Talk:Beta distribution

message: This page and many others now give an error when attempting to view with the Dictionary.app on MacOS:  "This page cannot be displayed in this application."
What's causing this, and how can it be fixed?  — Preceding unsigned comment added by Canavalia (talk • contribs) 14:35, 15 May 2023 (UTC)
thread_title: View from MacOS Dictionary.app

message: Consider the expression
:\ {n \choose i}p^i(1-p)^{n-i}
Fixing (n,p) it is the binomial distribution of i. Fixing (n,i) it is the (unnormalized) beta distribution of p. The article does not clarify this.
Bo Jacoby 10:02, 15 September 2005 (UTC)
replies: text: This is mentioned only implicitly in the current version, which describes the beta distribution as the conjugate prior for the binomial.  You could add a section on occurrence and uses of the beta distribution that would clarify this point further. --MarkSweep✍ 12:50, 15 September 2005 (UTC)
thread_title: why the difference in notation?

message: I added a section for the four parameter estimation case using the method of moments.  Elderton, (see section titled "History"), in his 1906 monograph "Frequency curves and correlation," fully discusses the four parameter case and contains the correct equations.  These equations have been repeated by other authors in other books.  It is curious that the classic reference by N.L.Johnson and S.Kotz, in their (1970) first edition of "Continuous Univariate Distributions Vol. 2" , Wiley, Chapter 21:Beta Distributions, page 44, equation (15) contains an important error. The support interval range (\hat{c}- \hat{a}) is given as follows:
: \frac{\sqrt{
                                               \text{(sample variance)}}}{2}\sqrt{
                                               (\hat{\nu}+2)^2(\text{sample kurtosis})^2+16(\hat{\nu}+1)
                                              }
Where Johnson and Kotz use the identical nomenclature used by Elderton in 1906: \beta_2 for the sample kurtosis, \mu_2 for sample variance and "r" for \hat{\nu}
This is incorrect.  The correct equation is given by Elderton in 1906:
: \frac{\sqrt{
                                               \text{(sample variance)}}}{2}\sqrt{
                                               (2+\hat{\nu})^2(\text{sample skewness})^2+16(1+\hat{\nu})
                                              }
This range can also be expressed in terms of the excess kurtosis or the kurtosis, as I have done in the article (but it will read very differently than in Johnson and Kotz, whether one uses the kurtosis or the excess kurtosis).  Dr. J. Rodal (talk) 19:15, 7 September 2012 (UTC)
thread_title: Method of moments - four parameter estimation case - another error in first edition of Johnson and Kotz

message: There are two forms for the Beta distribution. At present only the so-called 'Beta distribution of the first kind' is discussed. The Beta distribution of the second kind does not seem to be discussed in Wikipedia as I write. Rwb001 06:26, 30 September 2006 (UTC)
replies: text: As this article is written, it is, as you note, only about the "first kind". Perhaps it has been developed, but the "second kind" article is referenced in the lede as the Beta prime distribution, but the lede sentence needs to be clarified a bit. I will do that. Attic Salt (talk) 14:40, 7 October 2019 (UTC)
thread_title: Beta distribution of the second kind

message: Why is there SO much blank, and therefore wasted, space on this page? —The preceding unsigned comment was added by Algebra man (talk • contribs) 20:09, 8 December 2006 (UTC)
replies: text: The infobox on the right hand side can sometimes cause those problems.  Widen your browser window, and the blank space should go away.  Baccyak4H (Yak!) 20:11, 8 December 2006 (UTC)
thread_title:  Layout 

message: I dont'know their distribution?
What is F1 in the Char Function? —The preceding unsigned comment was added by 148.235.65.243 (talk) 18:40, 12 February 2007 (UTC)
replies: text: Retraction, I found this on Wolfram as the Kummer confluent hypergeometric function of the first kind.  Not too helpful, but if this is silent for awhile or if someone wants, I'll add a note and link to Confluent_hypergeometric_functions--Phays 22:25, 17 August 2007 (UTC)

