Talk:Akaike information criterion/Archive 1

Archive 1

Archive 2

Deviance Information Criterion

I've written a short article on DIC, please look it over and edit. Bill Jefferys 22:55, 7 December 2005 (UTC)

Great, thanks very much. I've edited a bit for style, no major changes though. Cheers, --MarkSweep (call me collect) 23:18, 7 December 2005 (UTC)

Travis Gee

I have sent an e-mail to this Mr. Gee who is cited as a possible reference, with the following text:

Dear Mr. Gee,

For some time now your name is mentioned in the Wikipedia article on the Akaike Information Criterion (http://en.wikipedia.org/wiki/Akaike_information_criterion). You are cited as having developed a pseudo-R2 derived from the AIC. However, no exact reference is given. I'd be glad to hear from you whether you actually developed this, and where, if anywhere, you have published this measure.

Thank you for your cooperation.

However, he has not answered. I will remove the reference. Classical geographer 12:18, 2 April 2007 (UTC)

That measurement ( R^2_{AIC}= 1 - \frac{AIC_0}{AIC_i} ) doesn't make sense to me. R^2 values range from 0-1. If the AIC is better than the null model, it should be smaller. If the numerator is larger than the denominator, the R^2_{AIC} will be less than 1. This is saying that better models will generate a negative R^2_{AIC}.

It would make sense if the model were: R^2_{AIC}= 1 - \frac{AIC_i}{AIC_0}

Denoting pronunciation

Please write the pronunciation using the International phonetic alphabet, as specified in Wikipedia:Manual of style (pronunciation). -Pgan002 05:09, 10 May 2007 (UTC)

Confusion

The RSS in the definition is not a likelihood function! However, it turns out that the log likelihood looks similar to RSS. —Preceding unsigned comment added by 203.185.215.144 (talk) 23:12, 7 January 2008 (UTC)

I agree. What's written is actually the special case of AIC with least squares estimation with normally distributed errors. (As stated in Burnham, Anderson, "Model selection and inference". p48) Furthermore you can factor out ln(2pi)*n and an increase in K due to using least squares. These are both constant when available data is given, so they can be ignored. The AIC as Burnham and Anderson present it, is really a tool for ranking possible models, with the one with the lowest AIC being the best, the actual AIC value is of less importance. EverGreg (talk) 15:27, 29 April 2008 (UTC)

Link spam?

I think the link that appeared at the bottom "A tool for fitting distributions, times series and copulas using AIC with Excel by Vose Software" is not too relevant and only one of many tools that may incorporate AIC. I am not certain enough to remove it myself. Dirkjot (talk) 16:36, 17 November 2008 (UTC)

Controversy?!

What on earth is this section? It should be properly explained, with real references, or permanently deleted! I would like to see a book on model selection which describes AIC in detail, but also points out these supposed controversies! True bugman (talk) 11:50, 7 September 2010 (UTC)

This is a good question, but there is probably something that does need to be said about properties of AIC, not necessarily under "Controversy". For example, in this online dissertation, I found "AIC and other constant penalties notoriously include too many irrelevant predictors (Breiman and Freedman, 1983)" with the reference being: L. Breiman and D. Freedman. "How many variables should be entered in a regression equation?" Journal of the American Statistical Association, pages 131–136, 1983. There are similar results for using AIC to select a model order in time series analysis. But these results just reflect the penalty on large models that is inherent in AIC, and arises from the underlying derivation of AIC as something to optimise. Melcombe (talk) 16:04, 7 September 2010 (UTC)

Time series data is only one of many data types where modelling and AIC are used together. Something like this should be included in a special section dedicated to time series data. True bugman (talk) 12:09, 8 September 2010 (UTC)

The point was that the supposed problem with AIC is known to occur for both regression and time series, in exactly the same way, so it would be silly to have to say it twice in separete sections. Melcombe (talk) 16:51, 8 September 2010 (UTC)

Regression, as an example, is not covered in this article. Neither is time series. But yes, AIC is not perfect, and yes this should probably be discussed. But in a neutral way, this is by no means a controversy. I believe the entire 'controversy' section should be deleted. These are all recent changes from different IP addresses (110.32.136.51, 150.243.64.1, 99.188.106.28, 130.239.101.140) unsupported by citation, irrelevant for the article (the controversial topics discussed are not even in the article), and it is very poorly written (again there is no connection to the article). What does "crossover design", "given to us a priori by pre-testing", and "Monte Carlo testing" even mean? This section is written as an attack on the technique rather than a non-biased source of information. It is not verifiable WP:V nor written with a neutral point of view WP:NPOV. It must go. True bugman (talk) 17:19, 8 September 2010 (UTC)

Takeuchi information criterion

I removed the part on Takeuchi information criterion (based on matrix trace), because this seemed to give credit to Claeskens & Hjort. There could be a new section on TIC, if someone wanted to write one; for now, I included a reference to the 1976 paper. Note that Burnham & Anderson (2002) discuss TIC at length, and a section on TIC should cite their discussion. TIC is rarely useful in practice; rather, it is an important intermediate step in the most-general derivation of AIC and AICc.  86.170.206.175 (talk) 16:24, 14 April 2011 (UTC)

BIC Section

The BIC section claims Akaike derived BIC independently and credits him as much as anyone else in discovering BIC. However, I have always read in the history books that Akaike was very excited when he first saw (Schwartz's?) a BIC derivation, and that after seeing that it inspired him to develop his own Bayesian version of AIC. I thought it was well-documented historically that this was the case, and that he was a very graceful man who didn't think of BIC as a competitor to him, but thought of it as just yet another very useful and interesting result. His only disappointment, many accounts do claim, was that he didn't think of it himself earlier. Isn't that the standard way that all the historical accounts read?

Your version of events seems right to me. Akaike found his Bayesian version of AIC after seeing Schwartz's BIC. BIC and the Bayesian version of AIC turned out to be the same thing. (Maybe you should edit the article?) — Preceding unsigned comment added by 86.156.204.205 (talk) 14:10, 14 December 2012 (UTC)

Removed confusing sentence

I removed the following sentence: "This form is often convenient, because most model-fitting programs produce $\chi ^{2}$ as a statistic for the fit." The $\chi ^{2}$ statistic produced with many model-fitting programs is in fact the RSS (e.g. Origin [1]). But the RSS cannot simply replace $\chi ^{2}$ in these equations. Either the σi has to be known or the following formula should be used AIC = n ln(RSS/n) + 2k + C. — Preceding unsigned comment added by 129.67.70.165 (talk) 14:34, 21 February 2013 (UTC)

Example?

