Talk:Maximum likelihood estimation

message: I guess the line "we may compute the probability associated with our observed data" is not correct. Because the probability of a continuous variable for any given point is always zero. The correct statement would be "we may compute the likelihood associated with our observed data".  For more argument please see 
:Your assertion is correct, but your header is not.  That is NOT the difference between the likelihood function and the density function.  One is a function of the parameter, with the data fixed; the other is a function of the data with the parameter fixed. Michael Hardy 02:16, 13 May 2006 (UTC)
replies: text: Maybe it was a temporary problem; they look OK to me now. Michael Hardy 15:06, 21 September 2006 (UTC)
thread_title:  Difference between likelihood function and probability  density  function 

message: Great article! Very, very clear. 18.243.6.178 04:08, 13 November 2006 (UTC)
thread_title:  Untitled

message: You really sould say something about the second derivative test.  You nonchalantly claimed you reached a maximum.  You could very well have found a minimum.  Don't reinforce bad habits. This is especially true for the Normal case.  Just because the gradient is zero does not mean you have a local maximum...and it's not especially trivial to just brush off.(ZioX 18:06, 19 March 2007 (UTC)
replies: text: I don't know which "you" is addressed, but I agree with the spirit of the comment.  But the details of the comment fall short.  The second-derivative test may prove a local maximum, but here we need a global maximum.  There are various ways to prove there is a global maximum, not all of them involving second derivatives.  For example, suppose you show that L(θ) increases as θ goes from 0 to 3, and decreases as θ goes from 3 to ∞, and the parameter space is the interval from 0 to ∞.  Then you've got a global maximum at 3, without benefit of second derivatives.  Or suppose you've shown that L(θ) is differentiable everywhere, and because the parameter space is compact, there must be a global maximum somewhere, and furthermore L(θ) is 0 on the boundary and positive in the interior.  Then the global maximum must be reached at a critical point in the interior.  If next it turns out that there is only one critical point in the interior, then you've got it again, and again without second derivatives.  This is not at all an unusual situation in elementary MLE problems. Michael Hardy 18:30, 19 March 2007 (UTC)
thread_title:  Sloppiness 

message: From the bias section:
"we can only be certain that it is greater than or equal to the drawn ticket number." Is this true? Wouldn't it be the less than or equal to?--Vince |Talk| 05:00, 15 May 2007 (UTC)
replies: text: disregard: It was the wording that confused me. The article Bias of an estimator phrases the problem in the manner I was thinking. I may try to make it clearer. --Vince |Talk| 06:37, 15 May 2007 (UTC)
replies: text: It was true but not relevant, so I dropped it when I expanded this section. RVS (talk) 00:17, 25 December 2008 (UTC)
thread_title: Is this true

message: Hi all,
The article is very well written from a statistics POV.
MLE is used ubiquitously in phylogenetic analysis and cladistics in genetics and evolutionary biology. It would be great if someone could include a section on how MLE can actually be applied to such studies, with some examples.  
Also, I, as an amateur biologist, know where MLE is used, but do not have an intuitive understanding of the technique. A section that would provide the layman with such a perspective (of whats actually happening) would be great...
Indiaman1 19:30, 30 June 2007 (UTC)
thread_title:  Some more applications please 

message: I have added a section on non-independent variables. I hope this proves useful to someone :) Velocidex (talk) 04:44, 19 March 2008 (UTC)
replies: text: I added a tie back to article topic by mentioning the likelihood function, which should possibly be the main thing being specifically discussed rather than the density function. Melcombe (talk) 10:45, 19 March 2008 (UTC)
thread_title:  Non-independent variables 

message: Someone wrote above that this article reads (read) like a 1980's textbook. In my opinion that was the case. For instance, the discussion of the asymptotic properties of maximum likelihood estimation could have been taken straight out of many standard textbooks, but an intelligent person can realise that the authors of those books either don't know what they are talking about or are hiding things from the reader. I added a sentence referring to modern mathematical results on the maximum likelihood estimator (modern: these results have been known since the 60's but still did not permeate into standard textbooks). I hope the result still makes sense to the non-expert.
Gill110951 (talk) 07:12, 5 May 2008 (UTC)
thread_title:  Mathematical precision 

