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. \frac{d^2\log p(y | \phi)}{d\phi^2} \frac{d^2\log p(y \mid \phi)}{d\phi^2} , n) is given by () qdet I() where the matrix I is the Fisher information, defined by Iij() = E ln Most texts I've read online make some comment to the effect that the Jeffreys prior is "invariant with respect to transformations of the parameters", and then go on to state its definition in terms of the Fisher information matrix without further motivation. 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. Rufus settings default settings confusing. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. The second line applies the definition of Jeffreys prior. Another trivial choice is $X=\emptyset$ and $\rho$ to be the empty map, however, this choice is also not at all useful or interesting. \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 & \propto & \frac{1}{| \varphi' (\theta) |} p (\theta) p (y| \theta)\\ Definition. This "Invariance" is what is expected of our solutions. Indeed this equation links the information of the likelihood to the information of the likelihood given the transformed model. Do Federal courts have the authority to dismiss charges brought in a Georgia Court? 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]. 503 South Saginaw Street, Suite 929 Flint Mi. You also need the product rule: \begin{align*} \end{eqnarray*} $\rho$ satisfies the equivariance property by construction. : () = 1 c c (12) for all c > 0. P(h(a)\le \phi \le h(b)) &= \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi\\ Understanding the Proof for why Jeffreys' prior is invariant \begin{eqnarray*} Also, it would help me a lot if you could expand on the distinction you make between "densities $p(x) dx$" and "the. WebANOREXIGENIC AGENTS, MISCELLANEOUS. $$. & \propto & \sqrt{I (\varphi (\theta))} |p (y| \theta)\\ \int_{\theta_1}^{\theta_2} \rho(\theta) d \theta = When we drop the bars, we can cancel $h'^{-1}$ and $h'$, giving, $$ \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))$$, Now, we need to show that a prior chosen as the square root of the Fisher Information admits this property. I will add some clarifications to my answer regarding your question about the invariance depending on the likelihood to my answer. $$\frac{\mathrm d\rho[(\mathsf P_\theta)_{\theta\in\Theta}]}{\mathrm d\lambda} &= \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} to finally come to an understanding. Jeff began his legal career in 1987 as an Assistant Prosecutor in Oakland County, Michigan. As did points out, the Wikipedia article gives a hint about this, by starting with Partly this is because there's just a lot left out of the Wikipedia sketch (e.g. Say that we have 2 experimenters who aim to find out the number of events that occurred in a specific time (Poisson dist.). Added an updated explanation in an edit. It is perfectly alright for them to do so because each and every problem of ours can be translated to their terms and vice-versa as long as the transform is a bijection. On applying the (dv/v) rule on the positive semi-infinite interval, we get the 1/p(1-p) dependence which Jeffreys accepts only for the semi-infinite interval. 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. About Jeffrey R. Saunders, JD, CFP RESPIRATORY AND CNS STIMULANTS. 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. We now define $$\rho:X\to\mathrm M^{\sigma}(\Theta,\mathcal B(\Theta))$$ as $$\rho[(\mathsf P_\theta)_{\theta\in\Theta}] =\begin{cases}h^{-1}_\# p, &\text{ if }(\mathsf P_{\theta})_{\theta\in\Theta}=(\mathrm Q_{h(\theta)})_{\theta\in\Theta} \text{ for some bijective }h\in C^\infty(\Theta;\Theta)\\0,&\text{otherwise}. $$ \begin{align*} The use of these "Uninformative priors" is completely problem-dependent and not a general method of forming priors. If we take $\theta(\phi)$ as a function of $\phi$, then, $$ What you need for Bayesian statistics (resp., likelihood-based methods) is the ability to integrate against a prior (likelihood), so really $p(x) dx$ is the object of interest. Site content copyright Jeffrey Clothier unless otherwise specified. re the second comment, the distinction is between functions and differential forms. Invariance under parameter transformation with the 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. Dr. Jef Getzinger, MD - Family Medicine | Cornerstone Medical Group Why do people generally discard the upper portion of leeks? Learn more about Stack Overflow the company, and our products. &= \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. Ask Question Asked 10 years, 10 months ago Modified 10 months ago Viewed 8k times 19 I've been trying to understand the motivation Invariance Property - University of South Carolina This happens through the relationship $ \sqrt{I (\theta)} = \sqrt{I (\varphi (\theta))} | \varphi' (\theta) | $. This means some local finite dimensional linear space of differential quantities at each point with linear maps between the before- and after- coordinate change spaces. Invariance of Posterior Distributions under Reparametrization Sorry but I absolutely completely do not care the least about bounties and points. The following ones are the derivation of that equation. Which