The example from U. Georgia is no longer found; so I deleted it. It was:

• Example Calculation (University of Georgia)
• ie, * http://coopunit.forestry.uga.edu/Coop_Wkshop/inference_effects/aic_reg.pdf Example Calculation (U.GA)

I added the best example I could find with a Google-search: [Akaike example filetype:pdf]  Done — Charles Edwin Shipp (talk) 13:31, 11 September 2013 (UTC)

AICc formula

Hi all, I recently made a small change to the AICc formula to add the simplified version of the AICc (i.e. not in terms of the AIC equation above, but the direct formula) so that readers could see both the AICc's relation to the AIC (as a "correction") and as the formula recommended by Burnham and Anderson for general use [edit number 618051554].

An unnamed user reverted the edit, citing "invalid simplification." Does anyone (the reverting editor from ip 2.25.180.99 included) have any reason to not want the edit I made to stand? It adds the second line to the equation below,

${\begin{aligned}AIC_{c}&=AIC+{\frac {2k(k+1)}{n-k-1}}\\&={\frac {2kn}{n-k-1}}-2\ln {(L)}\end{aligned}}$

The main point I have in favor of the change is that Burnham and Anderson point out in multiple locations that the AIC should be thought of as the asymptotic version of the AICc, rather than thinking of the AICc as a correction to the AIC. This second equation shows easily (by comparison with the AIC formula) how that asymptotic relationship holds, and it shows how to compute the AICc itself directly.

The point against it, as far as I can see, is just that there is now a second line of math (which I can imagine some people being opposed to...)

Any opinions? If no, I'll change my edit back in a few days/weeks. Dgianotti (talk) 18:03, 31 July 2014 (UTC)

The second equation is certainly valid. I do not see why the proposed new formula is a simplification though; for me, the first (current) formula is simpler. Regarding the asymptotic relationship, this is shown much more clearly by the first formula: as n→ ∞, the second term plainly goes to 0; so the formula becomes just AIC, as Burnham and Anderson state. Hence I prefer the current formula. 86.152.236.37 (talk) 21:11, 13 September 2014 (UTC)

Confusion 2

The equation given here for determining AIC when error terms are normally distributed does not match the equation given by Burnham and Anderson on page 63 of their 2002 book. Burnham and Anderson's equation is identical except that it does not include a term with pi. Anyone know why this is? Tcadam (talk) 03:13, 17 December 2008 (UTC)Tcadam (talk) 03:14, 17 December 2008 (UTC)

Hi, I took the liberty to format your question. this is touched on in the "confusion" paragraph above. I assume you mean this equation:

${\mathit {AIC}}=2k+n[\ln(2\pi {\mathit {RSS}}/n)+1]\,.$

We should really fix it. Since for logarithms ln(x*y) = ln(x) + ln(y), you can factor out the 2pi term so that AIC = Burnham and andersons equation + 2pi term. since the 2pi term is a constant, it can be removed. This is because AIC is used to rank alternatives as best, second best e.t.c. Adding or subtracting a constant from the AIC score of all alternatives can't change the ranking between them. EverGreg (talk) 12:18, 17 December 2008 (UTC)

Added the simplified version in the article and emphasized ranking-only some more. By the way, did Burnaham and Anderson skip the + 1 term too? EverGreg (talk) 12:39, 17 December 2008 (UTC)

I dont understand why you are using the term RSS/n. I dont see that in at least two of the references i am looking at. It is just RSS. 137.132.250.11 (talk) 09:29, 29 April 2010 (UTC)

Exactly. the 1/n term can be factored out and removed just like the 2pi term, using that ln(x/y) = ln(x) - ln(y). It makes no difference if it's there or not, so most books should really go with ${\mathit {AIC}}=2k+n[\ln({\mathit {RSS}})]\,$, as we have done in the article. The reason we see 2pi and 1/n at all is that they turn up when you take the general formula ${\mathit {AIC}}=2k-2\ln(L)\,$ and add the RSS assumption. We should probably add how this is derived, but I don't have a source on that nearby.EverGreg (talk) 13:19, 29 April 2010 (UTC)

Oh, I didn't check what you did on the article page. Thanks for spotting that! EverGreg (talk) 13:22, 29 April 2010 (UTC)

Further confusion: Is there a discrepancy between AIC defined from the $\chi ^{2}$: $AIC=2k-2C+\chi ^{2}$ and the RSS version: ${\mathit {AIC}}=2k+n[\ln({\mathit {RSS}})]\,$? Don't they differ with an extra $\ln$? —Preceding unsigned comment added by 152.78.192.25 (talk) 15:27, 13 May 2011 (UTC)

both formulas are valid, but the second one uses the additional assumption of a linear model. You can read that in . It is a bit confusing because they do not state the assumption of the linear model p. 63, but on p. 12 the derive the log-likelihood for the case of linear models and that makes it clear (I'm referring to page numbers in the second edition as found here Frostus (talk) 11:04, 16 September 2014 (UTC)

I reverted my change with the linear model. Although it is shown for a linear model in Burnham & Anderson, 2002, this assumption is not needed to derive the equation $AIC=n\ln(RSS/n)+2k+C$. I removed the part with "if the RSS is available", as it can always be calculated Frostus (talk) 14:12, 18 September 2014 (UTC)

It still looks wrong. There should be no ln in the ln(RSS) term after the "=" in this expression:$-{\frac {n}{2}}\ln(2\pi )-{\frac {n}{2}}\ln({\hat {\sigma }}^{2})-{\frac {1}{2{\hat {\sigma }}^{2}}}\mathrm {RSS} =-{\frac {n}{2}}\ln(\mathrm {RSS} /n)+C_{1}$ There might very well be such expressions in the literature, but that expression as it stands here is not mathematically valid.

I suspect that the whole derivation concerning chi-square is wrong, since it uses the likelihood function instead of the maximum of the likelihood function in the AIC. — Preceding unsigned comment added by 141.14.232.254 (talk) 19:22, 14 February 2012 (UTC)

I expect the maximum of the log of the likelihood function to be the same as the log of the maximum of the likelihood function - since log is a monotonically growing function?

