Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same {\displaystyle \mu } log -sided die with outcome probabilities ) . Formula (ii) is not correct in either the special case or in general. = When this property of "Uninformativeness" is needed, we seek priors that have invariance of a certain type associated with that problem. ( You can see that the use of Jeffreys prior was essential for $\frac{1}{| \varphi' (\theta) |}$ to cancel out. and deriving MathJax reference. In what sense is the Jeffreys prior invariant? That is, the Jeffreys prior for = In this case the Jeffreys prior is given by ( Sometimes the Jeffreys prior cannot be normalized, and is thus an improper prior. Is "wait" an exclamation in this context? This proof is clearly laid out in these lecture notes. 1 0 We will derive the prior on $\phi$, which we'll call $p_{\phi}(\phi)$. {\displaystyle [0,2\pi ]} The problem here is about the apparent "Principle of Indifference" considered by Laplace. The use of these "Uninformative priors" is completely problem-dependent and not a general method of forming priors. The link given by the OP contains the problem statement in good detail. p Transform characters of your choice into "Hello, world!". is the unnormalized uniform distribution on the non-negative real line. To learn more, see our tips on writing great answers. / Though his prior was perfectly alright, the reasoning used to arrive at it was at fault. 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. The prior does not lose the information. Where is the proof of uniqueness?) 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. is uniform on the whole circle Use of the Jeffreys prior violates the strong version of the likelihood principle, which is accepted by many, but by no means all, statisticians. The only difference is that the second line applies Bayes rule. {\displaystyle p_{\theta }({\vec {\theta }})} ( = As did points out, the Wikipedia article gives a hint about this, by starting with But using the "Principle of Indifference" violates this. 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. $$ Your answer is really clear, but I think is not quite there yet. However, the more I try to do this the more confused I get. p {\displaystyle p_{\theta }({\vec {\theta }})\propto {\sqrt {\det I_{\theta }({\vec {\theta }})}}} P(h(a)\le \phi \le h(b)) &= \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi\\ {\displaystyle \gamma } 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. & \propto & \frac{1}{| \varphi' (\theta) |} \sqrt{I (\theta)} p (y| with 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. {\displaystyle i} 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. Is the following parametrizations identifiable? {\displaystyle \gamma _{i}=\varphi _{i}^{2}} ( I will add some clarifications to my answer regarding your question about the invariance depending on the likelihood to my answer. For a coin that is "heads" with probability / Can I get a clock signal from a 4-pin crystal oscillator circuit by applying 5V to the input pin? [ and I suggest that you check carefully the formulas here (hint: $\left| \frac{d \Phi^{- 1}}{d y} \right|$ is the jacobian where $\Phi^{- 1}$ is the inverse transformation). ( { Equivalently, the Jeffreys prior for [ Finally, whatever the thing that's invariant is, it must surely depend in some way on the likelihood function! 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. 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. are related by the usual change of variables theorem. {\displaystyle \varphi } What we seek is a construction method $M$ with the following property: (I hope I have expressed this correctly) is. is "invariant" under a reparametrization if. (More info on this scale and location invariance can be found in Probability Theory the Logic of Science by E.T. T \rho(\theta) = \frac{1}{\pi\sqrt{\theta(1-\theta)}}, \qquad\qquad(i) ) {\displaystyle p_{\varphi }({\vec {\varphi }})\propto {\sqrt {\det I_{\varphi }({\vec {\varphi }})}}} |y)\\ for any smooth function $\varphi(\theta)$. The final line applies the definition of Jeffreys prior on $\varphi{(\theta)}$. This amounts to using a pseudocount of one half for each possible outcome. I suggest to start with $\varphi(\theta)=2\theta$ and $\varphi(\theta)=1-\theta$. But nonetheless, we can make sure that our priors are at least uninformative in some sense. 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. parameter even when the likelihood functions for the two experiments are the samea violation of the strong likelihood principle. p(\varphi)\propto\sqrt{I(\varphi)} 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. 