and deriving N {\displaystyle {\vec {\theta }}} 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. , {\displaystyle \theta } P(h(a)\le \phi \le h(b)) &= \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi\\ 2 ) are two possible parametrizations of a statistical model, and M\{ f(x\mid h(\theta)) \} = M\{ f(x \mid \theta) \}\circ h, I made some edits, I think it explains clearly now why the Wikipedia link is not a real answer. = Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same p ( In fact the desired invariance is a property of $M$ itself, rather than of the priors it generates. Just use the chain rule after applying the definition of the information as the expected value of the square of the score). log Then, start with some simple examples of some monotonic transformations in order to see the invariance. T ) 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. Connect and share knowledge within a single location that is structured and easy to search. , 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 $I$. Partly this is because there's just a lot left out of the Wikipedia sketch (e.g. N [ {\displaystyle p_{\theta }(\theta )} By the transformation of variables formula, $$p_{\phi}(\phi) = p_{\theta}( h^{-1} (\phi)) \Bigg| \frac{d}{d\phi} h^{-1}(\phi) \Bigg| $$. 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 $. is, This is the arcsine distribution and is a beta distribution with But let us say they were using some log scaled parameters instead of ours. ( , 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. 1 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. ( |y)\\ The following ones are the derivation of that equation. Here $| \varphi' (\theta) |$ is the inverse of the jacobian of the transformation. We will derive the prior on $\phi$, which we'll call $p_{\phi}(\phi)$. That is, the Jeffreys prior for 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. $$ The invariance of $|p dV|$ is the definition of "invariance of prior". To make sure that we are on the same page, let us take the example of the "Principle of Indifference" used in the problem of Birth rate analysis given by Laplace. Is the following parametrizations identifiable? 0 For example, the Jeffreys prior for the distribution mean is uniform over the entire real line in the case of a Gaussian distribution of known variance. N 1 I'm not sure I understand what you mean in your other comment, though - could you spell your counterexample out in more detail? I suggest to start with $\varphi(\theta)=2\theta$ and $\varphi(\theta)=1-\theta$. I will add some clarifications to my answer regarding your question about the invariance depending on the likelihood to my answer. & \propto & \sqrt{I (\varphi (\theta))} |p (y| \theta)\\ {\displaystyle \varphi } : your link is broken, I think you mean this one: @thc I've fixed the link. ( The use of these "Uninformative priors" is completely problem-dependent and not a general method of forming priors. 2 It is perfectly alright for them to do so because each and every problem of ours can be translated to their terms and vice-versa as long as the transform is a bijection. ( Now how do we define a completely "uninformative" prior? I In other words, on transforming the prior to a log-odds scale, the prior still says "See, I still consider no value of p1 to be preferable over another p2" and that is why the log-odds transform is not going to be flat. $$p_{\phi}(h(\theta)) = p_{\theta}(\theta) \Bigg| h'(\theta) \Bigg|^{-1} $$. By the way, I don't want to seem obstinate. ( . As did points out, the Wikipedia article gives a hint about this, by starting with d is "invariant" under a reparametrization if. 0 In the above case, the prior is telling us that "I don't want to give one value p$_1$ more preference than another value p$_2$" and it continues to say the same even on transforming the prior. To reiterate my question, I understand the above equations from Wikipedia, and I can see that they demonstrate an invariance property of some kind. = {\displaystyle \theta } Now, according to this Wikipedia page, the derivative the inverse gives: $$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. \end{eqnarray*} p If $h$ is decreasing, then $h(b) < h(a)$, which means the integral gets a minus in front of it. 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. What I want is to see a definition of the sought invariance property that. and {\displaystyle {\vec {\theta }}} re the second comment, the distinction is between functions and differential forms. As I explained earlier in the comments, it is essential to understand how jacobians work (or differential forms). {\displaystyle {\vec {\theta }}} p This "Invariance" is what is expected of our solutions. gives us the desired "invariance". This happens through the relationship $ \sqrt{I (\theta)} = \sqrt{I (\varphi (\theta))} | \varphi' (\theta) | $. 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. {\displaystyle {\vec {\theta }}} for each What was the purpose of those special user accounts in Unix? 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. Transform characters of your choice into "Hello, world!". Asking for help, clarification, or responding to other answers. Is it necessary to provide contact information for tens of co-authors when submitting a paper from a large collaboration? However, the link is helpful. ) p (\varphi (\theta) ) & = & \frac{1}{| \varphi' (\theta) |} p (\theta [ What I would like is to understand the sense in which this is invariant with respect to a coordinate transformation $\theta \to \varphi(\theta)$. 