part of the question is not dealt with. are the constants of proportionality the same in the two equations above, or different? M\{ f(x\mid h(\theta)) \} = M\{ f(x \mid \theta) \}\circ h, The problem here is about the apparent "Principle of Indifference" considered by Laplace. On a daily basis Jeffrey E. Clothier successfully defends clients on drunk driving offenses, all drug offenses, weapons assault, and domestic violence. Invariant Properties of Probability Distributions? Properties and Implementation of Jeffreys's Prior in Binomial I still think that your problem is with jacobians and the fact that the formula (ii) is correct for the special case I does not make correct in general. When in {country}, do as the {countrians} do, Any difference between: "I am so excited." Whatever priors they use must be completely uninformative about the scaling of time between the events. How to cut team building from retrospective meetings? The best answers are voted up and rise to the top, Not the answer you're looking for? I For a single parameter and data having joint density f(x|), the Securing Cabinet to wall: better to use two anchors to drywall or one screw into stud? I now want to show that, given any desired prior, there exists an equivariant method on a very large set $X$ producing this prior. 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. 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. Webstatistics - In what sense is the Jeffreys prior invariant? 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$. 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. Now, for the prior. $$ This shows that the invariant prior is very non-unique as there are many other ways to achieve the cancellation. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. &= \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} Asking for help, clarification, or responding to other answers. In fact, I will show that for any desired prior one can construct an "invariant" method that produces this prior. 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. Clearly something is invariant here, and it seems like it shouldn't be too hard to express this invariance as a functional equation. 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. Throughout this answer we fix a measurable space $(\Omega,\mathcal A)$, as well as a parameter space $\Theta\subset\mathbb R$, that, for simplicity, I assume to be an interval (the arguments here should also work for more general parameter spaces and the reader is invited to repeat them in a more general setting). Since that time he has been in private practice concentrating in Criminal Defense and Driver License Restorations. Dr. Getzinger practices primary care and preventative Learn more about Stack Overflow the company, and our products. 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 the binomial regression model, Jeffreys's By the transformation of variables formula, $$p_{\phi}(\phi) = p_{\theta}( h^{-1} (\phi)) \Bigg| \frac{d}{d\phi} h^{-1}(\phi) \Bigg| $$. Regarding your edit, that's not right. \end{cases}$$, This notion of "uninformative prior" is a different thing from Jeffreys priors though, isn't it? 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. p (\varphi (\theta) |y) & = & \frac{1}{| \varphi' (\theta) |} p (\theta It simply amounts to the chain rule of calculus. (I will let you verify this by deriving the information from the likelihood. WebDr. 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. 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. I'm not sure I understand what you mean in your other comment, though - could you spell your counterexample out in more detail? To reiterate my question, I understand the above equations from Wikipedia, and I can see that they demonstrate an invariance property of some kind. Is DAC used as stand-alone IC in a circuit? 1 Jereys Priors - University of California, Berkeley 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. = \frac{d^2 \log p(y\mid\theta(\phi))}{d \theta^2} \left|\frac{d\theta}{d\phi} \right|^2 where $\theta$ is the parameterisation given by $p_1 = \theta$, $p_2 = 1-\theta$. Your answer, @N. Virgo, has greatly improved my understanding of what the Jeffreys prior is and in what sense the word "invariant" is used. Lets derive a Jereys & = & \frac{1}{| \varphi' (\theta) |} \sqrt{I (\theta)} \\ the function $M\{ f(x\mid \theta )\}$ for some particular likelihood function $f(x \mid \theta)$) and trying to see that it has some kind of invariance property. |y)\\ University of Michigan What is the word used to describe things ordered by height? (Note that these equations omit taking the Jacobian of $I$ because they refer to a single-variable case.) 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). This is in particular true for Jeffreys method: If $\mathsf P_\theta$ doesn't depend on $\theta$, then neither does $f_\theta$ and therefore the Fisher information is always equal to $0$. 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. p (\varphi (\theta) ) & = & \frac{1}{| \varphi' (\theta) |} p (\theta \end{eqnarray*} This proof is clearly laid out in these lecture notes. 