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. , My silicone mold got moldy, can I clean it or should I throw it away? . {\displaystyle x}, with {\displaystyle \sigma } {\displaystyle {\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. $$ For the distribution $f_\theta (x) = \theta x^{\theta-1}$, what is the sufficient statistic corresponding to the Monotone Likelihood Ratio? Thanks for contributing an answer to Mathematics Stack Exchange! fixed, the Jeffreys prior for the standard deviation Let me know if you are stuck somewhere. M\{ f(x\mid h(\theta)) \} = M\{ f(x \mid \theta) \}\circ h, I I Connect and share knowledge within a single location that is structured and easy to search. $$ I still think that your problem is with jacobians and the fact that the formula (ii) is correct for the special case I does not make correct in general. for each It only takes a minute to sign up. Perhaps I can answer this myself now, but if you'd like to post a proper answer detailing it then I'd be happy to award you the bounty. , we say that the prior , {\displaystyle p_{\theta }(\theta )\propto {\sqrt {I_{\theta }(\theta )}}} {\displaystyle {\vec {\varphi }}} Then "$p_{L_{\varphi}}(\varphi) d\varphi \rm{\hskip2pt(claimed)} = p_{L_{\theta}}(\theta) d\theta = (\rm{Fisher \hskip3pt I \hskip3pt quantities)} d\varphi = \sqrt{I(\varphi)} d\varphi $. By the transformation of variables formula, $$p_{\phi}(\phi) = p_{\theta}( h^{-1} (\phi)) \Bigg| \frac{d}{d\phi} h^{-1}(\phi) \Bigg| $$. Diff-in-diff parallel trends with a positive outcome. : your link is broken, I think you mean this one: @thc I've fixed the link. It is trivial to define an. {\displaystyle \varphi } This "Invariance" is what is expected of our solutions. p (\varphi (\theta) |y) & = & \frac{1}{| \varphi' (\theta) |} p (\theta ) H Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. = is the Dirichlet distribution with all (alpha) parameters set to one half. ) ) The comments on this question make no sense if you don't already know that @did's comment was originally an answer, which was deleted by a moderator and made into a comment, and that the following two comments were originally, $$ Could one house of Congress completely shut down the other house by passing large amounts of frivolous bills? p(\theta)\propto\sqrt{I(\theta)} Clearly something is invariant here, and it seems like it shouldn't be too hard to express this invariance as a functional equation. I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. Furthermore, if {\displaystyle \theta } 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. re the second comment, the distinction is between functions and differential forms. This shows that the invariant prior is very non-unique as there are many other ways to achieve the cancellation. So there must be some other sense intended by "invariant" in this context. {\displaystyle \sigma >0} By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. $$ Equivalently, if we write zyx's answer is excellent but it uses differential forms. 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. 2 ] Perhaps I can, but it seems not at all trivial to me. 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. J This result holds if one restricts the parameter set to a compact subset in the interior of the full parameter space[citation needed]. This is ensured by the use of Jeffrey's prior which is completely scale and location-invariant. Since the Fisher information transforms under reparametrization as, defining the priors as I ( 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. 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 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. 1 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. & \propto & p (\varphi (\theta)) p (y| \theta) depend not just on the probability of the observed data as a function of The second line applies the definition of Jeffreys prior. , the Jeffreys prior for Yes, I think they are different. 0 does not depend upon 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. H ; it is the unnormalized uniform distribution on the real line the distribution that is 1 (or some other fixed constant) for all points. \begin{eqnarray*} {\displaystyle \lambda \geq 0} for any arbitrary smooth monotonic transformation $h$. Say that we have 2 experimenters who aim to find out the number of events that occurred in a specific time (Poisson dist.). N , 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. 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. = {\displaystyle {\vec {\theta }}} > & = & \sqrt{I (\varphi (\theta))} \\ By the way, I don't want to seem obstinate. Whatever priors they use must be completely uninformative about the scaling of time between the events. 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. & \propto & \sqrt{I (\varphi (\theta))} |p (y| \theta)\\ What is the rounding rule when the last digit is 5 in .NET 6? Is it possible to use a NAS to host videos? $$ Linked List implementation in c++ with all functions. 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. {\displaystyle \theta } ] $$ log What Jeffreys provides is a prior construction method $M$ which has this property. is the unnormalized uniform distribution on the real line, and thus this distribution is also known as the .