1 If $h$ is decreasing, then $h(b) < h(a)$, which means the integral gets a minus in front of it. So there must be some other sense intended by "invariant" in this context. When this property of "Uninformativeness" is needed, we seek priors that have invariance of a certain type associated with that problem. , Equivalently, What I'm looking for is something. 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. {\displaystyle 1-\gamma } Though his prior was perfectly alright, the reasoning used to arrive at it was at fault. Equivalently, the Jeffreys prior for Diff-in-diff parallel trends with a positive outcome. , each non-negative and satisfying [ 1 1 Perhaps I can answer this myself now, but if you'd like to post a proper answer detailing it then I'd be happy to award you the bounty. In particular, I remember him arguing in favour of an "uninformative" prior for a binomial distribution that's an improper prior proportional to $1/(p(1-p))$. The invariance of $|p dV|$ is the definition of "invariance of prior". is, This is the arcsine distribution and is a beta distribution with = 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. &= \int_{a}^{b} p_{\theta}(\theta) \Bigg| h'(\theta) \Bigg|^{-1} h'(\theta) d\theta, 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. T To make sure that we are on the same page, let us take the example of the "Principle of Indifference" used in the problem of Birth rate analysis given by Laplace. fixed, the Jeffreys prior for the mean {\displaystyle \varphi } ) P.S. 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. p {\displaystyle \mu } Asking for help, clarification, or responding to other answers. We call the prior By the transformation of variables formula, $$p_{\phi}(\phi) = p_{\theta}( h^{-1} (\phi)) \Bigg| \frac{d}{d\phi} h^{-1}(\phi) \Bigg| $$. For the Poisson distribution of the non-negative integer {\displaystyle n} and is "tails" with probability {\displaystyle \theta } , the Jeffreys prior for \begin{eqnarray*} the probability is {\displaystyle \alpha =\beta =1/2} log T N gives us the desired "invariance". ] $$ {\displaystyle \mu } / {\displaystyle \sigma >0} > & \propto & \frac{1}{| \varphi' (\theta) |} p (\theta) p (y| \theta)\\ where $\theta$ is the parameterisation given by $p_1 = \theta$, $p_2 = 1-\theta$. is the unnormalized uniform distribution on the real line, and thus this distribution is also known as the .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}logarithmic prior. that is, if the priors To use any other prior than this will have the consequence that a change in the time scale will lead to a change in the form of the prior, which would imply a different state of prior knowledge; but if we are completely ignorant of the time scale, then all time scales should appear equivalent. J Is it necessary to provide contact information for tens of co-authors when submitting a paper from a large collaboration? is. 0 is the unnormalized uniform distribution on the non-negative real line. be two possible parametrizations of a statistical model, with , then the Jeffreys prior for My answer is written as it is because yes, I believe. Can I get a clock signal from a 4-pin crystal oscillator circuit by applying 5V to the input pin? ( {\displaystyle p_{\theta }(\theta )} 2 Use MathJax to format equations. Partly this is because there's just a lot left out of the Wikipedia sketch (e.g. 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 presentation in Wikipedia is confusing, because. 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. is also uniform. As did points out, the Wikipedia article gives a hint about this, by starting with = From a practical and mathematical standpoint, a valid reason to use this non-informative prior instead of others, like the ones obtained through a limit in conjugate families of distributions, is that the relative probability of a volume of the probability space is not dependent upon the set of parameter variables that is chosen to describe parameter space. ( For a coin that is "heads" with probability Getting paid by mistake after leaving a company? & = & p (\varphi (\theta)) -sided die with outcome probabilities This seems to be rather an important question: if there is some other functional $M'$ that is also invariant and which gives a different prior for the parameter of a binomial distribution then there doesn't seem to be anything that picks out the Jeffreys distribution for a binomial trial as particularly special. Use of the Jeffreys prior violates the strong version of the likelihood principle, which is accepted by many, but by no means all, statisticians. 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. {\displaystyle \gamma \in [0,1]} . \end{aligned}, using the substitution formula from Wikipedia with $\phi = h(\theta)$, \begin{aligned} Yes, I think they are different. For example, the Jeffreys prior for the distribution mean is uniform over the entire real line in the case of a Gaussian distribution of known variance. i P(h(a)\le \phi \le h(b)) &= \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi\\ sin ) Also, it would help me a lot if you could expand on the distinction you make between "densities $p(x) dx$" and "the. 