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))$$. I like to understand things by approaching the simplest example first, so I'm interested in the case of a binomial trial, i.e. This is of course undesired (we want to generate any desired prior, not just $0$) and it doesn't seem very useful in practical problems to have multiple distinct parameters with the same probability distribution assigned to them. Making statements based on opinion; back them up with references or personal experience. the equations are between densities $p(x) dx$, but written as though for the density functions $p()$ that define the priors. Let me know if you are stuck somewhere. rev2023.8.22.43591. What I'm looking for is something. p(\theta)\propto\sqrt{I(\theta)} I was reviewing the section of Andrew Gelman's "Bayesian Data Analysis" on uninformative priors, and came across this explanation for why Jeffreys' prior is invariant to parameterization. : your link is broken, I think you mean this one: @thc I've fixed the link. &= \frac{d}{d\phi} \left( \frac{d \log p(y|\theta(\phi))}{d \theta} \frac{d\theta}{d\phi} \right) \tag{chain rule}\\ The timescale invariance problem is also mentioned there.). It is trivial to define an. $\mathsf P_\theta\in\mathrm M^1(\Omega,\mathcal A)$, $X\subset \mathrm M^1(\Omega,\mathcal A)^\Theta$, $(\mathsf P_\theta)_{\theta\in\Theta}\in X\implies (\mathsf P_{h(\theta)})_{\theta\in\Theta}\in X$, \begin{align*}\rho: X&\to \mathrm M^\sigma(\Theta, \mathcal B(\Theta))\\ (\mathsf P_\theta)_{\theta\in\Theta}&\mapsto\rho[(\mathsf P_\theta)_{\theta\in\Theta}]\end{align*}, $$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}. Just use the chain rule after applying the definition of the information as the expected value of the square of the score). As I explained earlier in the comments, it is essential to understand how jacobians work (or differential forms). Jef Getzinger is an ABMS board-certified doctor of Family Medicine working at Schoenherr Family Practice. The property of "Invariance" does not necessarily mean that the prior distribution is Invariant under " any " transformation. To make sure that we &= \int_{a}^{b} p_{\theta}(\theta) \Bigg| h'(\theta) \Bigg|^{-1} h'(\theta) d\theta, al. 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. It says that there is some prior information which is why this transformed pdf is not flat. 1. To me the term "invariant" would seem to imply something along the lines of for any smooth function $\varphi(\theta)$. \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi &= \int_{a}^{b} p_{\phi}(h(\theta)) h'(\theta) d\theta\\ Properties and Implementation of Jeffreys's Prior in Binomial . 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 $I$. I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. & = & p (\varphi (\theta)) I suggest to start with $\varphi(\theta)=2\theta$ and $\varphi(\theta)=1-\theta$. Say if the aliens used the same principle, they would definitely arrive at a different answer than ours. However, regardless what likelihood you use, the invariance will hold through. By the way, I don't want to seem obstinate. Thanks for the hints. Example. It is natural to ask for something local on the parameter space, so the invariant prior will be built from a finite number of derivatives of the likelihood evaluated at $\theta$. \end{aligned}, Now, we can drop the absolute value bars around $h'(\theta)$. First, it should be noted that if, for example, $\mathsf P_{\theta}=\mathsf P_{\vartheta}$ for all $\theta,\vartheta\in\Theta$, then we must have $\rho[(\mathsf P_\theta)_{\theta\in\Theta}]=0$. \begin{eqnarray*} Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 1 PMID: 19436775 PMCID: PMC2680313 DOI: 10.1198/016214508000000779 We study several theoretical properties of Jeffreys's prior for binomial regression models. This is simply because $0$ is the only $\sigma$-finite measure that remains unchanged when being pushforwarded by any smooth bijective map (actually I should prove this statement but I believe this is true). TV show from 70s or 80s where jets join together to make giant robot. WebWe also show that the prior and posterior normalizing constants under Jeffreys's prior are linear transformation-invariant in the covariates. Jeffrey's prior has only this type of invariance in it, not to all transforms (Maybe some others too, but not all for sure). First we show a probability density for which this is satisfied. So there must be some other sense intended by "invariant" in this context. M\{ f(x\mid h(\theta)) \} = M\{ f(x \mid \theta) \}\circ h, Perhaps I can, but it seems not at all trivial to me. What Jeffreys provides is a prior construction method $M$ which has this property. &= \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. WebThe prior on should be invariant to rescaling by any arbitrary positive constant, i.e. = \frac{d}{d\phi} \left( \frac{d \log p(y\mid\theta(\phi))}{d \theta} \frac{d\theta}{d\phi} \right) While at the Prosecutor's Office, Jeff found his work in protecting The invariance of $|p dV|$ is the definition of "invariance of prior". What is the intuition or motivation about Translation-invariant priors? The presentation in Wikipedia is confusing, because. The first line is only applying the formula for the jacobian when transforming between posteriors. \frac{d^2\log p(y | \phi)}{d\phi^2} This should be posted as a comment rather than an answer, since it is not an answer. The third line applies the relationship between the information matrices. 