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}logarithmic prior. P.S. {\displaystyle \log \sigma ^{2}=2\log \sigma } gives us the desired "invariance". 0 the case where the support is $\{1,2\}$. , ( Here $| \varphi' (\theta) |$ is the inverse of the jacobian of the transformation. That seems to be an open-ended question full of debates. This seems to be rather an important question: if there is some other functional $M'$ that is also invariant and which gives a different prior for the parameter of a binomial distribution then there doesn't seem to be anything that picks out the Jeffreys distribution for a binomial trial as particularly special. the first equality is a claim still to be proven. , , each non-negative and satisfying T I'm fairly certain it's a logical point that I'm missing, rather than something to do with the formal details of the mathematics. 2 The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Which part of the question is not dealt with. to finally come to an understanding. That is, the relative probability assigned to a volume of a probability space using a Jeffreys prior will be the same regardless of the parameterization used to define the Jeffreys prior. The key point is we want the following: If $\phi = h(\theta)$ for a monotone transformation $h$, then: $$P(a \le \theta \le b) = P(h(a) \le \phi \le h(b))$$. Use MathJax to format equations. {\displaystyle J} {\displaystyle \theta } ) $$ ) That's different from the Jeffreys prior, which is proportional to $1/\sqrt{p(1-p)}$. For the Poisson distribution of the non-negative integer It says that there is some prior information which is why this transformed pdf is not flat. and ) Also, it would help me a lot if you could expand on the distinction you make between "densities $p(x) dx$" and "the. First we show a probability density for which this is satisfied. Making statements based on opinion; back them up with references or personal experience. det . {\displaystyle \theta } {\textstyle {\sqrt {\lambda }}=\int d\lambda /{\sqrt {\lambda }}} If the full parameter is used a modified version of the result should be used. 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. 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. \int_{\varphi(\theta_1)}^{\varphi(\theta_2)} \rho(\varphi(\theta)) d \varphi \qquad\qquad(ii) However, regardless what likelihood you use, the invariance will hold through. For a parametric family of distributions one compares a code with the best code based on one of the distributions in the parameterized family. Now how do we define a completely "uninformative" prior? Suppose there was an alien race that wanted to do the same analysis as done by Laplace. 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. is uniform in the interval We call the prior 0 2 ) Jeffrey's prior has only this type of invariance in it, not to all transforms (Maybe some others too, but not all for sure). [3], Analogous to the one-parameter case, let 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. {\displaystyle p_{\theta }(\theta )} {\displaystyle \gamma \in [0,1]} Equivalently, the Jeffreys prior for {\displaystyle \gamma ^{H}(1-\gamma )^{T}} 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. p [ What I want is to see a definition of the sought invariance property that. That is, the Jeffreys prior for is the Jacobian matrix with entries, Since the Fisher information matrix transforms under reparametrization as. )\\ 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 I {\displaystyle {\vec {\theta }}} Getting paid by mistake after leaving a company? ) That is, $\rho(\theta) = M\{ f(x\mid \theta) \}$. Is there a device to plug in to dead socket to figure out which breaker? 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. What's a reasonable environmental disaster that could be caused by a probe from Earth entering Europa's ocean? fixed, the Jeffreys prior for the mean ( As I explained earlier in the comments, it is essential to understand how jacobians work (or differential forms). where $\theta$ is the parameterisation given by $p_1 = \theta$, $p_2 = 1-\theta$. \begin{eqnarray*} The Jeffreys prior for a parameter (or a set of parameters) depends upon the statistical model. Is there a way to add my Strength modifier to my Armor Class? ) {\displaystyle {\vec {\theta }}} My problem arose from looking at a particular example of a prior constructed by Jeffreys' method (i.e. 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. {\displaystyle N} Also, to answer your question, the constants of integration do not matter here. Understanding why the Uniform distribution does not make a good prior. Say if the aliens used the same principle, they would definitely arrive at a different answer than ours. ) Computationally it is expressed by Jacobians but only the power-of-$A$ dependences matter and having those cancel out on multiplication.
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