References

Relevance to $\chi ^{2}$ fitting

I have contributed a modified AIC, valid only for models with the same number of data points. It is quite useful though. Velocidex (talk) 09:17, 8 July 2008 (UTC)

Could you please supply some references on this one? Many variations of AIC have been proposed, but as e.g. Burnham and Anderson stresses, only a few of these are grounded in the likelihood theory that AIC is derived from. I'm away from my books and in "summer-mode" so it could very well be that I just can't see how the section's result follow smoothly from the preceding derivation using RSS. :-)

EverGreg (talk) 11:22, 8 July 2008 (UTC)

For $\chi ^{2}$ fitting, the likelihood is given by $L=\prod \left({\frac {1}{2\pi \sigma _{i}^{2}}}\right)^{1/2}\exp \left(-\sum {\frac {(y_{i}-f(\mathbf {x} ))}{2\sigma _{i}^{2}}}\right)$

i.e. $\ln L=\ln \left(\prod \left({\frac {1}{2\pi \sigma _{i}}}\right)^{1/2}\right)-{\frac {1}{2}}\sum {\frac {(y_{i}-f(\mathbf {x} )}{\sigma _{i}^{2}}}$

$=C-\chi ^{2}/2$, where C is a constant independent of the model used, and dependent only on the use of particular data points. i.e. it does not change if the data do not change.

The AIC is given by $AIC=2k-2\ln(L)=2k-2(C-\chi ^{2}/2)=2k-2C+\chi ^{2}$. As only differences in AICc are meaningful, this constant can be omitted provided n does not change. This is the result I had before, which was correct. Velocidex (talk) 19:39, 18 May 2009 (UTC)

I should also say RSS is used by people who can't estimate their errors. If any error estimate is available for the data points, $\chi ^{2}$ fitting should be used. Unweighted linear regression is dangerous because it uses the data points to estimate the errors by assuming a good fit. You get no independent estimate of the probability that your fit is good, Q. Velocidex (talk) 19:53, 18 May 2009 (UTC)

I am unhappy with this section. It says "where C is a constant independent of the model used, and dependent only on the use of particular data points, i.e. it does not change if the data do not change."

But this is only true if the $\sigma _{i}$:s are the same for the two models. And under "Equal-variances case" it explicitly saya that $\sigma$ is unknown, hence is estimated by the models. For instance, if we compare two nested linear models, then the larger will estimnate $\sigma$ to a smaller value than the smaller model. In this case it is the converse: the "constant" C will differ between models, whereas the term with the exponentials will cancel out (they will both be exp(−1).)

The formula with RSS is correct, but the derivation is wrong for the above reason.

All this needs to be fixed. (Harald Lang, 9/12/2015) — Preceding unsigned comment added by 46.39.98.125 (talk) 11:36, 9 December 2015 (UTC)

Your point seems valid to me. Additionally, it is notable that the subsection "General case" does not have any references (unlike the subsection "Equal-variances case"). Moreover, I have just skimmed through Burnham & Anderson (2002), and did not see any supportive discussion that could be cited.

The editor who first added the text for the "General case" has been inactive for over a year; so asking them would probably not lead anywhere. Does anyone have a justification for keeping the "General case" subsection? If not, I will delete that subsection, and revise the "Equal-variances case" subsection.

SolidPhase (talk) 22:43, 10 December 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Akaike information criterion/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Hello, I am not a statistician and therefore can only provide remark about stuff I was unable to understand. My concern is about "k" : In the paragraph "Definition" it is said that k is the number of parameters in the statistical model In the paragraph "AICc and AICu" it is said that k denotes the number of model parameters + 1. If these two ks are different, then why to give them the same name. If they are not there is a problem of definition somewhere ?

Last edited at 13:27, 7 October 2009 (UTC). Substituted at 19:44, 1 May 2016 (UTC)

Origin of Name

AIC was said to stand for "An Information Criterion" by Akaike, not "Akaike information Criterion" Yoderj 19:39, 16 February 2007 (UTC)

This is similar to Peter Ryom developing the RV-index for the works of Vivaldi - instead of the official Répertoire Vivaldi, his index of course became known as the Ryom Verzeichnis... I am inclined to believe this is not unintentional on the part of the developer; it's probably (false?) modesty. Classical geographer 12:18, 2 April 2007 (UTC)

This criterion is alternately called the WAIC: Watanabe-Akaike Information Criterion[2], or the widely-applicable information criterion [3][4]. — Preceding unsigned comment added by 167.220.148.12 (talk) 10:48, 29 April 2016 (UTC)

WAIC and AIC are not the same. I have now listed a paper for WAIC in the "Further reading" section. SolidPhase (talk) 07:31, 16 May 2016 (UTC)

Recent edits by Tayste

@SolidPhase: Explain yourself. The "relative quality of a model" is ungrammatical - relative to what? This must be "models" plural. As for measuring "quality" - this sounds like higher values of AIC mean greater quality, but the reverse is true, so this should be made clear up front in the lead. Why remove that? Thirdly, WP:HEADINGS states that "Headings should not refer redundantly to the subject of the article". Lastly, it seems (to me) better to talk about how AIC works before discussing its limitations. Tayste (edits) 18:07, 18 June 2015 (UTC)

@Tayste: Okay, how about removing the word "relative" from the first sentence?

About "models" plural, will you elaborate? AIC gives the value for a single model; so I think that singular is appropriate.

I disagree with mentioning about higher/lower values in the lead. Following WP:LEAD, this looks like clutter that someone who read only the lead would not benefit from. The issue is discussed in the second paragraph of the body: in the first sentence, and italicized.

Which heading referred redundantly to the subject of the article?

Does your last point pertain to my edit?