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)$. is uniform on the (N1)-dimensional unit sphere (i.e., it is uniform on the surface of an N-dimensional unit ball). Understanding why the Uniform distribution does not make a good prior. ) p The Jeffreys prior for a parameter (or a set of parameters) depends upon the statistical model. Perhaps I can, but it seems not at all trivial to me. 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. 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. 1 log {\displaystyle p_{\varphi }({\vec {\varphi }})\propto {\sqrt {\det I_{\varphi }({\vec {\varphi }})}}} I've read Jaynes' book, and quite a few of his papers on this topic, and I seem to remember him arguing. Use MathJax to format equations. The timescale invariance problem is also mentioned there.). does not depend upon {\displaystyle [0,2\pi ]} is also uniform. In what sense is the Jeffreys prior invariant? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. i Illustrate the invariance property of a noninformative prior. This notion of "uninformative prior" is a different thing from Jeffreys priors though, isn't it? fixed, the Jeffreys prior for the mean Where is the proof of uniqueness?) & = & p (\varphi (\theta)) 2 {\displaystyle {\vec {\varphi }}} It says that there is some prior information which is why this transformed pdf is not flat. When this property of "Uninformativeness" is needed, we seek priors that have invariance of a certain type associated with that problem. I To answer your question, the missing bit is the bit where I said "I'd like to understand this sense [of invariance] form of a functional equation similar to (ii), so that I can see how it's satisfied by (i)." , 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. Also, it would help me a lot if you could expand on the distinction you make between "densities $p(x) dx$" and "the. and {\displaystyle \gamma \in [0,1]} $$ , https://www2.stat.duke.edu/courses/Fall11/sta114/jeffreys.pdf, www2.stat.duke.edu/courses/Fall11/sta114/jeffreys.pdf. H 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 $\theta$. That is, the Jeffreys prior for 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. The third line applies the relationship between the information matrices. If the full parameter is used a modified version of the result should be used. It is trivial to define an. The Jeffreys prior for the parameter ) $$. The first line is only applying the formula for the jacobian when transforming between posteriors. Getting paid by mistake after leaving a company? ( $$ 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. In particular, I remember him arguing in favour of an "uninformative" prior for a binomial distribution that's an improper prior proportional to $1/(p(1-p))$. 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. You can see that the use of Jeffreys prior was essential for $\frac{1}{| \varphi' (\theta) |}$ to cancel out. ) The key point is we want the following: If $\phi = h(\theta)$ for a monotone transformation $h$, then: $$P(a \le \theta \le b) = P(h(a) \le \phi \le h(b))$$. 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 \mu } . {\displaystyle 1-\gamma } My silicone mold got moldy, can I clean it or should I throw it away? 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. Similarly, for a throw of an 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. & \propto & \frac{1}{| \varphi' (\theta) |} p (\theta) p (y| \theta)\\ 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. That's different from the Jeffreys prior, which is proportional to $1/\sqrt{p(1-p)}$. 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. the equations are between densities $p(x) dx$, but written as though for the density functions $p()$ that define the priors. For the Poisson distribution of the non-negative integer = are the constants of proportionality the same in the two equations above, or different? ] "invariant" under reparametrization if, where {\displaystyle {\vec {\varphi }}} The prior does not lose the information. 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. 0 {\displaystyle i} Computationally it is expressed by Jacobians but only the power-of-$A$ dependences matter and having those cancel out on multiplication. I But nonetheless, we can make sure that our priors are at least uninformative in some sense. As with the uniform distribution on the reals, it is an improper prior. Let me know if you are stuck somewhere. Repeat Hello World according to another string's length. Recalling that $\phi = h(\theta)$, we can write this as. ) P.S. 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. {\displaystyle p_{\theta }({\vec {\theta }})} and thus defining the priors as I do not currently know whether the particular prior construction method supplied by Jeffreys is unique in having this property. x , Estimation: what does it mean that the observable cancel from my equations? \rho(\theta) = \frac{1}{\pi\sqrt{\theta(1-\theta)}}, \qquad\qquad(i) {\displaystyle \mu } {\displaystyle {\vec {\theta }}} That is, $\rho(\theta) = M\{ f(x\mid \theta) \}$. ) {\displaystyle (H,T)\in \{(0,1),(1,0)\}} is the unnormalized uniform distribution on the non-negative real line. Maybe the problem is that you are forgetting the jacobian of the transformation in (ii). Indeed this equation links the information of the likelihood to the information of the likelihood given the transformed model. ( Jeffrey's prior has only this type of invariance in it, not to all transforms (Maybe some others too, but not all for sure). that is, if the priors {\displaystyle \theta } \end{aligned}, using the substitution formula from Wikipedia with $\phi = h(\theta)$, \begin{aligned} p [ the probability is where $\theta$ is the parameterisation given by $p_1 = \theta$, $p_2 = 1-\theta$. 