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. {\displaystyle \gamma =\sin ^{2}(\theta )} Maybe the problem is that you are forgetting the jacobian of the transformation in (ii). ( = This is an improper prior, and is, up to the choice of constant, the unique translation-invariant distribution on the reals (the Haar measure with respect to addition of reals), corresponding to the mean being a measure of location and translation-invariance corresponding to no information about location. For the Gaussian distribution of the real value {\displaystyle p_{\theta }({\vec {\theta }})\propto {\sqrt {\det I_{\theta }({\vec {\theta }})}}} , but also on the universe of all possible experimental outcomes, as determined by the experimental design, because the Fisher information is computed from an expectation over the chosen universe. Recalling that $\phi = h(\theta)$, we can write this as. are two possible parametrizations of a statistical model, and Let's say were working with the binomial distribution and two possible parameterizations: success probability (theta) and odds ratio (phi) where, Thanks for the hints. Then, start with some simple examples of some monotonic transformations in order to see the invariance. 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)$. My silicone mold got moldy, can I clean it or should I throw it away? ( When using the Jeffreys prior, inferences about 0 Estimation: what does it mean that the observable cancel from my equations? 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. does not depend upon ( [2], If & = & \sqrt{I (\varphi (\theta))} \\ ( Equivalently, the Jeffreys prior for H 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$. is a continuously differentiable function of ( 0 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. \int_{\varphi(\theta_1)}^{\varphi(\theta_2)} \rho(\varphi(\theta)) d \varphi \qquad\qquad(ii) {\displaystyle {\vec {\gamma }}} 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. 1 I want to first understand the desired invariance property, and then see that the Jeffrey's prior (hopefully uniquely) satisfies it, but the above equations mix up those two steps in a way that I can't see how to separate. I suggest to start with $\varphi(\theta)=2\theta$ and $\varphi(\theta)=1-\theta$. ) [ 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. i ( Where is the proof of uniqueness?) p Look again at what happens to the posterior ($y$ is obviously the observed sample here) 1 p If the full parameter is used a modified version of the result should be used. rev2022.8.1.42699. That is, the Jeffreys prior for Why won't this electromagnet home experiment work? / gives us the desired "invariance". Clearly something is invariant here, and it seems like it shouldn't be too hard to express this invariance as a functional equation. {\displaystyle {\vec {\gamma }}=(\gamma _{1},\ldots ,\gamma _{N})} \begin{aligned} Is the following parametrizations identifiable? ) . But using the "Principle of Indifference" violates this. The link given by the OP contains the problem statement in good detail. This should be posted as a comment rather than an answer, since it is not an answer. Illustrate the invariance property of a noninformative prior. This shows that the invariant prior is very non-unique as there are many other ways to achieve the cancellation. Repeat Hello World according to another string's length. In this case the Jeffreys prior is given by al. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. If $h$ is increasing, then $h'$ is positive and we don't need the absolute value. That is, the Jeffreys prior for \begin{eqnarray*} As with the uniform distribution on the reals, it is an improper prior. : your link is broken, I think you mean this one: @thc I've fixed the link. To reiterate my question, I understand the above equations from Wikipedia, and I can see that they demonstrate an invariance property of some kind. \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi &= \int_{a}^{b} p_{\phi}(h(\theta)) h'(\theta) d\theta\\ ; it is the unnormalized uniform distribution on the real line the distribution that is 1 (or some other fixed constant) for all points. p \end{eqnarray*} p {\displaystyle \varphi } and However, the link is helpful. 2 ) Because changes of coordinate alter $dV$, an invariant prior has to depend on more than $p(\theta)$. To me the term "invariant" would seem to imply something along the lines of , we say that the prior It only takes a minute to sign up. Understanding why the Uniform distribution does not make a good prior. {\displaystyle \mu } $$ I've read Jaynes' book, and quite a few of his papers on this topic, and I seem to remember him arguing. to finally come to an understanding. parameter even when the likelihood functions for the two experiments are the samea violation of the strong likelihood principle. ( 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. Also, to answer your question, the constants of integration do not matter here. The problem here is about the apparent "Principle of Indifference" considered by Laplace. This notion of "uninformative prior" is a different thing from Jeffreys priors though, isn't it? a continuously differentiable function of This makes it of special interest for use with scale parameters. ( with What I want is to see a definition of the sought invariance property that. In the minimum description length approach to statistics the goal is to describe data as compactly as possible where the length of a description is measured in bits of the code used. I like to understand things by approaching the simplest example first, so I'm interested in the case of a binomial trial, i.e. . {\displaystyle {\vec {\theta }}} p(\varphi)\propto\sqrt{I(\varphi)} p for any smooth function $\varphi(\theta)$. then. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 1 $$. The main result is that in exponential families, asymptotically for large sample size, the code based on the distribution that is a mixture of the elements in the exponential family with the Jeffreys prior is optimal. = p Locally the Fisher matrix $F$ transforms to $(J^{-1})^TFJ^{-1}$ under a change of coordinates with Jacobian $J$, and $\sqrt{\det}$ of this cancels the multiplication of volume forms by $\det J$. {\displaystyle {\vec {\varphi }}} The prior does not lose the information. Note that if I start with a uniform prior and then transform the parameters, I will in general end up with something that's not a uniform prior over the new parameters. $$ {\displaystyle [0,\pi /2]} Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same } Whatever priors they use must be completely uninformative about the scaling of time between the events. Could one house of Congress completely shut down the other house by passing large amounts of frivolous bills? This proof is clearly laid out in these lecture notes. ) Just use the chain rule after applying the definition of the information as the expected value of the square of the score). {\displaystyle \sigma } , and 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. {\displaystyle N} The final line applies the definition of Jeffreys prior on $\varphi{(\theta)}$. & \propto & \sqrt{I (\varphi (\theta))} |p (y| \theta)\\ "invariant" under reparametrization if, where ) I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. 1 log 0 The Jeffreys prior for the parameter Equivalently, if we write What is the rounding rule when the last digit is 5 in .NET 6? depend not just on the probability of the observed data as a function of That is, $\rho(\theta) = M\{ f(x\mid \theta) \}$. is uniform on the (N1)-dimensional unit sphere (i.e., it is uniform on the surface of an N-dimensional unit ball). & \propto & p (\varphi (\theta)) p (y| \theta) Let me know if you are stuck somewhere. \end{eqnarray*} is the Jacobian matrix with entries, Since the Fisher information matrix transforms under reparametrization as. (More info on this scale and location invariance can be found in Probability Theory the Logic of Science by E.T. Now, for the prior. The clearest answer I have found (ie, the most blunt "definition" of invariance) was a comment in this Cross-Validated thread, which I combined with the discussion in "Bayesian Data Analysis" by Gelman et. ( 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. Would a spear ever out perform a bow when wielded by an insanely powerful person? $$ What I would like is to understand the sense in which this is invariant with respect to a coordinate transformation $\theta \to \varphi(\theta)$. You can see that the use of Jeffreys prior was essential for $\frac{1}{| \varphi' (\theta) |}$ to cancel out. 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 contributing an answer to Mathematics Stack Exchange! Here $| \varphi' (\theta) |$ is the inverse of the jacobian of the transformation. {\displaystyle {\vec {\theta }}} 2 . To subscribe to this RSS feed, copy and paste this URL into your RSS reader. is the Dirichlet distribution with all (alpha) parameters set to one half. But unfortunately, if their clocks were running at different speeds (say, t' = qt) then their results will definitely be conflicting if they did not consider this difference in time-scales. {\displaystyle p_{\varphi }({\vec {\varphi }})\propto {\sqrt {\det I_{\varphi }({\vec {\varphi }})}}} In (i), it is $\pi$. It is the unique (up to a multiple) prior (on the positive reals) that is scale-invariant (the Haar measure with respect to multiplication of positive reals), corresponding to the standard deviation being a measure of scale and scale-invariance corresponding to no information about scale. 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. 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. n Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. MathJax reference. Now how do we define a completely "uninformative" prior? 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. 0 and Indeed this equation links the information of the likelihood to the information of the likelihood given the transformed model. and 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. 