message: I see mixed use of \mathcal{L} and L for the likelihood function. The likelihood function page itself only uses L. Which is it? (Or do they mean slightly different things?) —Ben FrantzDale (talk) 12:56, 14 August 2008 (UTC)
thread_title:  Typography 

message: there is a pw requirement on the link to the tutorial   —Preceding unsigned comment added by 193.157.202.64 (talk) 13:04, 20 August 2008 (UTC)
thread_title:  link has pw 

message: The current article lists a mish-mash of applications.  I think a distinction needs to be made between application areas that are various statistical methodology methods (univariate models, structural equation models, etc.) vs. application areas that are different fields (agriculture, communications, business research, etc.).  Either type is perhaps too broad to list in this article, but i don't think the types should be mixed.
The article currently mentions
"...applications in various fields, including:
* linear models and generalized linear models;
* communication systems;
* exploratory and confirmatory factor analysis;
* structural equation modeling;
* psychometrics and econometrics;
* time-delay of arrival (TDOA) in acoustic or electromagnetic detection;
* data modeling in nuclear and particle physics;
* computational phylogenetics;
* origin/destination and path-choice modeling in transport networks;
* many situations in the context of hypothesis testing and confidence interval formation.
Hope this comment is helpful. doncram (talk) 21:28, 30 December 2008 (UTC)
thread_title:  applications list 

message: (discussion section separated out under this title later)
I think the second sentence in the "Principles" section is ambiguous.  The sentence currently reads, in part, "We draw a sample x_1,x_2,\dots,x_n of n values from this distribution [...]".  I think it would be much clearer to specify exactly which "distribution" is being referred to by "this distribution".  The current wording makes it unclear whether we're looking at one member of the family of distributions, or instead the PDF itself.  —Preceding unsigned comment added by Diracula (talk • contribs) 21:08, 27 January 2009 (UTC)
replies: text: Your point escapes me.  It is of course referring to one member of the family, and that one member does have a pdf.  And that one could be ANY of the members of the family.  "Amibiguous" normally means there are two ways to construe the sentence.  Just what those two ways are is unclear.  Your way of phrasing it seems to suggest that the two ways are (1) We're looking at one member of the family of distributions; and (2) We're looking at the pdf.  I don't see how the sentence can be read as giving us that choice.  Before that, you say the ambiguity is about the question of WHICH DISTRIBUTION it is.  That's quite a different thing from choosing between the statements I labeled (1) and (2) above, since the choice between (1) and (2) is not a choice of which distribution it is.  Nor can I find any ambiguity about which distribution it is.  So I find your comments completely cryptic. Michael Hardy (talk) 22:31, 27 January 2009 (UTC)
thread_title: about which distribution

message: Why was this article renamed "Maximum likelihood" instead of "Maximum likelihood estimation"? The latter seems more appropriate and I suggest renaming it back. -Roger (talk) 15:52, 25 March 2009 (UTC)
replies: text: I agree - why the change? The article even starts "Maximum likelihood estimation (MLE) ..." 128.61.125.131 (talk) 14:55, 28 January 2010 (UTC)
replies: text: Notionally, maximum likelihood is more general, with some things worth saying that are not about "maximum likelihood estimation", but there probably not be very much. However, none of this appears (yet). There is also a question of whether it should be "estimate" or "estimation" .... this would preferably aggree with other article titles, such as "minimum distance". Melcombe (talk) 10:51, 29 January 2010 (UTC)
replies: text:  We have M-estimator, Extremum estimator, Bayes estimator, but Maximum spacing estimation, Maximum a posteriori estimation, Minimum distance estimation, and then there is also Generalized method of moments. I’d vote for “Maximum likelihood estimation”, because that’s how it is usually defined in econometric textbooks (MLE). Although the abbreviation “ML” can also be seen occasionally.  … stpasha »  18:42, 29 January 2010 (UTC)
thread_title:  Article title 