2 Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. = Math Proofs - why are they important and how are they useful? {\displaystyle {\vec {\varphi }}} Also, to answer your question, the constants of integration do not matter here. M\{ f(x\mid h(\theta)) \} = M\{ f(x \mid \theta) \}\circ h, 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). 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. , = / Equivalently, ( ) Clearly something is invariant here, and it seems like it shouldn't be too hard to express this invariance as a functional equation. To me the term "invariant" would seem to imply something along the lines of {\displaystyle \mu } Announcing the Stacks Editor Beta release! rev2022.8.1.42699. This amounts to using a pseudocount of one half for each possible outcome. Linked List implementation in c++ with all functions. ( Equivalently, the Jeffreys prior for 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. Thanks for contributing an answer to Mathematics Stack Exchange! Sometimes the Jeffreys prior cannot be normalized, and is thus an improper prior. . I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. {\displaystyle {\vec {\gamma }}=(\gamma _{1},\ldots ,\gamma _{N})} ] , the Jeffreys prior for $$ I like to understand things by approaching the simplest example first, so I'm interested in the case of a binomial trial, i.e. In the univariate case, does the expression in your first sentence reduce to $p(\theta) d\theta$? What's a reasonable environmental disaster that could be caused by a probe from Earth entering Europa's ocean? 0 {\displaystyle \log \sigma ^{2}=2\log \sigma } = Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. {\displaystyle \gamma =\sin ^{2}(\theta )} gives us the desired "invariance". 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. a continuously differentiable function of 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. Since the Fisher information transforms under reparametrization as, defining the priors as ( We call the prior In (i), it is $\pi$. 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. 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. p(\varphi)\propto\sqrt{I(\varphi)} = ( When using the Jeffreys prior, inferences about This result holds if one restricts the parameter set to a compact subset in the interior of the full parameter space[citation needed]. The problem here is about the apparent "Principle of Indifference" considered by Laplace. Though his prior was perfectly alright, the reasoning used to arrive at it was at fault. {\displaystyle p_{\theta }(\theta )\propto {\sqrt {I_{\theta }(\theta )}}} , \theta)\\ is the Jacobian matrix with entries, Since the Fisher information matrix transforms under reparametrization as. and 1 (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\}$. is the Dirichlet distribution with all (alpha) parameters set to one half. the first equality is a claim still to be proven. n {\displaystyle N} 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. $$ {\textstyle \log \sigma =\int d\sigma /\sigma } The preference for Jeffreys form of invariant prior is based on other considerations. be two possible parametrizations of a statistical model, with / Is "wait" an exclamation in this context? {\displaystyle \alpha =\beta =1/2} is uniform in the interval For the distribution $f_\theta (x) = \theta x^{\theta-1}$, what is the sufficient statistic corresponding to the Monotone Likelihood Ratio? What we seek is a construction method $M$ with the following property: (I hope I have expressed this correctly) MathJax reference. To learn more, see our tips on writing great answers. Would a spear ever out perform a bow when wielded by an insanely powerful person? \int_{\varphi(\theta_1)}^{\varphi(\theta_2)} \rho(\varphi(\theta)) d \varphi \qquad\qquad(ii) is uniform on the whole circle Do the calculations with $\pi$ in there to see that point. If so I don't think that can be the thing that's invariant. &= \int_{a}^{b} p_{\theta}(\theta) \Bigg| h'(\theta) \Bigg|^{-1} h'(\theta) d\theta, The second line applies the definition of Jeffreys prior. Yes, I think they are different. al. ) This is ensured by the use of Jeffrey's prior which is completely scale and location-invariant. = H with Formula (ii) is not correct in either the special case or in general. \begin{eqnarray*} 1 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. Let $p_{\theta}(\theta)$ be the prior on $\theta$. to finally come to an understanding. 2 = In this case the Jeffreys prior is given by For a parametric family of distributions one compares a code with the best code based on one of the distributions in the parameterized family. [2], If {\displaystyle \gamma ^{H}(1-\gamma )^{T}} What Jeffreys provides is a prior construction method $M$ which has this property. Can I get a clock signal from a 4-pin crystal oscillator circuit by applying 5V to the input pin? and Say if the aliens used the same principle, they would definitely arrive at a different answer than ours. Say that we have 2 experimenters who aim to find out the number of events that occurred in a specific time (Poisson dist.). ) \end{eqnarray*} p(\theta)\propto\sqrt{I(\theta)} \int_{h(a)}^{h(b)} p_{\phi}(\phi) d\phi &= \int_{a}^{b} p_{\phi}(h(\theta)) h'(\theta) d\theta\\ d 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. 1 ) is a continuously differentiable function of > . 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. } This proof is clearly laid out in these lecture notes. 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. Suppose there was an alien race that wanted to do the same analysis as done by Laplace. ) 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 {\displaystyle \varphi } 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. So there must be some other sense intended by "invariant" in this context. i $$ 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.