SolidPhase (talk) 18:28, 18 June 2015 (UTC)

As an interim measure (only), I have restored the body to my last edit, but kept your lead section. SolidPhase (talk) 19:47, 18 June 2015 (UTC)

It has now been over four days.
Regarding the sentence "Lower values of AIC indicate higher quality and therefore better models", as above I think that including this is clutter, which will be especially distracting for people who only read the lead. Additionally, there are many activities where the minimum is the optimum, e.g. golf. Moreover, in the field of Optimization, the canonical examples are minimization. I definitely believe that the sentence should be removed; so I have now done that.
Regarding the grammatical changes that you made, I do not agree. Back in March, though, you found a grammatical error: and you were correct, of course. Hence I get the impression that you have a really good grammatical knowledge. I do not understand what you find grammatically wrong about the previous version, though, or why your version is correct. Simply put, I am confused about this(!). Your edits to the grammar remain as you made them, but I would really appreciate it if we could discuss this issue further. Will you explain the reasons for your grammatical change more?
SolidPhase (talk) 19:19, 22 June 2015 (UTC)

I've stayed away (partly) to give other editors an opportunity to chip in. The AIC value for a single model is completely meaningless in isolation. It tells absolutely nothing about the quality of that model. AIC values are only useful when the differences in values is taken for pairs of models fitted to the same data set. So the word "relative" must be there. Thank you for retaining the plural in the first sentence.

Despite your specific counter examples, the generally understood meaning of the verb to measure is that it assigns higher numbers for greater amounts of the aspect being measured. In terms of relative measurement, AIC measures not the quality of models but their lack of quality, since higher values mean worse. I disagree that it clutters the lead to state the direction in which AIC works. It is a fundamentally important point to get across early for anyone wishing to understand what AIC is. Tayste (edits) 20:56, 22 June 2015 (UTC)

To me (a far-from-expert in grammar), the phrase “relative quality of statistical models” seems inappropriate, because “quality” is singular and “models” is plural.

The lead currently states that AIC “offers a relative estimate of the information lost”; so the less information lost, the better. Regarding measure, this is a formal term in mathematics, and the definition requires that all measures be nonnegative. What about replacing the term “measure” by something else?—e.g. “AIC provides a means for assessing the relative quality of statistical models”. Could something like that be okay?

SolidPhase (talk) 22:17, 22 June 2015 (UTC)

"Quality" is indeed singular, but "relative quality" necessitates a comparison involving at least a pair of models. I'd be happy with "means" but I think "measure" here was being used in the more general sense (anywhere on the Real line) rather than that specific mathematical definition. The point about the information lost is actually a better definition than "quality". Tayste (edits) 22:44, 22 June 2015 (UTC)

Quick comments. The lead is confusing and misrepresents. AIC is a number that calculated without reference to other models. It is a metric that does not depend on other models. That is, the value of AIC ignores all other models. AIC's value is not "relative to each of the other models".

AIC can be used to rank different models under the AIC metric. Using that metric does not guarantee the earlier statement that "Lower values of AIC indicate higher quality and therefore better models." The notion of "better" is tempered by the metric. AIC might deprecate an exact model due to its complexity.

I don't care much about which scores are better. For fits, lower chi square values are better, so smaller is better is not a foreign concept.

I don't know if AIC has some value as an absolute metric. For example, if input variances are known, then reduced chi square near 1 suggests a good model.

Glrx (talk) 05:46, 23 June 2015 (UTC)

I agree that the lead is confusing. I have spent 1–2 hours trying to come up with something better, but so far I have got nothing constructive to propose. Including the sentence about lower values indicating higher quality makes the lead more confusing, which is why I have been advocating keeping the sentence out.

I agree that metric is more appropriate than measure, considering the formal mathematical definitions. I also think that the mathematical definitions are highly relevant, given that AIC is part of some fairly advanced mathematical statistics. One problem with "metric", though, is that the word is not commonly known. Hence, if the word were used, people without the requisite mathematical background would be confused.

SolidPhase (talk) 09:42, 23 June 2015 (UTC)

Are you quite sure that AIC is ranked from best is lowest? Here are rankings from a Mathematica case of the 5 best models for a problem that uses BIC for ranking:

BIC	AIC	HQIC
3.841	3.857	3.845
3.815	3.825	3.818
3.735	3.746	3.738
3.732	3.742	3.735
3.458	3.468	3.461

Note that they go from highest as best to lowest as worst. I checked on this and is seems that some programs output -AIC, not AIC. However, the word is "index." In addition to being accurate, it is in common usage and people with no higher mathematical training understand it. Please change this, post an objection to the change or otherwise I will change it. If you then change it back without discussion, which is typical, we will have a dispute, as I will keep changing it back until there is a dispute settlement. CarlWesolowski (talk) 14:21, 11 July 2016 (UTC)CarlWesolowski (talk) 18:37, 14 July 2016 (UTC)

I strongly oppose using the word "index", because the word would be confusing here.

Your claim that I undo your edits "without discussion, which is typical" is false, and slanderous.

Your threat to start an edit war is in violation of Wikipedia norms.

SolidPhase (talk) 14:30, 15 July 2016 (UTC)

No surprise that you object to the word "index." However, not only is AIC an index, but as it is based on Shannon entropy, it is a data specific index namely Self-information. So, most indices would smoke it, and they tend to be at least somewhat comparable between data sets. I am just asking for a more objective presentation. Moreover, I am convinced that you cannot take this article to the next level. I have found some of the advocates for AIC use to be unusually partisan, which given AIC's very limited applicability due to the restrictive assumptions not being met as a common occurrence, is somewhat difficult for me to reconcile with objectivity.CarlWesolowski (talk) 22:36, 12 September 2016 (UTC)

Biased Tone of BIC Comparison Section

I made a few minor edits in the BIC section to try to keep it a *little* more neutral, but it still reads with a very biased tone. I imagine a bunch of AIC proponents had a huge argument with BIC proponents and then decided to write that section as pro-AIC propaganda. You can find just as many papers in the literature that unjustifiably argue that BIC is "better" than AIC, as you can find papers that unjustifiably argue AIC is "better" than BIC. Furthermore, if AIC can be derived from the BIC formalism by just taking a different prior, then one might argue AIC is essentially contained within "generalized BIC", so how can BIC, in general, be "worse" than AIC if AIC can be derived through the BIC framework?