text: \begin{align}\, _1F_1(\alpha;2\alpha; it)&= e^{\frac{it}{2}}\, _0F_1 (; \alpha+\tfrac{1}{2}; \tfrac{1}{16}(it)^2)= e^{\frac{it}{2}} \left(\tfrac{it}{4}\right)^{\tfrac{1}{2}-\alpha} \Gamma\left(\alpha+\tfrac{1}{2}\right) I_{\alpha-\frac 1 2}\left(\tfrac{it}{2}\right).\end{align}
I also included accompanying plots, the real part (Re) of the characteristic function of the beta distribution displayed for symmetric (α = β) and skewed (α ≠ β) cases.Dr. J. Rodal (talk) 11:39, 16 August 2012 (UTC)
thread_title:  Characteristic Function (Sum of independent betas) 

message: Is anyone able to add a section how you would draw random samples from the beta distribution?  Is there a direct method like a transform from uniform variates, or do you have to use rejection sampling? 195.157.136.194 08:50, 22 January 2007 (UTC)
thread_title:  Generating beta-distributed random variates

message: For  \alpha = 1  the beta distribution:
:: \frac{1}{\mathrm{B}(\alpha,\beta)}\, x
^{\alpha-1}(1-x)^{\beta-1}\!
has the form:
:: \beta(1-x)^{\beta-1}\!
Has this form also a special name?  —Preceding unsigned comment added by 84.83.33.64 (talk) 09:13, 3 October 2008 (UTC)
replies: text: Standard power function  —Preceding unsigned comment added by Mochan Shrestha (talk • contribs) 15:59, 9 March 2010 (UTC)
replies: text: Almost, but not quite.  For me a standard power function distribution is when \beta = 1, so with pdf of the form  \alpha x^{\alpha -1}\!.  That is similar but reversed. --Rumping (talk) 15:39, 12 March 2010 (UTC)
thread_title:  The name of a special case? 

message: Is it worth mentioning that for large values of α + β the beta distribution converges towards a normal distribution?
in terms of a weighted-coin toss: if heads are worth 1 and tails are worth 0:
 beta(N_{heads}+1,N_{tails}+1) \approx \mathcal{N}(mean(samples),var(samples)/N_{samples})
Dividing the sample-variance by the number of samples gives the expected variance of our sample's mean.  —Preceding unsigned comment added by Sukisuki (talk • contribs) 22:41, 10 April 2010 (UTC)
thread_title:  related distributions? 

message: The present version has a supposed formula for the "kurtosis excess" that starts off with the expession for the 4th central moment only, so that part is wrong. But is the rest of the expression correct for either the excess kurtosis or the 4th moment? It doesn't coincide with any formula I can conveniently find. .... and it would be good provide explicit citations for this stuff. Melcombe (talk) 13:03, 25 October 2010 (UTC)
thread_title:  Kurtosis 

message: In contingency table analysis, and I'm sure in many other situations, the need to test the difference between two beta-distributed random variables arises.  The posterior distribution of the difference is the convolution of the two original beta distributions.  However I have seen unsubstantiated claims in a couple of places that the convolution of two beta distributions has no analytical form.  Has it been proved that an analytical form doesn't exist, or is there just no known analytical form?  I'm assuming you can still estimate the mean of the posterior as the difference of the means of the two beta distributions, and the variance of the posterior as the sum of the variances (if independent), or the sum of the variance plus twice the covariance (if the covariance is non-zero)?  Is there a closed form for covariance between two beta distributions?  I *think* that the 1st order Taylor series approximation of the covariance is given by \frac{\alpha_1 \times \alpha_2 + (1 + \beta_1) \times (1 + \beta_2)}{(\alpha_1 + \beta_1) \times (\alpha_2 + \beta_2) \times (1 + \alpha_1 + \beta_1) \times (1 + \alpha_2 + \beta_2)} as shown in Eq (4) of "An algorithm for generating positively correlated Beta-distributed random variables with known marginal distributions and a specified correlation", but I'm not sure if it's OK to drop the \delta_1 and \delta_2 variables.
It would be great if there could be some treatment of these topics in the main article. LukeH (talk) 01:34, 27 October 2010 (UTC)
replies: text: There is a paper on this very topic by Pham-Gia and Turkkan (1993), "Bayesian analysis of the difference of two proportions", Commun. Statist.-Theory Meth, 22(6), 1755-1771.  There is an expression in the paper for the posterior. IapetusWave (talk) 18:08, 29 January 2012 (UTC)
thread_title:  Difference between two beta-distributed RVs 

