior, Compute $\pi(H_0|x)$ with Jeffreys prior for a family $N(\theta,1)$, Obtaining Jeffreys prior by taking the limit of a particular prior density on $(\mu, \Sigma)$, Understanding definition of informative and uninformative prior distribution, Significance of parameterisation invariance of Jeffreys prior, Jeffreys' prior invariance under reparametrization. Invariance of Posterior Distributions under Reparametrization About Jeffrey Clothier Thanks for contributing an answer to Cross Validated! & \propto & \frac{1}{| \varphi' (\theta) |} p (\theta) p (y| \theta)\\ This explanation is of course restricted to the unidimensional case. &= \frac{d}{d\phi} \left( \frac{d \log p(y|\theta(\phi))}{d \theta} \frac{d\theta}{d\phi} \right) \tag{chain rule}\\ $$ But using the "Principle of Indifference" violates this. rule} \\ &= \left(\frac{d^2 \log p(y|\theta(\phi))}{d \theta^2 }\right)\left( \frac{d\theta}{d\phi}\right)^2 + \left(\frac{d \log p(y|\theta(\phi))}{d \theta}\right) \left( \frac{d^2\theta}{d\phi^2}\right) \tag{chain rule} This should be posted as a comment rather than an answer, since it is not an answer. Clearly something is invariant here, and it seems like it shouldn't be too hard to express this invariance as a functional equation. Ask Question Asked 10 years, 10 months ago Modified 10 months ago Viewed 8k times 19 I've been trying to understand the motivation Your answer is really clear, but I think is not quite there yet. We denote by $\mathrm M^1(\Omega,\mathcal A)$ the set of probability measures on $(\Omega,\mathcal A)$ and by $\mathrm M^\sigma(\Omega,\mathcal A)$ the space of all $\sigma$-finite measures on $(\Omega,\mathcal A)$. Connect and share knowledge within a single location that is structured and easy to search. In what sense is the Jeffreys prior invariant? . $$ On the other hand, if this is not the case then the Jeffreys prior does have a special property, in that it's the only prior that can be produced by a prior generating method that is invariant under parameter transformations. Properties and Implementation of Jeffreys's Prior in Binomial It would therefore seem rather valuable to find a proof that Jeffrey's prior construction method is unique in having this invariance principle, or an explicit counterexample showing that it is not. Thanks for the hints. Webstatistics - In what sense is the Jeffreys prior invariant? He is a retained attorney that does not do any court-appointed work so he has the time, experience, expertise and attitude to fight for you. (Say they were reasoning in terms of log-odds ratios). Harold Jeffreyss default Bayes factor hypothesis tests: Explanation Is the following parametrizations identifiable? How to combine uparrow and sim in Plain TeX? The property of "Invariance" does not necessarily mean that the prior distribution is Invariant under "any" transformation. By the way, I don't want to seem obstinate. Hi! For the binomial regression model, Jeffreys's Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. \int_{\theta_1}^{\theta_2} \rho(\theta) d \theta = - Mathematics Stack Exchange In what sense is the Jeffreys prior invariant? \begin{align*} However, regardless what likelihood you use, the invariance will hold through. I now want to show that, given any desired prior, there exists an equivariant method on a very large set $X$ producing this prior. $\mathsf P_\theta\in\mathrm M^1(\Omega,\mathcal A)$, $X\subset \mathrm M^1(\Omega,\mathcal A)^\Theta$, $(\mathsf P_\theta)_{\theta\in\Theta}\in X\implies (\mathsf P_{h(\theta)})_{\theta\in\Theta}\in X$, \begin{align*}\rho: X&\to \mathrm M^\sigma(\Theta, \mathcal B(\Theta))\\ (\mathsf P_\theta)_{\theta\in\Theta}&\mapsto\rho[(\mathsf P_\theta)_{\theta\in\Theta}]\end{align*}, $$h_\# \rho[(\mathsf P_{h(\theta)})_{\theta\in\Theta}] = \rho[(\mathsf P_\theta)_{\theta\in\Theta}]$$, $f_\theta=\frac{\mathrm d\mathsf P_\theta}{\mathrm d\nu}$, $\frac{\partial^2}{\partial\theta^2}\ln f_\theta\in L^1(\Omega,\mathcal A, \mathsf P_\theta)$, $\mathrm M^\sigma(\Theta, \mathcal B(\Theta))$, $\rho[(\mathsf P_\theta)_{\theta\in\Theta}]$, $\mathsf P_{\theta}=\mathsf