The truth is that neither AIC nor BIC is inherently "better" or "worse" than the other until you define a specific application (and by AIC, I include AICc and minor variants, and by BIC I include variants also to be fair). You can find applications where AIC fails miserably and BIC works wonderfully, and vice versa. To argue that this or that method is better in practice, because of asymptotic results or because of a handful of research papers, is flawed since, for most applications, you never get close to the fantasy world of "asymptopia" where asymptotic results can actually be used for justification, and you can almost always find a handful of research papers that argue method A is better than method B when, in truth, method A is only better than method B for the specific application they were working on. — Preceding unsigned comment added by 173.3.109.197 (talk) 17:44, 15 April 2012 (UTC)

The difference between AIC and BIC is not explored in this biased article. To see some of these differences viewed by rather more knowledgeable people, including, for example, Rob Hyndman, who relates:

   AIC is best for prediction as it is asymptotically equivalent to cross-validation.
   BIC is best for explanation as it is allows consistent estimation of the underlying data generating process.

Please follow the link http://stats.stackexchange.com/questions/577/is-there-any-reason-to-prefer-the-aic-or-bic-over-the-other CarlWesolowski (talk) 17:39, 1 October 2016 (UTC)

Recent edit war by anonymous IPs

This section is created to discuss the recent edits by anonymous IPs. SolidPhase (talk) 10:28, 31 October 2016 (UTC)

Bayes' Factor

Query - should this page not also link to Bayes' Factor[5]?

I'm not an expert in model selection but in my field (molecular phylogenetics) model selection is an increasingly important problem in methods involves Bayesian inference (e.g. MyBayes, BEAST) and AIC is apparently 'not appropriate' for these models [6]

Any thoughts anyone? I've also posted this on the model selection[7] page. Thanks.--Comrade jo (talk) 12:19, 19 December 2007 (UTC)

I agree. The opening statement: "Hence, AIC provides a means for model selection." should read "Hence, AIC provides a means for model selection, in certain circumstances." Circumstances in which it is not appropriate abound, see introduction in — Preceding unsigned comment added by CarlWesolowski (talk • contribs) 21:49, 12 November 2016 (UTC)

Number of parameters?

In the article, it is written "If the model under consideration is a linear regression, k {\displaystyle k} k is the number of regressors, including the intercept". This is wrong, isn't it? What about the error variance? Shouldn't the error variance also count as a parameter? — Preceding unsigned comment added by 193.174.15.2 (talk) 09:30, 3 January 2017 (UTC)

Personal attack by SolidPhase upon CarlWesolowski with reversal of all edits

Some of the edits are ungrammatical, e.g. "AIC use as one means of model selection". Some of the edits introduce technical invalidity, e.g. "each candidate model has residuals that are normal distributions". I have undone the edits.

CarlWesolowski has been sporadically making edits to this article since at least 20 March 2015. Each time, those edits have been undone. My suggestion is this: if CarlWesolowski wants to make changes to the article, then he should discuss those changes on this Talk page, and get a consensus of editors to agree to the changes.
SolidPhase (talk) 06:18, 8 July 2016 (UTC)

That my edits can and have been reversed is not surprising. That SolidPhase takes exception to my person is inexcusable, calling me "ignorant" on my talk page. This article is misleading. The premise "Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Hence, AIC provides a means for model selection." is not logical. For example, BIC will yield better quality of ft than AIC. BIC is more appropriate for model selection than AIC. When a model is already selected, AIC will provide better estimates of that model's parameter values than BIC.

AIC is only one of many criteria for model selection, and often suggested for use when it is inappropriate. "The AIC is not a consistent model selection method"-point 10 of Facts and fallacies of the AIC by Rob J Hyndman . The introduction reads like a commercial for cigarettes. Mathematica uses BIC, as one of several tests whose combined score ranks models, and, there are lots of good folks who may compute AIC but not use it to rank. In addition to AIC, other methods used include BIC, step-wise partial probability ANOVA, HQIC, log likelihood, complexity error, factor analysis, and goodness of fit testing with Pearson Chi-squared, Cramer Von Mises probabilities and others. And, without looking at those other measurements, any pronouncements made with respect to model selection using AIC should be ignored. It is said that AIC does not assume that there is a true model, but BIC does. BIC is also more self-consistent. Neither of these maximum likelihood approaches is appropriate for model selection when the objective is extrapolation, not interpolation, as the goodness-of-extrapolation makes goodness-of-fit irrelevant.

In the section that says in rather poor quality English "Sometimes, each candidate model assumes that the residuals are distributed according to independent identical normal distributions (with zero mean). That gives rise to least squares model fitting." Let us take this one statement at a time.

Candidate models do not "assume." People assume. The requirement for normally distributed residuals is unnecessary, that happens approximately 10% of the time. Normally distributed residuals are not a requirement for AIC any more than they are for maximum likelihood. Again quoting Hyndman-point 3-"The AIC does not assume the residuals are Gaussian. It is just that the Gaussian likelihood is most frequently used. But if you want to use some other distribution, go ahead. The AIC is the penalized likelihood, whichever likelihood you choose to use." Again with inanimate objects making assumptions, tisk, tisk.

"That gives rise to least squares model fitting." Well, no it doesn't. Other assumptions for OLS can include homoscedasticity, and fixed intervals on the x-axis. Otherwise, OLS fit parameters are biased and only approximate. Summarizing, I really think that one should consider pulling back from the claims herein and injecting some perspective into this sloppy article. You will not let me fix this article, so fix it yourselves. — (talk • contribs) 03:08, 9 July 2016 (UTC) CarlWesolowski (talk) 07:51, 29 January 2017 (UTC)

A more general treatment of the fit problem that may be worth mentioning is QML, Quasi-Maximum Likelihood, based upon . It is currently totally unclear what the statistical use of AIC is in the article, so fix it. IThe current article is dangerous, it promotes AIC without sufficient insight as to appropriate usage. CarlWesolowski (talk) 17:09, 10 July 2016 (UTC)CarlWesolowski (talk) 07:51, 29 January 2017 (UTC)

The word "a" is used as an indefinite article. Thus, it is clear that there might be more than one. The paragraph also links to model selection, which lists 13.

Some of your remarks about model assumptions might be appropriate for the article on model selection. They are, however, not specific to AIC.

It is colloquial to talk about models (rather than people) assuming something.

It is common to assume that "the residuals are distributed according to independent identical normal distributions (with zero mean)". That assumption "gives rise to least squares model fitting", as the article states. Other assumptions can also give rise to least squares model fitting, but that is irrelevant in the context.