message: To grasp the meaning of the beta distribution, an intuitive interpretation and/or an illustratory example of application from everyday life would be very helpful for the layperson. --Typofier 08:23, 22 June 2011 (UTC)
replies: text: The applications section lists a couple good ones, although framed in mathematical language.  The Bayesian interpretation can be described in concrete terms as follows.  Suppose you have an Internet ad you place on your webpages and keep track of click-throughs.  Each occurrence can be thought of as a Bernoulli trial, and a click-through can be considered a "success".  After gathering some amount of observations/data, e.g. 1000 trials and 50 click throughs, you'd like to know what the probability p of a click-through is.  This probability p is itself distributed probabilistically as a beta distribution, if you assume that your prior knowledge was zero (mathematically, the prior distribution of p is assumed to be uniform on (0,1) ).  --C S (talk) 18:53, 4 July 2012 (UTC)
thread_title: Intuitive interpretation?

message: The article currently defines the Dirichet distribution as "the multivariate generalization of the beta distribution." Certainly, there are other multivariate distributions, such as those for random matrices. Refer, for example, to C. G. Khatri, "On the mutual independence of certain statistics." Annals of Mathematical Statistics, 30 : 4 : 1258--1262 (1959). Accordingly, I have changed "the" to "a" in the quoted portion of the article. Lovibond (talk) 03:54, 22 September 2011 (UTC)
thread_title:  Multivariate beta distribution and Dirichlet 

message: Currently the Mean and the Variance based on Ln[X] are in the property box for the Beta Distribution:
* :\operatorname{E}[\ln X] = \psi(\alpha) - \psi(\alpha + \beta)\!
* :\operatorname{Var}[\ln X] = \psi^{(1)}(\alpha) - \psi^{(1)}(\alpha + \beta)\!
I presume that these have been added because the Ln[X] transformation extends the distribution from a bounded [0,1] domain for X to a semi-infinite Ln[X] domain (for X approaching zero) ?.  
QUESTION: Should we have these Ln[X] properties in the statistical property box rather than just in the text? 
The same could be done for other statistical distributions on a bounded domain. However, I could not find such an addition in the property box for other distributions. 
I don't feel strongly about this one way or another, but it would be nice to arrive at some convention as to what properties should be included in the statistical property box for distributions in Wikipedia.
Also, the Ln[X] transformation naturally leads to a discussion of Johnson distributions, which I have not found presently in Wikipedia.
Thanks,
Dr. J. Rodal (talk) 14:53, 11 August 2012 (UTC)
thread_title:  Should the Mean and the Variance based on Ln[''X''] (rather than ''X'') be included in the statistical property box? 

message: I added a new section titled "Kurtosis bounded by the square of the skewness," which contains the following equation:
<< 
The region occupied by the beta distribution is bounded by the following two lines in the (skewness2,kurtosis) plane, or the (skewness2,excess kurtosis) plane:
:(\text{skewness})^2+1< \text{kurtosis}< \frac{3}{2} (\text{skewness})^2 + 3
>>
I am aware that the (otherwise excellent) reference "Gupta (Editor), Arjun K. (2004). Handbook of Beta Distribution and Its Applications. CRC Press. pp. 42. ." quoted in the rest of this Wikipedia article instead has this equation on page 42 (in section VII of the chapter "Mathematical properties of the Beta distribution" by Gupta and Nadarajah):
:(\text{skewness})^2+1< \text{kurtosis}< (\text{skewness})^2 + 3
Gupta et.al quotes Karian (1996) as the source of this equation.  The lower bound for this equation is correct: it is the "impossible region" previously found by K. Pearson.  However, the upper bound equation in Gupta:
 \text{kurtosis}< (\text{skewness})^2 + 3
is incorrect.  The correct equation is: 
 \text{kurtosis}< \frac{3}{2} (\text{skewness})^2 + 3
as correctly found by K.Pearson practically a century ago (one can also verify this by numerical examples, as I included in the Wikipedia article).Dr. J. Rodal (talk) 22:20, 12 August 2012 (UTC)
thread_title:  Kurtosis bounded by the square of the skewness 