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. )\\ That's different from the Jeffreys prior, which is proportional to $1/\sqrt{p(1-p)}$. {\displaystyle p_{\varphi }(\varphi )\propto {\sqrt {I_{\varphi }(\varphi )}}} Gaussian distribution with mean parameter, Gaussian distribution with standard deviation parameter, "An invariant form for the prior probability in estimation problems", "Harold Jeffreys's Theory of Probability Revisited", https://en.wikipedia.org/w/index.php?title=Jeffreys_prior&oldid=1078504366, Short description is different from Wikidata, Articles with unsourced statements from September 2018, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 21 March 2022, at 22:34. $$ 2 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. = {\textstyle {\sqrt {\lambda }}=\int d\lambda /{\sqrt {\lambda }}} In what sense is the Jeffreys prior invariant? \end{aligned}, Now, we can drop the absolute value bars around $h'(\theta)$. We will derive the prior on $\phi$, which we'll call $p_{\phi}(\phi)$. (+1) Your answer is perhaps one of the cleareast I've found so far, together with the lecture you mention. & \propto & \frac{1}{| \varphi' (\theta) |} \sqrt{I (\theta)} p (y| {\displaystyle {\vec {\varphi }}} What we seek is a construction method $M$ with the following property: (I hope I have expressed this correctly) {\displaystyle \log \sigma ^{2}=2\log \sigma } Is "wait" an exclamation in this context? p(\theta)\propto\sqrt{I(\theta)} is uniform in the interval Sorry but I absolutely completely do not care the least about bounties and points. , {\displaystyle [0,2\pi ]} The first line is only applying the formula for the jacobian when transforming between posteriors. The following ones are the derivation of that equation. Computationally it is expressed by Jacobians but only the power-of-$A$ dependences matter and having those cancel out on multiplication. / Finally, whatever the thing that's invariant is, it must surely depend in some way on the likelihood function! ( It is trivial to define an. 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. But whatever we estimate from our priors and the data must necessarily lead to the same result. (Note that these equations omit taking the Jacobian of $I$ because they refer to a single-variable case.) the case where the support is $\{1,2\}$. = p What Jeffreys provides is a prior construction method $M$ which has this property. 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). is "invariant" under a reparametrization if. ) . and thus defining the priors as {\displaystyle p_{\theta }({\vec {\theta }})} Math Proofs - why are they important and how are they useful? {\displaystyle \mu } However, the more I try to do this the more confused I get. The key point is we want the following: If $\phi = h(\theta)$ for a monotone transformation $h$, then: $$P(a \le \theta \le b) = P(h(a) \le \phi \le h(b))$$. 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. re the second comment, the distinction is between functions and differential forms. = I'm not sure I understand what you mean in your other comment, though - could you spell your counterexample out in more detail? Announcing the Stacks Editor Beta release! det Then "$p_{L_{\varphi}}(\varphi) d\varphi \rm{\hskip2pt(claimed)} = p_{L_{\theta}}(\theta) d\theta = (\rm{Fisher \hskip3pt I \hskip3pt quantities)} d\varphi = \sqrt{I(\varphi)} d\varphi $. {\displaystyle \theta } fixed, the Jeffreys prior for the standard deviation The only difference is that the second line applies Bayes rule. Is it possible to use a NAS to host videos? the equations are between densities $p(x) dx$, but written as though for the density functions $p()$ that define the priors. {\displaystyle {\vec {\theta }}} , {\displaystyle \gamma _{i}=\varphi _{i}^{2}} 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. Transform characters of your choice into "Hello, world!". In fact the desired invariance is a property of $M$ itself, rather than of the priors it generates. x This "Invariance" is what is expected of our solutions. are the constants of proportionality the same in the two equations above, or different? $$ But let us say they were using some log scaled parameters instead of ours. Do the calculations with $\pi$ in there to see that point. H That is where this "Invariance" comes into the picture. Is there a way to add my Strength modifier to my Armor Class? I think I found out why I considered them the same, Jaynes in his book refers only to the (dv/v) rule and it's consequences as Jeffreys' priors. What was the purpose of those special user accounts in Unix? ( . In Bayesian probability, the Jeffreys prior, named after Sir Harold Jeffreys,[1] is a non-informative (objective) prior distribution for a parameter space; its density function is proportional to the square root of the determinant of the Fisher information matrix: It has the key feature that it is invariant under a change of coordinates for the parameter vector Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. {\displaystyle x}, with M\{ f(x\mid h(\theta)) \} = M\{ f(x \mid \theta) \}\circ h, is uniform on the whole circle $$ , for a given for any arbitrary smooth monotonic transformation $h$. ) If so I don't think that can be the thing that's invariant. 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.

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invariance of jeffreys prior