SolidPhase (talk) 19:15, 10 July 2016 (UTC)

Thank you for responding. However, the "a" is too soft. In the matter of implication, hinting at something is not as good as saying it. This article has that problem throughout, and it is less useful in that form than it would be if it were more clearly written. For example, let us take the infamous sentence "Sometimes, each candidate model assumes that the residuals are distributed according to independent identical normal distributions (with zero mean). That gives rise to least squares model fitting." It took me a very long time to figure out what you are trying to say and, BTW, do not. Consider for "that gives rise to least squares..." it is unclear that it does, and most people having studied least squares would still not know what you are getting on about. Consider saying something relevant rather than making the reader study the phrase to make any sense out of it, namely, note that "There are several different frameworks in which the linear regression model can be cast in order to make the OLS technique applicable. Each of these settings produces the same formulas and same results. The only difference is the interpretation and the assumptions which have to be imposed in order for the method to give meaningful results. The choice of the applicable framework depends mostly on the nature of data in hand, and on the inference task which has to be performed." You do not say what you are assuming, and that is not a problem for the editors, but, it is a big problem for the readers.

When you use the "colloquialism" as you call it, you depreciate not only the language, but mask the fact that you have imposed an assumption, which does not help the reader understand what you are saying. The phrase "the residuals are distributed according to independent identical normal distributions (with zero mean)" is so inaccurate that it is nearly unintelligible. I think perhaps that you obliquely referring to ML $X$~ND, where $X=\left\{x_{1},x_{2},x_{3},{\text{...}}x_{n-1},x_{n}\right\}$ and ${\text{Maximize}}\left[p(X)=\prod _{i=1}^{n}p_{i}\left(x_{i}\right)\right]$, or some such. Take a look at . It is much more clearly written than this Wikipedia entry. It is not misleading, it is not oversold, and it give a much better indication of where AIC is in the universe of methods. Try to emulate that level of clarity, please. What happens when the residuals are not ND. Surely you realize that that is most of the time. AIC, BIC and maximum likelihood can be, and should be defined in that broader context, in which case there is no direct relationship to OLS, such that the relationship to OLS for normal residuals is an aside that does more to confuse than to clarify. CarlWesolowski (talk) 23:50, 10 July 2016 (UTC)CarlWesolowski (talk) 00:54, 11 July 2016 (UTC)CarlWesolowski (talk) 18:51, 14 July 2016 (UTC)CarlWesolowski (talk) 21:23, 6 February 2017 (UTC)

@SolidPhase The sentence "Sometimes, each candidate model assumes that the residuals are distributed according to independent identical normal distributions (with zero mean). That gives rise to least squares model fitting." is incorrect, because 1) Models do not make assumptions and when you do you confuse not only the reader but also yourself. To wit 2) When the residuals are actually normally distributed only then are OLS and AIC as both applied to normal residuals the same. However, 3) AIC does not assume normal residuals because A) AIC can be applied to non-normal residual structure, and B) The assumption of normal residuals is clearly yours as it is not a requirement for AIC.CarlWesolowski (talk) 22:52, 10 November 2016 (UTC)

References

AIC asymptotics, misleadingly phrased?

@BetterMath:, could you please explain why the sentence “Asymptotically, AIC selects the model that minimizes the mean squared error of (out-of-sample) prediction,” is wrong and/or misleading? By definition, an asymptotically efficient (information) criterion chooses the (candidate) model that minimises the mean squared error of prediction, and by Stone (1977) AIC is asymptotically efficient. --bender235 (talk) 23:56, 16 January 2018 (UTC)

The statement is true in some cases, but it is not known to be true in general, e.g. for panel data. Stone (1977) considers regression only, and the article does say that "in the context of regression … AIC is asymptotically optimal for selecting the model with the least mean squared error". Even for univariate time series, the status of the statement is only partially known (Ing & Wei, Annals of Statistics, 2005). If you (or someone else) want to study this issue more, perhaps also consider the cited work of Akaike (1985).  BetterMath (talk) 00:50, 17 January 2018 (UTC)

Discussion of lead

This section is created pursuant to WP:BRD, to discuss recent proposed changes to the lead. (Consider also WP:Lead.)

Having the lead use the word "score" in this context seems wrong, because score has a technical meaning in statistics that does not apply in this context. Having the lead mention that a lower AIC is better seems inappropriate to me, because it is a technical detail—a detail, moreover, that is well discussed in the Definition section; it seems much more appropriate to have the lead tell what AIC does, rather than tell how AIC does things. Having the lead claim "Thus, AIC provides a means for model selection that deals with the trade-off between the goodness of fit of the model and the simplicity of the model" seems wrong, because the "Thus" is not logically supported by the context. Having the first paragraph of the lead worded as proposed does not even suggest that AIC can only evaluate relative quality, which is surely inappropriate and confusing.

For the above reasons, I have reverted to the prior version.  SolidPhase (talk) 16:19, 29 June 2018 (UTC)

Hi, thanks for the remarks about the change I proposed! I've made a new edit that incorporates the points you made. Please do improve it directly (plus talk discussion), rather than reverting -- I prefer discussion combined with a WP:BOLD, BOLD, BOLD cycle. As WP:ROWN says: "It is usually preferable to make an edit that retains at least some elements of a prior edit than to revert the prior edit."

Changes with respect to my first proposal:

• I've replaced the word 'scores' with 'numbers'. 'Scores' might still be the better word, though: in statistics 'scoring' also has the general meaning of 'assigning a number'. Compare the z-score a.ka. standard score, propensity score matching, test scores, and the notion of 'scoring questions'. What do you think?
• I've kept the higher/lower words. If somebody who does not know the AIC encounters one and looks up the article, 'lower AICs are better' is the first thing they will want to know.
• I've kept the 'Thus', because that sentence describes the high-level consequences of the low-level properties described in the preceding sentence.
• You are right that AIC is only a relative measure. I've made the last sentence refer to multiple models, to make this more clear. The word 'relative' already appears in the first sentence, and in the third paragraph of the intro, so now it's triply clear.

Let me know what you think!

--Sietse (talk) 15:17, 12 July 2018 (UTC)

I believe that the changes you made overall make the lead more confusing. For example, a sentence with two parenthesized phrases is awkward. Overall, the first paragraph is bordering on incomprehensible.