message: I added a new section titled "Mean absolute deviation around the mean".  A few notes:
* At the time of this writing, Wikipedia "Mean deviation" redirects to the article Absolute deviation which very succinctly discusses only the discrete case for the mean absolute deviation, and does not contain formulas for the continuous case.
*The term "absolute deviation" does not uniquely identify a measure of statistical dispersion, as there are several measures that can be used to measure absolute deviations, and there are several measures of central tendency that can be used as well.  Thus, to uniquely identify the absolute deviation it is necessary to specify both the measure of deviation and the measure of central tendency. Unfortunately, the statistical literature has not yet adopted a standard notation, as both the mean absolute deviation around the mean and the median absolute deviation around the median have been denoted by their initials "MAD" in the literature, which may lead to confusion, since in general, they may have values considerably different from each other.
*The classic reference by N.L.Johnson and S.Kotz, in their (1970) first edition of "Continuous Univariate Distributions Vol. 2" , Wiley, Chapter 21:Beta Distributions, page 40, Eq. (10) contains an error for the mean absolute deviation: the exponent in the denominator should be (p+q+1) instead of (p+q).  That is why I quote Gupta instead of Johnson and Kotz for this formula.
* Weisstein, Eric W. "Mean Deviation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MeanDeviation.html, contains a number of expressions for the mean absolute deviations for a number of distributions.  Unfortunately, as of this writing, the expression for the mean absolute deviation in  Weisstein, Eric W. "Mean Deviation." From MathWorld--A Wolfram Web Resource (see http://mathworld.wolfram.com/images/equations/MeanDeviation/Inline19.gif) is unnecessarily lengthy and complicated: it involves the (unnecessary) computation of three different expressions for the beta function (instead of just one).  As of this writing, Weisstein, Eric W. "Mean Deviation" article does not quote either Johnson and Kotz or Gupta.  It can be shown that Weisstein's expression (although unnecessarily complicated) is completely identical to Gupta's shorter expression and Johnson and Kotz's (correcting their above-mentioned errata).  The expression by Weisstein was apparently obtained by using Mathematica (as he includes a Mathematica notebook with the derivation).  As of this writing, Mathematica version 8 does not "know" the beta function identities necessary to transform Weisstein's expression into the simpler expression, when using Mathematica's "FullSimplify" standard function.  However, it is easy to use Mathematica to derive the simpler expression (by including the relevant beta function identity).
Dr. J. Rodal (talk) 15:36, 23 August 2012 (UTC)
thread_title:  Mean absolute deviation around the mean 

message: I wrote a section on the Fisher information matrix for the beta distribution.  I framed its derivation in terms of the log likelihood function (as done for example by E.T.Jaynes in "Probability theory, the logic of science", A.W.F. Edwards in "Likelihood", and several others) instead of the probability density function conditional on the value of a parameter as done in the Wikipedia article on Fisher Information, to emphasize its main role in parameter estimation.  These two ways to frame it are equivalent and whether to chose one or another is a matter of preference and the context in which one is writing.  For the four parameter case I quote the recent article by Aryal and Nadarajah because it may be more readily available to an Internet audience, since its ".pdf" is presently freely available through the Internet.  This article by Aryal and Nadarajah contains an error for the Fisher information component: the component given by the log likelihood function's second order derivative with respect to the minimum "a" (or "c" in Aryal and Nadarajah's nomenclature). The expression in their article is incorrect in several ways, and it appears incorrectly twice (first as the second order partial derivative of the likelihood function and then again as the expected value for the Fisher information component). I have corrected their error and placed the correct equation in the Wikipedia article.  One can verify this by obtaining the correct equation by differentiation of the log likelihood function and carrying out the Integration for the expected value, one can readily see that this Fisher information component is incorrect in Aryal and Nadarajah from symmetry arguments, since the Fisher information terms given by the second order differentiation with respect to the minimum "a" and the Fisher information component given by the second order differentiation with respect to the maximum "c" (or "d" in Aryal and Nadarajah's nomenclature) should be symmetric.  It is curious that Aryal and Nadarajah do not use the trigamma function in their expression for the first three Fisher information matrix components, and instead express it in terms of the (more lengthy expressions for the) derivatives of the gamma function. Dr. J. Rodal (talk) 18:11, 13 September 2012 (UTC)
thread_title:  Fisher information matrix 