message: This is one of the rare articles that gives synopses of the related articles: 
Isn't it better just to give the links?
I humbly suggest these alternatives, which are usually shorter and which in some cases (e.g. abuction) avoid objectionable statements. 
* Abductive reasoning, the logic of generating hypotheses.??
* Censoring (statistics) 
* Delta method, a method for finding the distribution of differentiable functions of a random variable.
* Generalized method of moments, a method related to maximum likelihood estimation. ??
* Inferential statistics, for alternatives to the maximum likelihood estimator. ??
* Likelihood function.
* Maximum a posteriori (MAP) estimator, for the estimator selecting a posterior mode.
* Maximum spacing estimation, a robust estimator.
* Mean squared error, a measure of how 'good' an estimator of the parameter is.??
* Method of moments (statistics), another estimation method. ??
* Method of support, another inferential method.??
* Minimum distance estimation
* Quasi-maximum likelihood estimator, a consistent estimator for misspecified models.
* The Rao–Blackwell theorem, a result which yields a process for finding the best possible unbiased estimator (in the sense of having minimal mean squared error). An MLE can be the unconditional estimate.??
* Sufficient statistic.
I put ?? after synopses that should be removed imho. 
Kiefer.Wolfowitz (talk) 18:59, 24 May 2009 (UTC)
replies: text: "Isn't it better just to give the links?" No. It is at least helpful to have some indication of relevance. For example, why is there is link to censoring? But it might be better to reduce the number of "see alsos" by creating brief subsection with headings like "alternative estimation methods", so as to better group the topics. Melcombe (talk) 09:42, 27 May 2009 (UTC)
thread_title:  See also 

message: As i understand, consistency is acually weak consistency,
i would like to know if there is proof about strong consistency of the maximum likelihood?  —Preceding unsigned comment added by 132.72.51.67 (talk) 16:38, 25 August 2009 (UTC)
thread_title:  Weak consistency or strong consistency? 

message: The article provides a great layman description of the concept of maximum likelihood, however the rigorous mathematical treatment of the same topic is also necessary. What use is the claim “under certain fairly weak conditions MLE is consistent” if those conditions are never spelled out. What use is the “asymptotic normality” property if the article doesn’t even state the asymptotic variance of the estimator.
The “Maximum spacing estimation” article lists several examples when the MLE fails: certain heavy-tailed distributions, certain mixture distributions, certain 3-parameter models. It would be nice to know why those examples don’t work (at least they are not the case of boundary parameters). ... stpasha » talk » 10:31, 2 September 2009 (UTC)
thread_title:  Insufficient content 

message: As a motivational example, let X be a binomial random variable with n = 10 and an unknown parameter p. Further suppose that x = 3, a realization of the random variable X, is observed. As in all estimation problems, the goal is to estimate the unknown parameter p. The fundamental idea in maximum likelihood estimation is to calculate the probability of generating the observed value x = 3, that is, P(X = 3), under the different possible values that p can take. By construction of the binomial distribution, p is a probability and therefore p ∈ [0, 1]. In order to calculate these probabilities, we view the binomial distribution as a function of p, holding x fixed at 3:
: 
    L : [0, 1] \longrightarrow \mathbf{R}, \; p \longmapsto \frac{10!}{3!(10 - 3)!} p^3 (1 - p)^{10 - 3} = 120 p^3 (1 - p)^7.
  
In this example, L is the likelihood function. Here is a plot of L:
As can be seen on the graph, it turns out that the probability of generating x = 3 is the largest when p = 0.3. Hence  \hat{p} = 0.3  will be the maximum likelihood estimate of p. Of course, in general, inference will not be based on a single observed value x but on a random sample  X_1, \ldots, X_n  yielding observations  x_1, \ldots, x_n . Supposing that the random variables  X_1, \ldots, X_n  are independent and identically distributed, the likelihood function is a product of binomial densities, regarded as a function of p.
So there used to be a section called “Motivational example”. I've looked at it for a long time, and finally concluded that the best way to improve it would be to remove it altogether. This example simply breaks the flow. It starts using the MLE method without explaining how and why it is used so. It uses an example with 1 observation, which is although legal, but quite counterintuitive since no properties of such estimator can be stated. I think we have enough examples in the “Examples” section, so that we don't need this one too.  … stpasha »  07:50, 24 January 2010 (UTC)
thread_title:  (Dis?)motivational example 