An additional issue is that you flagged your edit as minor. A minor edit should follow WP:Minor: an edit that the editor believes requires no review and could never be the subject of a dispute. Plainly, your edit could be subject to dispute.  SolidPhase (talk) 19:17, 12 July 2018 (UTC)

I did not mark my main edit, at 16:09, 12 July 2018, as minor. The edit I marked as minor was a subsequent one that fixed a typo. I would hate for you to accidentally think I was engaging in bad faith :-)

As for your opinion that my changes make the lead more confusing and 'bordering on incomprehensible': I have reread my last proposal with an open mind, and I respectfully disagree. So let's work on the text. Here is the last text I proposed, and which you reverted:

The Akaike information criterion (AIC) is an estimator of the relative quality of statistical models for a given set of data. It rewards (gives lower AICs to) models with high likelihood, but penalizes (gives higher AICs to) models with more parameters. Thus, AIC provides a means for choosing between models that deals with the trade-off between the goodness of fit of the models and the simplicity of the models.

AIC is founded on information theory: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data.

I prefer this text (which incorporates your suggestions) over the status quo ante because:

• it makes clear to the layman how to interpret an AIC (lower is better),
• it gives the layman an idea of how AIC functions (goodness of fit is rewarded, parameters are penalized)

Your latest objections were only to the phrasing. Let's avoid falling into an cycle where I only propose and you only reject. Could you propose a better phrasing which also incorporates this content?

Cheers, Sietse (talk) 23:18, 12 July 2018 (UTC)

My apologies about the minor comment; you are correct.

There was some discussion about the lead back in June 2015 (see above). Then, User:Glrx said that "The lead is confusing" and I replied "I agree". We could not come up with anything that we thought was better though. Since then, the lead has improved a little, but it is still confusing.

Your proposed changes clearly make the confusion worse. I previously commented about how a sentence with two parenthesized phrases is awkward. There is also the use of "likelihood"—not even wikilinked—even though not everyone will understand what that means. Your proposed first paragraph contains far more technical information than the current paragraph. Despite that, it does not convey the essence of what AIC does nearly as well as the current version.

Additionally, your proposed use of "Thus" is not logically proper, as I commented before.

SolidPhase (talk) 15:11, 13 July 2018 (UTC)

Since we agree that the current lede could be better, but disagree about the relative quality of the status quo and the quality of the proposal, and since your opinion is as good as mine and mine as good as yours, shall we ask for a third opinion? I'd be curious to hear what an outside party would recommend. Sietse (talk) 19:27, 13 July 2018 (UTC)

My belief is that you should first address the points that I have raised. SolidPhase (talk) 17:39, 14 July 2018 (UTC)

1. Wikilinking 'likelihood' is fine, that's a good suggestion! It's a good word, though: one that statisticians understand precisely, and laymen understand approximately.
2. The proposal does contain more technical information. It is, however, a description of the mechanism in simple language. I therefore believe that it does a better job of presenting the AIC, by making it easy for the reader to understand what it does. I realise we disagree on this: hence my proposal to get a third opinion.
3. The word 'thus' also means 'in this way or manner'. Again, I realise that you claim it will be interpreted as a logical 'therefore', while I claim readers will realise the lede is not a formal argument. Let's avoid the discussion entirely by writing 'In this way'. (You could have proposed that, too!)
4. Remember WP:BRD-NOT: "BRD is not a valid excuse for reverting good-faith efforts to improve a page simply because you don't like the changes. BRD is never a reason for reverting. Unless the reversion is supported by policies, guidelines or common sense exists, the reversion is not part of BRD cycle."
5. I don't think you and I are any closer to agreeing on the fundamental 'is this lede clearer, or more difficult to read' point. So let's get somebody else in, and see what they think. I've added a subsection below, and will post a request on WP:3O.

-- Sietse (talk) 14:49, 15 July 2018 (UTC)

1. The problem with "likelihood" is that some people confuse it with probability. Your claim that "laymen understand approximately" is, in general, false. Worse, many laymen will erroneously believe that they understand better than they actually do. Thus, the proposed paragraph would mislead laymen.
2. You acknowledge that your proposed paragraph contains more technical information. Thus, it is more difficult for most people to understand. You claim that the proposed paragraph "does a better job of presenting the AIC"; I believe that the opposite is true. In particular, I cannot comprehend the proposed first paragraph on a single reading. The lead, and especially the first paragraph, should be readily comprehensible to non-specialists, as per WP:LEAD; your proposal does not do that. Additionally, as I noted earlier, it seems much more appropriate to have the lead tell what AIC does, rather than tell how AIC does things: one of the reason for this is that non-specialists will not be able to understand how. The proposed paragraph is so confusing that even reading it a few times, I find it difficult to tell what AIC does.
3. Both "thus" and "in this way" fail with your proposal. The current logic is clear: a higher quality implies preferred selection. The logic in your proposal is not clear; if you are going to claim otherwise, please give details.

The general issue of readability has been studied extensively. There are automatic estimates of readability that have proved helpful in practice. The estimate that is most commonly used, by far, is the Flesch–Kincaid grade level. (This estimate has been shown to correlate with actual readability at about 90%; it is used in the military, in some legal situations, by many businesses, etc.; it is also incorporated in programs such a Microsoft Word—which I have used here.) The grade level for the current first paragraph is 11.6; the grade level for the proposed first paragraph is 13.2. Thus, there is strong objective evidence that the proposed paragraph is substantially more confusing than the current paragraph.

Moreover, the grade levels here surely underestimate the difference in actual readability. First, because the Flesch–Kincaid grade level does not consider the double parenthesization of phrases, which obviously complicates the sentence. Second, because the Flesch–Kincaid grade level assumes that readers well understand the meanings of all the words, which is not true for "likelihood".

SolidPhase (talk) 18:31, 16 July 2018 (UTC)

Third opinion

Sietse (talk)'s request for a third opinion on the following (see discussion above): of the two proposed lede paragraphs below, which do you recommend we use (with or without changes)? (SolidPhase, I think that is also what the dispute is about from your point of view. But if you'd also like the third opinion to address a different question, feel free to add that here.)

• The current version:

The Akaike information criterion (AIC) is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

AIC is founded on information theory: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data. (In doing so, it deals with the trade-off between the goodness of fit of the model and the simplicity of the model.)