message: "It has also been" is used 5 times in a row in the lead. Boooring.... :-) I guess the intro is too loong and descriptive and most of those uses (all of them?) could go into a (new and first) section of their own. - Nabla (talk) 14:02, 18 September 2012 (UTC)
replies: text: Introduction "has also been" :-) modified to accommodate your request to eliminate repetition of "has also been"Dr. J. Rodal (talk) 14:12, 19 September 2012 (UTC)
thread_title:  "It has also been" excessively used 

message: revision 527168631 by 99.241.86.114 placed the Mean and Sample Size Parametrization in the introductory paragraph.  This can be justified by the fact that this parametrization is discussed throughout the text in different sections.  However, other more (historically) important parametrizations (for example the four parameter parametrization) are also discussed throughout the text.  Is there a reason why (only) the Mean and Sample Size should be discussed in the introductory paragraph ? Should all parametrizations be discussed in the introductory paragraph? Isn't it adequate to discuss the parametrizations under their own section? Alternatively, should the parametrization section be moved near the top of the article?  Dr. J. Rodal (talk) 17:56, 9 December 2012 (UTC)
thread_title:  Parametrizations 

message: Ferrari and Cribari-Neto (2004) propose a Beta regression algorithm. This approach is not very popular (doesn't have a wikipedia page) but I think it's worth mentioning it here.  — Preceding unsigned comment added by 134.191.232.71 (talk) 08:29, 19 February 2013 (UTC)
thread_title:  Beta regression 

message: Unless you already have enough grounding in mathematics to already know what the Beta Distribution is this article never actually explains it. The first sentence (which is supposed to be accessible to lay people) is just a jargon filled way of saying "its an equation with some variables in it".  — Preceding unsigned comment added by 167.206.48.220 (talk) 21:02, 20 February 2013 (UTC)
thread_title:  What is the Beta Distribution? 

message: From one point of view, the article is great as a quite detailed description of the beta distribution and its properties. However, considering wikipedia an encyclopedia, I have serious doubts about the article's usefulness. If I need to find something quite general, I skim through encyclopedia (e.g. wikipedia). For details, I never do this, because it's not the purpose of any encyclopedia to be exhaustive in any entry. I believe that focused books/papers are appropriate for that. In this respect, the beta distribution article on wikipedia fails to deliver the "encyclopedic message". I honestly respect the great effort of the author and I'm not sure about any wiki-consistent modification. Maybe to move the detailed content to another article like "Beta distribution (details)"?  — Preceding unsigned comment added by 77.78.117.1 (talk) 09:23, 7 September 2013 (UTC)
replies: text: This is a philosophical point. Should an encyclopedia be encyclopedic - be far reaching and encompasing as much as possible of a topic - or merely a simple reference work outlining the main points of a topic. My personal bias is towards the former. WP allows for the aggregation of information in a way that is impossible in encyclopedias that have been created before. Another question that might better adress what I think the P was suggesting here would be: could this article be divided into multiple sub articles? Again I am not sure. I actually like long articles as I may find material in them that I would not have thought connected to the original matter. But I do know others have different opinions on these matters so additional opinions would be useful here. DrMicro (talk) 12:21, 17 September 2013 (UTC)
thread_title:  Length and intelligibility of the article 

message: This section is unnecessarily wordy, and ends up repeating a lot of the same information. While I appreciate that it's worth pointing out the difference between the Jeffreys, Bayes and Haldane priors is it really necessary to specify their posterior distributions individually? And after that there is a huge block of text which deals with the s=0 and n=0 cases in such a way that there is lots of text that says nothing. Most of this could go.
118.208.48.83 (talk) 20:04, 24 October 2013 (UTC)
thread_title:  Sensitivity of posterior different priors 

message: At User_talk:Jimbo_Wales#Search_recent_changes.3F, someone has pretended that page Beta distribution is owned by a terrific dragon that delete any contribution that tries to make the article clearer and shorter (see https://en.wikipedia.org/w/index.php?title=User_talk%3AJimbo_Wales&type=revision&diff=710796589&oldid=710789244). Let us try if this assertion contains any part of truth. Pldx1 (talk) 17:32, 19 March 2016 (UTC)
thread_title:  To own or not to own 

message: The inbox says x ∈ (0,1) while the first paragraph says it is [0,1]. which is it?107.133.211.254 (talk) 12:05, 6 August 2016 (UTC)
replies: text: [0,1], as the support (mathematics) for a distribution on the real line is always a closed interval, not an open interval. I've fixed the infobox. Qwfp (talk) 13:05, 7 August 2016 (UTC)
thread_title:  range of support 