message: This section () has been an ugly duckling from the beginning, but despite the selfless attempts of Michael Hardy, it can never grow in a swan… There is just so much wrong with that piece, for example:
# It silently assumes that θ is a scalar, although the article states otherwise;
# It concludes the “proof” with Pr[Sn(θ)>0] → 0, while in fact this probability is equal to 1 for all n.
# The general idea to prove a theorem without stating it, and without specifying the assumptions needed for it, is faulty.
Sentence: removed.  … stpasha »  08:11, 24 January 2010 (UTC)
thread_title:  “Convergence of maximum likelihood estimator” gone 

message: Earlier versions of this article were fairly readable (for example this version of early January), but the recent addition of so-called "proofs" has ruined this. Can we improve things by just moving most of this over-technical stuff towards the end of the article (after the existing "see also" section, which could be renamed) under a heading like "Theoretical details". This would be in line with the guidance for maths articles, that it is OK to start simple and become more thechnical later, even at the risk of some repetition. Melcombe (talk) 12:39, 26 January 2010 (UTC)
replies: text: This is a good idea. Another idea would be to have a self-standing article entitled "Proofs of properties of MLE". Some articles on technical subjects have informal and formal versions; for example, see the gentle Introduction to M-theory and the high fallutin' M-theory.  Kiefer.Wolfowitz (talk) 15:42, 26 January 2010 (UTC)
replies: text:  I agree that the subsections “Consistency” and “Asymptotic normality” still require lots of work. And that’s why I put the tags  there. In particular, the material from the “Asymptotics” subsection should be distributed between the first two. But I’m not sure if moving the material all the way out of the article is a good idea… Perhaps that guidance for mathematical articles can be interpreted section-wise, like start the section simple, and then proceed to more advanced theory.  … stpasha »  18:52, 26 January 2010 (UTC)
replies: text: The previous strtucture of the model was that there was a readable outline of all the properties of MLE, which must be retained but which is being destroyed by these supposed "proof" intrusions. My own view is that "proofs" should typically not appear on wikipedia, particularly when they are text book stuff, but that the theory necessary for a proper description of what is being discussed should appear. Thus I am against "Proofs of ... articles" when they have no encylopedic content (ie no proofs for the sake of having proofs), as they would here. Proofs would be much better dwealt with by just giving appropriate citations. Melcombe (talk) 10:23, 27 January 2010 (UTC)
thread_title:  Proposed restructure 

message: I object to the blatantly sexist and ageist example being used as an example of a normal distribution in nature, namely the height of adult female giraffes. I would like to suggest a less controversial example: the mass of mouse turds divided by the mass of the mouse producing them. This will be approximately normally distributed, and has the added advantage of being independent of age and (very probably) sex, thus avoiding an unhealthy obsession with age and sex. It does however suffer from a particularly mouse-o-centric view of the universe, and is therefore not perfect. I am wracked with guilt over this flaw, but lacking a better example, I offer it as perhaps a temporary improvement. PAR (talk) 22:50, 24 March 2010 (UTC)
thread_title:  I object 

message: One of the references is to Pfanzagl.  Is this a made-up name or something?  It shows up once in the entire page, so it's not a valid reference/citation.  —Preceding unsigned comment added by 130.58.92.245 (talk) 03:04, 29 June 2010 (UTC)
replies: text: The inadequate reference has been improved, now reading "Pages 207-208 in ". Kiefer.Wolfowitz (talk) 11:37, 29 June 2010 (UTC)

text: Who is Pfanzagl? Professor Dr. Johann Pfanzagl may have been the leading mathematical statistician in Austria and perhaps all of the German speaking countries from 1970-1995, say. In his early work, Pfanzagl solved the outstanding problem of von Neumann and Morgnenstern of showing that expected utility theory could be axiomatized with subjective probability (essentially compatible with von Neumann's approach), and wrote a related monograph on measurement theory. He then worked on statistical inference, with deep results on median-unbiased estimation and exponential families. Following Le Cam and Hajek, Pfanzagl has been one of the architects of asymptotic theory, including both parametric and semiparametric approaches; Le Cam credits Pfanzagl with introducing tangent cones and spaces, and these objects are now standard in advanced graduate books. Although Pfanzagl clearly states that MLE has no good finite-sample properties, he has contributed one of the best convergence analyses of maximizing the conditional likelihood; Pfanzagl also includes simulation studies for assessing the performance of MLE and other methods, which often show that the MLE behaves quite well in moderately large samples. Kiefer.Wolfowitz (talk) 11:37, 29 June 2010 (UTC)
thread_title:  Who is Pfanzagl? 