P_{\vartheta}$, $\rho[(\mathsf P_\theta)_{\theta\in\Theta}]=0$, $p\in\mathrm M^\sigma(\Theta,\mathcal B(\Theta))$, $$\rho:X\to\mathrm M^{\sigma}(\Theta,\mathcal B(\Theta))$$, $$\rho[(\mathsf P_\theta)_{\theta\in\Theta}] =\begin{cases}h^{-1}_\# p, &\text{ if }(\mathsf P_{\theta})_{\theta\in\Theta}=(\mathrm Q_{h(\theta)})_{\theta\in\Theta} \text{ for some bijective }h\in C^\infty(\Theta;\Theta)\\0,&\text{otherwise}. Jeffrey grew up in Flint, attended Flint Public Schools and graduated from Flint Southwestern High School in 1987. Concentrated only in the area of Criminal Defense, he has handled cases as simple as a Speeding Ticket and as complex as First Degree Murder. How to cut team building from retrospective meetings? I suggest to start with $\varphi(\theta)=2\theta$ and $\varphi(\theta)=1-\theta$. His justication was one of While at the Prosecutor's Office, Jeff found his work in protecting To read the Wikipedia argument as a chain of equalities of unsigned volume forms, multiply every line by $|d\varphi|$, and use absolute value of all determinants, not the usual signed determinant. $$ United States Federal Court in the Eastern District of Michigan, 2000, United States Federal Court in the Western District of Michigan, 2007, TOP 10 Criminal Law Attorney for Michigan by American Jurist Institute, TOP 100 Trial Lawyer recognized by the National Trial Lawyers Association, 10 Best Attorney Client Satisfaction American Institute Criminal Attorneys, TOP 100 OWI Attorney recognized by National Advocacy for DUI Defense, Distinguished High Legal Ability and Ethical Standards-Martindale Hubbell. WebUnderstanding the Proof for why Jeffreys' prior is invariant Ask Question Asked 6 years, 6 months ago Modified 4 years, 7 months ago Viewed 2k times 5 I was reviewing the We saw previously that a at prior () 1 does not have this property. The first line is only applying the formula for the jacobian when transforming between posteriors. Why do people generally discard the upper portion of leeks? p (\varphi (\theta) |y) & = & \frac{1}{| \varphi' (\theta) |} p (\theta TV show from 70s or 80s where jets join together to make giant robot. are the constants of proportionality the same in the two equations above, or different? & \propto & \sqrt{I (\varphi (\theta))} |p (y| \theta)\\ To reiterate my question, I understand the above equations from Wikipedia, and I can see that they demonstrate an invariance property of some kind. : () = 1 c c (12) for all c > 0. In (i), it is $\pi$. Whatever priors they use must be completely uninformative about the scaling of time between the events. rev2023.8.22.43591. Here the argument used by Laplace was that he saw no difference in considering any value p$_1$ over p$_2$ for the probability of the birth of a girl. Added an updated explanation in an edit. However, I can't see how to express this invariance property in the form of a functional equation similar to $(ii)$, which is what I'm looking for as an answer to this question. By the transformation of variables formula, $$p_{\phi}(\phi) = p_{\theta}( h^{-1} (\phi)) \Bigg| \frac{d}{d\phi} h^{-1}(\phi) \Bigg| $$. The third line applies the relationship between the information matrices. So there must be some other sense intended by "invariant" in this context. However, none of them then go on to show that such a prior is indeed invariant, or even to properly define what was meant by "invariant" in the first place. Notation. (+1) Your answer is perhaps one of the cleareast I've found so far, together with the lecture you mention. In this case the Jeffreys prior is given by WebThe Jereys Prior Uniform priors and invariance Recall that in his female birth rate analysis, Laplace used a uniform prior on the birth rate p2[0;1]. p (\varphi (\theta) ) & = & \frac{1}{| \varphi' (\theta) |} p (\theta Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. The prior does not lose the information. About Jeffrey R. Saunders, JD, CFP I think I found out why I considered them the same, Jaynes in his book refers only to the (dv/v) rule and it's consequences as Jeffreys' priors. That seems to be an open-ended question full of debates. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. This is ensured by the use of Jeffrey's prior which is completely scale and location-invariant. Asking for help, clarification, or responding to other answers. WebThe prior on should be invariant to rescaling by any arbitrary positive constant, i.e. That is where this "Invariance" comes into the picture. Site content copyright Jeffrey Clothier unless otherwise specified. Here $| \varphi' (\theta) |$ is the inverse of the jacobian of the transformation. Fix now any "privileged" family of distributions $(\mathrm Q_\theta)_{\theta\in\Theta}$ (in the language of Bayesians this would be a "privileged parametrization") and the "privileged" prior $p\in\mathrm M^\sigma(\Theta,\mathcal B(\Theta))$ that you want to obtain. For the [0,1] interval he supports the square root dependant term stating that the weights over 0 and 1 are too high in the former distribution making the population biased over these 2 points only. for any (smooth, differentiable) function $\varphi$ -- but it's easy enough to see that this is not satisfied by the distribution $(i)$ above (and indeed, I doubt there can be any density function that does satisfy this kind of invariance for any transformation). Finally, whatever the thing that's invariant is, it must surely depend in some way on the likelihood function! WebWe want to choose a prior () that is invariant under reparameterizations. What is the word used to describe things ordered by height? By clicking Post Your Answer, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct. Where is the proof of uniqueness?) Use MathJax to format equations. Your answer, @N. Virgo, has greatly improved my understanding of what the Jeffreys prior is and in what sense the word "invariant" is used. Plotting Incidence function of the SIR Model. Another trivial choice is $X=\emptyset$ and $\rho$ to be the empty map, however, this choice is also not at all useful or interesting. |y)\\ Lets derive a Jereys How to combine uparrow and sim in Plain TeX? What's the meaning of "Making demands on someone" in the following context? If $h$ is increasing, then $h'$ is positive and we don't need the absolute value. Say that we have 2 experimenters who aim to find out the number of events that occurred in a specific time (Poisson dist.). This is of course undesired (we want to generate any desired prior, not just $0$) and it doesn't seem very useful in practical problems to have multiple distinct parameters with the same probability distribution assigned to them. SELECTIVE SEROTONIN Invariance Property - University of South Carolina The Jeffreys prior is proportional to the square root of We now define $$\rho:X\to\mathrm M^{\sigma}(\Theta,\mathcal B(\Theta))$$ as $$\rho[(\mathsf P_\theta)_{\theta\in\Theta}] =\begin{cases}h^{-1}_\# p, &\text{ if }(\mathsf P_{\theta})_{\theta\in\Theta}=(\mathrm Q_{h(\theta)})_{\theta\in\Theta} \text{ for some bijective }h\in C^\infty(\Theta;\Theta)\\0,&\text{otherwise}. (I will let you verify this by deriving the information from the likelihood. This "Invariance" is what is expected of our solutions. In fact the desired invariance is a property of $M$ itself, rather than of the priors it generates. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. \begin{eqnarray*} Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. \begin{eqnarray*} The preference for Jeffreys form of invariant prior is based on other considerations. We then define Jeffreys prior (not-normalized) $\rho[(\mathsf P_\theta)_{\theta\in\Theta}]$ as the measure over $\Theta$ whose density with respect to the Lebesgue measure $\lambda$ is the square root of the Fisher information, i.e. Amazing answer. RESPIRATORY AND CNS STIMULANTS. Perhaps I can, but it seems not at all trivial to me. zyx's answer is excellent but it uses differential forms. Why do people generally discard the upper portion of leeks? Is DAC used as stand-alone IC in a circuit? My key stumbling point seems to be that the phrase "the Jeffreys prior is invariant" is incorrect - the