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). 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. You can see that the use of Jeffreys prior was essential for $\frac{1}{| \varphi' (\theta) |}$ to cancel out. Why do Airbus A220s manufactured in Mobile, AL have Canadian test registrations? This is ensured by the use of Jeffrey's prior which is completely scale and location-invariant. 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. 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 The best answers are voted up and rise to the top, Not the answer you're looking for? What I would like is to understand the sense in which this is invariant with respect to a coordinate transformation $\theta \to \varphi(\theta)$. Suppose there was an alien race that wanted to do the same analysis as done by Laplace. Determinants appear because there is a factor of $\det J$ to be killed from the change in $dV$, and because we will want the changes of the local quantities to multiply and cancel each other as is the case in Jeffreys prior, which practically requires a reduction to one dimension where the coordinate change can act on each factor by multiplication by a single number. Do the calculations with $\pi$ in there to see that point. \int_{\varphi(\theta_1)}^{\varphi(\theta_2)} \rho(\varphi(\theta)) d \varphi \qquad\qquad(ii) Plotting Incidence function of the SIR Model. \end{align*} By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. MathJax reference. Henceforth I will use the word equivariant instead of invariant since it is a better fit in my opinion. If $h$ is increasing, then $h'$ is positive and we don't need the absolute value. To give an attempt at fleshing this out, let's say that a "prior construction method" is a functional $M$, which maps the function $f(x \mid \theta)$ (the conditional probability density function of some data $x$ given some parameters $\theta$, considered a function of both $x$ and $\theta$) to another function $\rho(\theta)$, which is to be interpreted as a prior probability density function for $\theta$. (Say they were reasoning in terms of log-odds ratios). This explanation is of course restricted to the unidimensional case. the case where the support is $\{1,2\}$. Harold Jeffreyss default Bayes factor hypothesis tests: Explanation =\sqrt{-\int_{\Omega} \frac{\partial^2}{\partial\theta^2}\ln f_\theta(x)\,\mathrm d\mathsf P_\theta(x)}.$$. Here $| \varphi' (\theta) |$ is the inverse of the jacobian of the transformation. Edit: The dependence on the likelihood is essential for the invariance to hold, because the information is a property of the likelihood and because the object of interest is ultimately the posterior. What is invariant is the volume density $|p_{L_{\theta}}(\theta) dV_{\theta}|$ where $V_\theta$ is the volume form in coordinates $\theta_1, \theta Jeffrey Clothier immediately entered law school and graduated from Michigan State University College of Law in 1995. My problem arose from looking at a particular example of a prior constructed by Jeffreys' method (i.e. INTRODUCTION Jeffreys's prior is perhaps the most widely used noninformative prior in Bayesian analysis. I was looking for an invariance property that would apply to a particular prior generated using Jeffreys' method, whereas the desired invariance principle in fact applies to Jeffreys' method itself. & \propto & p (\varphi (\theta)) p (y| \theta) What we seek is a construction method $M$ with the following property: (I hope I have expressed this correctly) Amazing answer. I would like to understand this sense in the form of a functional equation similar to $(ii)$, so that I can see how it's satisfied by $(i)$. This choice is not at all useful or interesting. \theta)\\ & = & \sqrt{I (\varphi (\theta))} \\ Let $p_{\theta}(\theta)$ be the prior on $\theta$. $$ But let us say they were using some log scaled parameters instead of ours. 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. WebJeffrey Clothier immediately entered law school and graduated from Michigan State University College of Law in 1995. The only difference is that the second line applies Bayes rule. the first equality is a claim still to be proven. and deriving We therefore restrict our attention to $X\subset\mathrm M^1(\Omega,\mathcal A)$ containing all $(\mathsf P_\theta)_{\theta\in\Theta}$ such that $\theta\mapsto\mathsf P_\theta$ is an injective map. zyx's answer is excellent but it uses differential forms. 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. Is DAC used as stand-alone IC in a circuit? 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. Use MathJax to format equations. Look again at what happens to the posterior ($y$ is obviously the observed sample here) Your answer is really clear, but I think is not quite there yet. (More info on this scale and location invariance can be found in Probability Theory the Logic of Science by E.T. But whatever we estimate from our priors and the data must necessarily lead to the same result.
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