message: The vandalism intensity has increased lately. Could the level of protection be increased to prevent damage by IP editors? Thanks!  Kiefer.Wolfowitz  (Discussion) 13:11, 13 March 2011 (UTC)
thread_title:  Vandalism: Administrators, consider raising the protection level 

message: It basically starts something like "MLE is a method for making estimates of ...". Well, are there other ways of doing so? Other generic ways? Do these methods all give the same estimates, or is making estimates not an exact science? 80.162.194.33 (talk) 09:22, 4 August 2011 (UTC)
replies: text: Please ask such questions at the help desk or at an internet chat site. Talk pages are for discussing improvements to the article.  Kiefer.Wolfowitz 10:55, 4 August 2011 (UTC)
replies: text:  The anon user might have meant that we should address these concerns in the article itself. If so, that's a legitimate use of a talk page. - dcljr (talk) 00:53, 3 April 2013 (UTC)
thread_title:  Are there other ways of estimating? 

message: The introduction section provides a very useful starting point for non-experts reading the page, with the exception of the following sentence: "In the Kolmogorov structure function one deals with individual strings." I personally find it a bit strange that this appears here. If I understood what it is dealing with, I'd relegate it to further down in the article, but I don't. Would anyone care to move this sentence or improve it to make it more congruent with the rest of the introduction?Jimjamjak (talk) 15:15, 25 January 2012 (UTC)
replies: text: Good point. I've just removed that sentence. It was added last November by , who heavily edited 'Kolmogorov structure function'. It's relationship to maximum likelihood isn't clear, and it's certainly not suitable for the lead. If someone else understands it better and thinks it deserves a mention further down the article, that's fine with me. Qwfp (talk) 15:39, 25 January 2012 (UTC)
thread_title:  Kolmogorov in the introduction 

message: In the article it says:
'Consistency: a subsequence of the sequence of MLEs converges in probability to the value being estimated.'
I don't agree with it. The sequence converges in probability to the value. Why do you have to take the subsequence? 95.113.187.130 (talk) 21:05, 29 October 2012 (UTC)
replies: text: Agreed. I have changed the article. As discussed in the Consistent estimator article, it is the sequence of estimators that converges in probability to the parameter. If only "a subsequence" did so, that would be a hideously weak result. Now, if it said "all subsequences", that would be different: that would imply that the sequence itself converges... but there's no reason to say it that way, AFAIK. - dcljr (talk) 00:49, 3 April 2013 (UTC)
thread_title:  Subsequence of the sequence converges in probability 

message: Hey!
There is an error in the conditions that are demanded for the MLE to be consistent. The second condition states that the parameter space must be compact, which is not true. The parameter space must be an open subset of \mathbb{R}^p, which implies completeness, but not compactness.
For example the parameter space of \sigma in the normal distribution is (0, \infty). That subset is open, and thus complete but not compact.
Repairing this would call for replacing the whole part about consistency. Suggested sources are Lehmann (1999) Elements of Large Sample Theory, or Knight (2000) Mathematical Statistics.  — Preceding unsigned comment added by Emogstad (talk • contribs) 10:09, 25 May 2013 (UTC)
thread_title:  Error in the article: Consistency 

message: All the literature calls it "maximum likelihood" without a hyphen.  Why have you moved it?  It is not a compound adjective requiring hyphenation (we don't say this estimation is maximum-likelihood).  It is a compound noun and therefore should not be hyphenated.  Tayste (edits) 02:30, 20 June 2016 (UTC)
replies: text: I moved maximum likelihood to maximum-likelihood estimation, because estimation was part of the acronym (MLE) mentioned in the lead. The hyphen was not intentional. I've now requested a technical move to maximum likelihood estimation. fgnievinski (talk) 02:50, 20 June 2016 (UTC)
replies: text:  Oh I see, thank you.  Tayste (edits) 20:09, 20 June 2016 (UTC)
thread_title:  Move to hyphenated title 