invariance in question is not a property of any given prior, but rather it's a property of a method of constructing priors from likelihood functions. The only difference is that the second line applies Bayes rule. Maybe the problem is that you are forgetting the jacobian of the transformation in (ii). \theta)\\ I agree with William Huber. First we show a probability density for which this is satisfied. I For a single parameter and data having joint density f(x|), the & = & \frac{1}{| \varphi' (\theta) |} \sqrt{I (\theta)} \\ Why is there no funding for the Arecibo observatory, despite there being funding in the past? Was Hunter Biden's legal team legally required to publicly disclose his proposed plea agreement? Connect and share knowledge within a single location that is structured and easy to search. Securing Cabinet to wall: better to use two anchors to drywall or one screw into stud? Jereys priors - University of California, Berkeley Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. A trivial choice is $X=\mathrm M^1(\Omega,\mathcal A)$ and $\rho=0$, because the measure assigning $0$ to all measurable sets is invariant under push-forward by any map. INTRODUCTION Jeffreys's prior is perhaps the most widely used noninformative prior in Bayesian analysis. What is the best way to say "a large number of [noun]" in German? If $h$ is decreasing, then $h(b) < h(a)$, which means the integral gets a minus in front of it. $$ &= \left(\frac{d^2 \log p(y|\theta(\phi))}{d \theta d\phi}\right)\left( \frac{d\theta}{d\phi}\right) + \left(\frac{d \log p(y|\theta(\phi))}{d \theta}\right) \left( \frac{d^2\theta}{d\phi^2}\right) \tag{prod. What we seek is a construction method $M$ with the following property: (I hope I have expressed this correctly) WebANOREXIGENIC AGENTS, MISCELLANEOUS. We will derive the prior on $\phi$, which we'll call $p_{\phi}(\phi)$. While his office is located in Flint, Michigan his reputation for obtaining outstanding results has led him to almost every county in the State of Michigan. Definition. where $\theta$ is the parameterisation given by $p_1 = \theta$, $p_2 = 1-\theta$. Since, as you say, $p(\varphi)d\varphi \equiv p(\theta)d\theta$ is an identity, it holds for every pdf $p(\theta)$, not just the Jeffreys prior. My own party belittles me as a player, should I leave? Invariant Properties of Probability Distributions? The goal of this answer is to provide a rigorous mathematical framework of the "invariance" property and to show that the prior obtained by Jeffreys method is not unique. Understanding why the Uniform distribution does not make a good prior. Why not say ? The lack of evidence to reject the H0 is OK in the case of my research - how to 'defend' this in the discussion of a scientific paper? The clearest answer I have found (ie, the most blunt "definition" of invariance) was a comment in this Cross-Validated thread , which I combined w Then, start with some simple examples of some monotonic transformations in order to see the invariance. The problem is not that I don't understand those equations. 1 PMID: 19436775 PMCID: PMC2680313 DOI: 10.1198/016214508000000779 We study several theoretical properties of Jeffreys's prior for binomial regression models. Computationally it is expressed by Jacobians but only the power-of-$A$ dependences matter and having those cancel out on multiplication. The problem here is about the apparent "Principle of Indifference" considered by Laplace. For the distribution $f_\theta (x) = \theta x^{\theta-1}$, what is the sufficient statistic corresponding to the Monotone Likelihood Ratio? Invariance under parameter transformation with the \end{eqnarray*} By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. My problem arose from looking at a particular example of a prior constructed by Jeffreys' method (i.e. Indeed this equation links the information of the likelihood to the information of the likelihood given the transformed model. How can select application menu options by fuzzy search using my keyboard only? Because changes of coordinate alter $dV$, an invariant prior has to depend