• An alternative proposal:

The Akaike information criterion (AIC) is an estimator of the relative quality of statistical models for a given set of data. It rewards (gives lower scores to) models with high likelihood, but penalizes (gives higher scores to) models with more parameters. In this way, AIC provides a means for choosing between models that deals with the trade-off between the goodness of fit of the models and the simplicity of the models.

AIC is founded on information theory: it offers an estimate of the relative information lost when a given model is used to represent the process that generated the data.

3O Response: I feel that I understand the topic better after reading both the current and proposed leads. I agree that:

• It's probably not a bad idea to communicate that 'a lower result is better'. Some readers may visit the article because AIC estimates are used with little explanation in another article – or a sister project – and that's information they'd want to obtain from the lead.
• Multiple parenthetic statements do make that one sentence a bit complex for the lead. And though likelihood is wikilinked, suggesting a specific meaning, that meaning is not clear and its preferable to not make readers chase links.

The 3O asks which I recommend, and my choice would be the current version. But that's the sort of revert/no-revert choice that the requester doesn't like. I feel that there is room to build. The lead is short and some information could be added. I feel the 'lower is better' concept could be added as well as summarizing other parts of the article like advantages/disadvantages of using AIC and its history (mentioning Hirotugu Akaike and the year he published). This information should probably not go in the MOS:LEADPARAGRAPH. I suspect that given a little more room outside that first paragraph, it'd be easier to state it without that awkwardness.
For what it's worth, that's my non-binding opinion. I'll try to keep an eye open for any follow-up. – Reidgreg (talk) 21:36, 17 July 2018 (UTC)

Thank you for your time and your advice! You have written many helpful things: advice on which version to start working from, advice on the strengths and weaknesses of both versions, and advice on lead paragraphs in general. We're going to do good things with this :-D

Kind regards, Sietse (talk) 08:55, 19 July 2018 (UTC)

Proposed revision

A proposed revision is below. The first paragraph is the same as before; the second paragraph is expanded; the third paragraph is the same as before; the fourth paragraph is new (and it mentions Hirotugu Akaike, as recommended by User:Reidgreg).

The Akaike information criterion (AIC) is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.

AIC is founded on information theory: it estimates the relative information lost when a given model is used to represent the process that generated the data. The less information is lost, the higher is the quality of the model. (In making an estimate of the information lost, AIC deals with the trade-off between the goodness of fit of the model and the simplicity of the model.)

AIC does not provide a test of a model in the sense of testing a null hypothesis. It tells nothing about the absolute quality of a model, only the quality relative to other models. Thus, if all the candidate models fit poorly, AIC will not give any warning of that.

AIC was formulated by the statistician Hirotugu Akaike. It now forms the basis of a paradigm for the foundations of statistics; as well, it is widely used for statistical inference.

Perhaps the third paragraph should be moved to the Definition section.

@Glrx:
SolidPhase (talk) 09:45, 19 July 2018 (UTC)

Further discussion of lead

The prior section was for discussion of the wording of the lead. This section is for further discussion. It was created pursuant to a comment from User:Glrx that "the [second] paragraph is difficult and should be rewritten". As per the prior section, recommendations for rewording the lead are solicited. (Note that, in the prior section, I twice wikilinked Glrx.)

Additionally, Glrx has now changed a word in the lead: "representation" was changed to "model". The context is this clause: "When a statistical model is used to represent the process that generated the data, the representation will almost never be exact" versus "When a statistical model is used to represent the process that generated the data, the model will almost never be exact". My preference is for "representation".

SolidPhase (talk) 23:08, 2 December 2018 (UTC)

I don't have time for this debate; too much is going on in RL. Your task here was to garner consensus for your edit before reinserting it. See WP:BRD. I do not support the edit and you have not obtained any other editor's support.

The paragraph is extremely poor and is using "represent" as a synonym for model. The contorted sentence says "a statistical model is used to model the process that generated the data". See my edit comments. The sentence is awkward and verbose. Simpler, clearer, statement is "a statistical model will almost never be exact". Flush "represent" completely; it is not needed. It's only there to hide the circularity of the statement.

Glrx (talk) 03:33, 7 December 2018 (UTC)

Your suggestion, “a statistical model will almost never be exact”, does not make sense to me: exact what? The sentence needs to say what is inexact. Your claim that the statement has “circularity” is not something I understand.

You still do not seem to realize what a statistical model is. Please read the article statistical model. In particular, as stated in that article, a statistical model is a representation. Your repeated failure to grasp that basic aspect of the definition has led to your comments so far having been repeatedly nonsensical. For example, your comment above complains about “using "represent" as a synonym for model”. It is ironic that you would fail that way given that your first edit summary was “model is technical term”: yes, model is indeed a technical term, and you should find out what it means!

As for WP:BRD, you did not participate in the discussion (which I opened). Hence your complaint seems ill-founded.

You also complain that the second “paragraph is extremely poor”. I am much welcoming of (sensible) recommendations, as I have strongly demonstrated in the prior section. I spent hours writing the second paragraph. I am sure that it can be improved further though; so if you can recommend something, please do.

(An idea that I had was to take the last, parenthetical, sentence and put that in a separate paragraph. I wanted to avoid a single-sentence paragraph though. Perhaps you have an idea around that?)

Note that AIC is used for selecting a statistical model: that is stated, repeated and clearly, in the first paragraph. As such, it is assumed afterward that the reader has some familiarity with statistical models. Your not knowing even the definition of a statistical model is presumably part of the reason that you find the second paragraph difficult to understand.

It would also be helpful if you accurately understood what “representation” means. There are definitions at e.g. Wiktionary and OxfordDictionaries.com.

SolidPhase (talk) 23:47, 10 December 2018 (UTC)

splitting long equation

There is a long equation in the section "Replicating Student's t-test". The equation is so long that it is split over two lines. Should the equals sign go on the first line (at the end) or on the second line (at the beginning)?

I did not find a relevant policy or guideline on this. I prefer that the equals sign be on the first line, for two reasons. First, because it makes the first line clearer: when someone reads the first line, they know immediately that the next line is the other side of an equation, and that eases reading of the whole equation. Second, because it lessens the overall display width.

@Michael Hardy:

BetterMath (talk) 10:43, 27 July 2019 (UTC)