Talk:Fisher information

message: The first line in 'Example' miss a left-hand parenthesis ")".  Thank you for a nice article!
:There seem to be some superfluous brackets in the expectation notation:  Neither \mathbb E X ^2 \, nor \left[\mathbb E X\right]^2\, is ambiguous, and in fact their difference should be the variance of X if I remember right.   \mathbb E X ^2 \, does not need to be disambiguated according to standard order of operations. -- 130.94.162.61 04:45, 10 February 2006 (UTC)
replies: text: Also there are some philosophical issues not discussed in the article. Without some prior probability distribution on \theta, how can we ever hope to extract information about it?  For example, take a person's height.  We usually start with some a priori idea or expectation of what a person's height ought to be before taking any measurements at all.  If we measure a person's height to be 13 feet, we would normally assume the measurement was wrong and probably discard it (as a so-called "outlier").  But if more and more measurements gave a result in the vicinity of 13 feet, it might dawn on us that we are measuring a giant.  On the other hand, a single measurement of 5 feet 5 inches would probably convince us of someone else's height to a reasonable degree of accuracy.  Fisher information doesn't say anything about a priori probability distributions on θ.  A maximum likelihood estimator which assumes a "uniform" distribution over all the reals (w.r.t. the Lebesgue measure) is an absurdity.  I'm not sure I'm making any sense (and feel free to delete this comment if I'm not), but I don't believe any information can be extracted about an unknown parameter without having beforehand some rough estimate of the a priori probability distribution of that parameter. -- 130.94.162.61 13:54, 10 February 2006 (UTC)
thread_title: Untitled

message: I should admit that I have studied mathematical statistics. Even so, by Wiki standards, this entry is not unduly technical. I've added some links (and am sure more could be added) that should help the novice reader along.
The first person to contribute to this talk page is an unwitting Bayesian, when (s)he calls for a "prior distribution" on θ. Information measures and entropy are bridges connecting classical and Bayesian statistics. This entry should sketch bits of those bridges, if only by including a few links.
This entry should say more comparing and constrasting Fisher information with the measures of Shannon, Kullback-Leibler, and possibly others.
Wiki should also say more, somewhere, about the extraordinary work of Roy Frieden. Frieden, a respectable physicist, has written a nearly 500pp book arguing that a great deal of theoretical physics can be grounded in Fisher information and the calculus of variations. This should not come as complete surprise to anyone who has mastered Hamiltonian mechanics and has thought about the principle of least action, but even so, Frieden's book is a breathtaking high wire act. It appears that classical mechanics, electromagnetism, and thermodynamics, general relativity, and quantum electrodynamics are all merely different applications of a few core information-theoretic and variational principles. Frieden (2004) also includes a chapter on what he thinks his EPI approach could contribute to unsolved problems, such as quantum gravitation, turbulence, and topics in particle physics. Could EPI even prove to be the eventual gateway to that Holy Grail of contemporary science, the unification of the three fundamental forces, electroweak, strong, and gravitation? I should grant that EPI doesn't answer everything; for example, it sheds no light on why the fundamental dimensionless constants take on the values that they do. Curiously, Frieden says little about optics even though that was his professional specialty.202.36.179.65 13:19, 11 April 2006 (UTC)
replies: text: A number of links to articles about Frieden and his work are already in this article. Michael Hardy 20:31, 11 April 2006 (UTC)

text: The physical and mathematical correctness of Frieden's ideas have been characterized as highly dubious by several knowledgeable observers; see, for example, Ralph F. Streater's ``Lost Causes in Theoretical Physics: Physics from Fisher Information, and Cosma Shalizi's review of Physics from Fisher Information. QuispQuake 14:55, 12 July 2006 (UTC)
replies: text:  Hey, 202.36.179.65 don't be coy. You must be the man himself!81.178.157.195 (talk) 11:40, 31 January 2012 (UTC)
thread_title:  Say more about Roy Frieden's work 

message: B. Roy Frieden claims to have developed a "universal method" in physics, based upon Fisher information.  He has written a book about this.  Unfortunately, while Frieden's ideas initially appear interesting, his claimed method has been characterized as highly dubious by knowledgeable observers (Google for a long discussion in sci.physics.research from some years ago.)
Note that Frieden is Prof. Em. of Optical Sciences at the University of Arizona.  The data.optics.arizona.edu anon has used the following IPs to make a number of questionable edits:
#
##20 May 2005 confesses to being Roy Frieden in real life
##6 June 2006: adds cites of his papers to Extreme physical information
##23 May 2006 adds uncritical description of his own work in Lagrangian and uncritically cites his own controversial book
## 22 October 2004 attributes uncertainty principle to Cramer-Rao inequality in Uncertainty Principle, which is potentially misleading
##21 October 2004 adds uncritical mention of his controversial claim that Maxwell-Boltzmann distribution can be obtained via his "method"
## 21 October 2004 adds uncritical mention of his controversial claim that the Klein-Gordon equation can be "derived" via his "method"
#
##9 September 2004 adds uncritical description of his work to Fisher information
## 8 September 2004 adds uncritical description of his highly dubious claim that EPI is a general approach to physics to Physical information
##16 August 2004 confesses IRL identity
##13 August 2004 creates uncritical account of his work in new article, Extreme physical information
These POV-pushing edits should be modified to more accurately describe the status of Frieden's work.---CH 21:54, 16 June 2006 (UTC)
replies: text: Hear,hear! I totally agree with the first few sentences of this talk section, and perhaps it should appear in the article as a health warning.81.178.157.195 (talk) 11:39, 31 January 2012 (UTC)
thread_title: [[B. Roy Frieden]]'s anonymous POV-pushing edits