on more than $p(\theta)$. &= \int_{a}^{b} p_{\theta}(\theta) \Bigg| h'(\theta) \Bigg|^{-1} h'(\theta) d\theta, In the above case, the prior is telling us that "I don't want to give one value p$_1$ more preference than another value p$_2$" and it continues to say the same even on transforming the prior. Those equations (quoted from Wikipedia) omit the Jacobian because they refer to the case of a binomial trial, where there is only one variable and the Jacobian of $I$ is just $I$. Do Federal courts have the authority to dismiss charges brought in a Georgia Court? Suppose there was an alien race that wanted to do the same analysis as done by Laplace. = \frac{d}{d\phi} \left( \frac{d \log p(y\mid\theta(\phi))}{d \theta} \frac{d\theta}{d\phi} \right) That is, we can either apply $h$ to transform the likelihood function and then use $M$ to obtain a prior, or we can first use $M$ on the original likelihood function and then transform the resulting prior, and the end result will be the same. The second line applies the definition of Jeffreys prior. What happens if you connect the same phase AC (from a generator) to both sides of an electrical panel? $$ So they will use the $\lambda^{-1}d\lambda$ prior, the Jeffrey's prior (because it is the only general solution in the one-parameter case for scale-invariance). The following lecture notes were helpful in coming to this conclusion, as they contain an explanation that is clearer than anything I could find at the time of writing the question: https://www2.stat.duke.edu/courses/Fall11/sta114/jeffreys.pdf. Illustrate the invariance property of a noninformative prior. Jeffrey Clothier immediately entered law school and graduated from Michigan State University College of Law in 1995. Having come back to this question and thought about it a bit more, I believe I have finally worked out how to formally express the sense of "invariance" that applies to Jeffreys' priors, as well as the logical issue that prevented me from seeing it before. & = & p (\varphi (\theta)) Confirming my understanding of posterior, marginal, and conditional distributions. ANOREXIGENICS;RESPIRATORY,CNS STIMULANTS. During law school, Clothier was the Judicial Law Clerk But nonetheless, we can make sure that our priors are at least uninformative in some sense. Locally the Fisher matrix $F$ transforms to $(J^{-1})^TFJ^{-1}$ under a change of coordinates with Jacobian $J$, and $\sqrt{\det}$ of this cancels the multiplication of volume forms by $\det J$. Jeffrey E. Clothier has been practicing law in the State of Michigan for the past twenty-four years and is the criminal defense attorney that Genesee County residents trust most. (Note that these equations omit taking the Jacobian of $I$ because they refer to a single-variable case.) But whatever we estimate from our priors and the data must necessarily lead to the same result. $\rho$ satisfies the equivariance property by construction. Just use the chain rule after applying the definition of the information as the expected value of the square of the score). That is, we want something that will take a likelihood function and give us a prior for the parameters, and will do it in such a way that if we take that prior and then transform the parameters, we will get the same result as if we first transform the parameters and then use the same method to generate the prior. However, the more I try to do this the more confused I get. : your link is broken, I think you mean this one: @thc I've fixed the link. The Jeffreys prior is a product of two locally defined quantities one of which scales by $\sqrt{A^{-2}}$ and the other by $A$ where $A(\theta)$ is a local factor that depends on $\theta$ and on the coordinate transformation. )\\ University of Michigan Making statements based on opinion; back them up with references or personal experience. = \frac{d^2 \log p(y\mid\theta(\phi))}{d \theta^2} \left|\frac{d\theta}{d\phi} \right|^2 How can you spot MWBC's (multi-wire branch circuits) in an electrical panel. Dr. Getzinger practices primary care and preventative p(\varphi)\propto\sqrt{I(\varphi)} This