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thread_title:  External links modified 

message: The equation in the Probability Density Function section in the beginning of the article uses the parameter "u" in the denominator but fails to mention what "u" is.  Clarification would be appreciated.  — Preceding unsigned comment added by Pwvirgo (talk • contribs) 15:23, 29 July 2017 (UTC)
thread_title:  Undefined parameter in PDF 

message: I believe that there is an error in the following formula:
:-\frac{1}{N} \frac{\partial^2\ln \mathcal{L} (\alpha, \beta, a, c\mid Y)}{{\partial \alpha}{\partial \beta}} = \operatorname{cov}[\ln {X,(1-X)}]  = -\psi_1(\alpha+\beta) ={\mathcal{I}}_{\alpha, \beta}=  \operatorname{E} \left [- \frac{1}{N}\frac{\partial^2\ln \mathcal{L} (\alpha, \beta, a, c\mid Y)}{\partial \alpha \partial \beta} \right ] = \ln(\operatorname{cov_{G{X,(1-X)}}})
It should be the covariance between ln(X) and ln(1-X) instead of between ln(X) and 1-X:
:-\frac{1}{N} \frac{\partial^2\ln \mathcal{L} (\alpha, \beta, a, c\mid Y)}{{\partial \alpha}{\partial \beta}} = \operatorname{cov}[\ln X, \ln(1-X)]  = -\psi_1(\alpha+\beta) ={\mathcal{I}}_{\alpha, \beta}=  \operatorname{E} \left [- \frac{1}{N}\frac{\partial^2\ln \mathcal{L} (\alpha, \beta, a, c\mid Y)}{\partial \alpha \partial \beta} \right ] = \ln(\operatorname{cov_{G{X,(1-X)}}})
This would be consistent with the formula in Section 3.3.1 for the 2 parameter case. However, I am not entirely sure, so perhaps someone else can also have a look at this.
2A00:E040:C0C:4501:0:0:0:167 (talk) 10:25, 24 January 2018 (UTC)
thread_title:  Error in section 3.3.2 Fisher information matrix, Four parameters? 

message: The first sentence suggests that the distribution is a function of some random variable, which surely is not correct. I propose to remove "appear as exponents of the random variable and".  — Preceding unsigned comment added by Prof. Globi (talk • contribs) 22:31, 20 March 2019 (UTC)
thread_title:  Error in first sentence 

message: The following two sections should be merged:
* Beta_distribution#Rule_of_succession
* Beta_distribution#Bayesian_prior_probability_(Beta(1,1))
Tal Galili (talk) 11:40, 30 November 2019 (UTC)
thread_title:  Two sections needs merging 

message: The article states that in a particular limit, the Beta distribution approaches the Gamma distribution. Is there a source or derivation for this? Also, should it also mention approaching the Delta distribution in another limit? Cesiumfrog (talk) 00:17, 4 August 2020 (UTC)
thread_title:  Limiting towards Gamma 

message: I tried to translate the article into Czech with the Wikipedia built-in translator, but I was forced to stop after a few sentences. The article is so incredibly long and equation/graphics heavy that it breaks the tool. Can please somebody knowledgeable divide the article into a set of shorter ones? There is a lot of material here which is of interest only for very specialized experts.Jan Spousta (talk) 11:24, 22 December 2020 (UTC)
replies: text: Tal, Normal distribution has about 1/2 of text compared with this, and much less pictures. This article has 247,277 bytes of wiki markup size and more than 200,000 bytes of readable prose size. WP:TOOBIG says that articles over 100,000 bytes "Almost certainly should be divided".Jan Spousta (talk) 09:16, 23 December 2020 (UTC)
replies: text:  Hey, interesting and fair point. I'm not sure what to tell you. Each of the sections I'm looking at are reasonable. I'm wondering how they should be moved. For example, the properties section has information on many measures and moments on the beta distribution. Should there be a new article "Beta distribution - properties"?! (obviously it shouldn't be called properties, but maybe something else?). Is there another example of such an article for other distributions? I'm asking to understand what prior example we might follow here. Tal Galili (talk) 09:36, 24 December 2020 (UTC)
thread_title:  Too heavy article