message: In addressing the technical accessibility tag above, I would recommend the addition of some graphs.  For example, this concept could be related to the widely understood concept of the Gaussian bell curve. -- Beland 21:35, 4 November 2006 (UTC)
thread_title:  Graphs to improve technical accessibility 

message: In the one-dimensional equation, there is a minus sign in the equation linking the second derivative of the log likelihood to the variance of theta. This stands to reason, as we want maximum, not minimum likelihood, so the second derivative becomes negative. In the matrix formulation below, there is no minus sign. Should it not be there, too? In practice, of course, one often minimizes sums of squares, or other "loss" functions, instead. This already is akin to -log(L).
I am not a professional statistician, but I use statistics a lot in my profession, microbiology. I did not find the article too technical. After all, the subject itself is somewhat technical. Wikipedia does a great job of making gems such as this accessible.
82.73.149.14 19:51, 30 December 2006 (UTC)
thread_title:  Minus sign missing? 

message: I think that the style in which parts of this article are written is more appropriate for a textbook than for an encyclopedia article.  For example: "To informally derive the Fisher Information, we follow the approach described by Van Trees (1968) and Frieden (2004)"  This type of comment is only really appropriate in a textbook where a single author or a few authors are writing a book with a coherent theme.  An encyclopedia article ought to adopt a different style: in particular, I object to the use of the term "we", as on wikipedia, with so many authors and with anonymous authors, it is not clear who the word "we" refers to.  Instead, I think we should word things "Van Trees (1968) and Frieden (2004) provide the following method of deriving the Fisher information informally:".  I am going to rewrite this to try to eliminate these sorts of comments.  But...I think this style problem goes beyond just the use of the word "we"...it's pretty pervasive and it needs deep changes.  Cazort (talk) 18:14, 10 January 2008 (UTC)
thread_title: Style

message: This derivation doesn't seem to be a derivation of the Fisher information, but rather, a derivation of the relationship between Fisher information and the bound on the variance of an estimator.  Does everyone agree with me that this should be renamed?  Also, this remark relates to the definition of Fisher information.  For example, the comment "The Fisher information is the amount of information" is loaded, because it is not defined what information means.  I am going to weaken this statement accordingly.  If we can come up with a more rigorous and more precise definition then we should include it!  Cazort (talk) 18:22, 10 January 2008 (UTC)
thread_title: Informal Derivation & Definition

message: I've heard mention of "mutual information" and "joint information" (bivariate discrete random variables); shouldn't these terms be discussed?199.196.144.13 (talk) 21:08, 29 May 2008 (UTC)
thread_title:  How about putting in 'Mutual Information' and 'Joint Information' discussion 

message: Nowhere in the article it says that the Fischer Information Matrix is the inverse of the Covariance matrix in the multivariate normal case. Yet this information is used in many sources especially in the context of Bayesian Networks (e.g. see http://en.wikipedia.org/wiki/Kalman_filter#Information_filter)  — Preceding unsigned comment added by 89.204.138.242 (talk) 12:34, 23 January 2013 (UTC)
thread_title:  Inverse of FIM for multivariate Gaussian 

message: This article is virtually useless to any reader who is not already familiar with the field. I came here simply to find out what a 'Fisher matrix' is and there is nothing here which clearly answers that question. There's a sentence stating the general idea but it then dives directly into the full derivation with no simple example. The page appears to be trying to be a postgrad textbook rather than an article for a reader who has come across a term and would like to know what it means. Sadly I'm nowhere near able to do so myself, but I would suggest this article needs: 1) A simple example of what a Fisher matrix is. 2) A beginner-friendly description of what its components actually mean. The priority for any page should be to give someone who has never heard of the subject before a general idea of what the subject is, this article seems to fail on that.  — Preceding unsigned comment added by 86.153.104.154 (talk) 09:47, 4 April 2014 (UTC)
thread_title:  Far too technical 

message: In the section "Single-parameter Bernoulli experiment," could you explain why the variance of the mean of successes in "n Bernoulli trials" is \frac{\theta(1 - \theta)}{n}?  This is what you imply when you say that the variance in question is the inverse of the additive Fisher information.  Everywhere I looked, the variance of the mean of successes in "n Bernoulli trials," with probability of success p, is np(1-p).
Also, why did you drop the word "independent" in the last sentence of that section?  — Preceding unsigned comment added by 174.192.30.141 (talk) 04:21, 3 May 2018 (UTC)
thread_title:  Variance of <math>n</math> independent Bernoulli trials is <math>\frac{p(1-p)}{n}</math>? 

message: I added a subsection that clarifies a commonly seen discrepancy in the definition of Fisher information.  That is, some textbooks and notes define Fisher information with respect to one observation while some others define it using likelihood for all observations.
A critical problem is the lack of clarification which version is used in each scenario.  I hope someone can help with adding short phrases after some frequently used important results that clarifies which version of Fisher information definition is used.  For example, Cramer-Rao lower bound (writing only $I(\theta)$ rather than $nI(\theta)$ on the denominator may cause a misunderstanding that this lower bound doesn't depend on $n$, and it will be much better if it's clarified immediately that this $I(\theta)$ is defined using the joint log-likelihood so it is linear in $n$) -- this might seem a bit repetitive but it's really not (it may save a lot of time for beginners from confusion, especially when they compare the $I(\theta)$ defined in C-R lower bound to the $I(\theta)$ that appears in the asymptotic normal variance of MLE, where the tradition is almost unanimously defining $I(\theta)$ for only one observation).
Yzhanghf0700 (talk) 18:52, 8 March 2019 (UTC)
thread_title:  Discrepancy in definition