happens through the relationship $ \sqrt{I (\theta)} = \sqrt{I (\varphi (\theta))} | \varphi' (\theta) | $. One thing I would like to note that if you look at the proof for this invariance, it is only important that we have the variance of a (differentiable) function of the density function of the sampling distribution. STA 114: Statistics Notes 12. The Je reys Prior - Duke University Also, to answer your question, the constants of integration do not matter here. This paper considers a generalization of the connection between Jeffreys prior and the Kullback-Leibler divergence as a procedure for generating a wide class of Why is the town of Olivenza not as heavily politicized as other territorial disputes? This is genuinely very helpful, and I'll go through it very carefully later, as well as brushing up on my knowledge of Jacobians in case there's something I've misunderstood. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. $$. What is the intuition or motivation about Translation-invariant priors? Now, according to this Wikipedia page, the derivative the inverse gives: $$p_{\phi}(\phi) = p_{\theta}( h^{-1} (\phi)) \Bigg| h'(h^{-1}(\phi)) \Bigg|^{-1} $$, We will write this in another way to make the next step clearer. I'm fairly certain it's a logical point that I'm missing, rather than something to do with the formal details of the mathematics. The best answers are voted up and rise to the top, Not the answer you're looking for? But still, it seems like having a better understanding of how to go from $p(\theta)$ to $p(\varphi(\theta))$ isn't automatically giving me a grasp of what the "XXX" is. What I want is to see a definition of the sought invariance property that. Though his prior was perfectly alright, the reasoning used to arrive at it was at fault. When this property of "Uninformativeness" is needed, we seek priors that have invariance of a certain type associated with that problem. the equations are between densities $p(x) dx$, but written as though for the density functions $p()$ that define the priors. It only takes a minute to sign up. WebThe Jeffreys prior probability density for a set of parameters = (1, . Properties and Implementation of Jeffreys's Prior in Binomial To learn more, see our tips on writing great answers. Level of grammatical correctness of native German speakers. & = & \sqrt{I (\varphi (\theta))} \\ & \propto & \frac{1}{| \varphi' (\theta) |} \sqrt{I (\theta)} p (y| The timescale invariance problem is also mentioned there.). What is invariant is the volume density $|p_{L_{\theta}}(\theta) dV_{\theta}|$ where $V_\theta$ is the volume form in coordinates $\theta_1, \theta_2, \dots \theta_n$ and $L_\theta$ is the likelihood parametrized by $\theta$. Properties and Implementation of Jeffreys's Prior in Binomial On a daily basis Jeffrey E. Clothier successfully defends clients on drunk driving offenses, all drug offenses, weapons assault, and domestic violence. $$ The best answers are voted up and rise to the top, Not the answer you're looking for? rule} \\ I would like to understand this sense in the form of a functional equation similar to $(ii)$, so that I can see how it's satisfied by $(i)$. (More info on this scale and location invariance can be found in Probability Theory the Logic of Science by E.T. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. In other words, on transforming the prior to a log-odds scale, the prior still says "See, I still consider no value of p1 to be preferable over another p2" and that is why the log-odds transform is not going to be flat. \begin{align*}\rho: X&\to \mathrm M^\sigma(\Theta, \mathcal B(\Theta))\\ (\mathsf P_\theta)_{\theta\in\Theta}&\mapsto\rho[(\mathsf P_\theta)_{\theta\in\Theta}]\end{align*} satisfying the equivariance property I was reviewing the section of Andrew Gelman's "Bayesian Data Analysis" on uninformative priors, and came across this explanation for why Jeffreys' prior is invariant to parameterization. Look again at what happens to the posterior ($y$ is obviously the observed sample here)
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