message: A recent edit:  suggest that ML is a "special case" of maximum a posteriori estimation (MAP) for a uniform prior. Is this true? If so, it would be good to have a citation to a source. Thanks. Isambard Kingdom (talk) 13:24, 19 September 2016 (UTC)
replies: text: this is not true. MLEs enjoy several properties such as invariance under measurable transformations (i.e g(MLE) is the mle for g(theta)). This property does not hold for all measurable transformations g for the corresponding bayesian MAP estimator. Perhaps we could say the value of MLE will match the value of the map estimator after evaluation, assuming one does not change the underlying measure. Chjacamp (talk) 20:29, 6 May 2023 (UTC)
thread_title:  Maximum a posteriori estimation 

message: This section began with main|Likelihood principle, even though MLE does not depend upon the likelihood principle. The section then made the following claim.
Suppose there is a sample  of n independent and identically distributed observations, coming from a distribution with an unknown probability density function f0(·). It is however surmised that the function f0 belongs to a certain family of distributions } (where θ is a vector of parameters for this family), called the parametric model, so that . The value θ0 is unknown and is referred to as the true value of the parameter vector.
The claim about a "surmise" and a true value is false: all models are wrong and so the "true model" does not exist. Additionally, θ could be a scalar, matrix, or something else.
The section then talked a lot about the special case of when the observations are iid, even though it is supposed to be a general exposition of principles. The section then made the following claim. 
In the exposition above, it is assumed that the data are independent and identically distributed. The method can be applied however to a broader setting, as long as it is possible to write the joint density function , and its parameter θ has a finite dimension which does not depend on the sample size n.
That is erroneous: consider the likelihood function for an arbitrary autoregressive model and for a proportional hazards model.
There were other technical problems too. As well, much of the rest of the section seemed like blather. 
I have removed most of the section, and revised much of the remainder. It could benefit from further improvement though. For the Bayesian aspects, I moved the discussion to a new subsection.  BetterMath (talk) 19:00, 6 January 2018 (UTC)
thread_title:  Section "Principles" 

message: What is the purpose of the subsection "Asymptotic normality"? The subsection defines asymptotic normality as follows.
maximum likelihood parameter estimates exhibit asymptotic normality – that is, they are equal to the true parameters plus a random error that is approximately normal (given sufficient data)
The "true parameter", however, does not exist with real-world data: because all models are wrong. Thus, this subsection seems to describe something that is relevant only for simulated data. Such a long a complicated subsection that is relevant only for simulated data would not appear to benefit the article.  BetterMath (talk) 00:56, 13 January 2018 (UTC)
thread_title:  Subsection "Asymptotic normality" 

message: For an MLE to be "the same regardless of whether we maximize the likelihood or the log-likelihood," we need the log to be "a strictly increasing function."  Injective is not enough.  Monotonic is not enough.  If I replace the log(likelihood) with [-log(likelihood)], then I'd have to minimize [-log(likelihood)] to maximize the likelihood.  
Injective is even worse.  Suppose I have an observation from a Poisson distribution with a parameter space \Theta restricted to {1, 2, 3}.  This is a contrived example, but with more work I believe I could find one more realistic.  Besides, this is mathematics, and we want principles that will work based only on the assumptions.  
Suppose further my observation is 2.  Then the likelihood \mathcal{L}(\theta;\mathbf{y}) = \theta^2 \exp(-\theta)/2.  For \theta in \Theta = {1, 2, 3}, we get \mathcal{L}(\theta;\mathbf{y}) = \exp(-1)/2 = 0.184, 4 \exp(-2)/2 = 0.271, and 9 \exp(-3)/2 = 0.224, respectively.  
On this restricted parameter space, [\mathcal{L}(\theta;\mathbf{y}) - 0.3]^2 is injective and is maximized when \theta = 1, not 2.  
Conclusion:  I don't see a way to allow the use of only injective transformations and still get a transformation that is necessarily maximized in the same place.  Monotonically increasing seems to be necessary and sufficient.  
Accordingly, I've reverted "injective" to "monotonically increasing".  DavidMCEddy (talk) 02:55, 11 April 2019 (UTC)
replies: text: Correct. No objection from my side. --bender235 (talk) 03:54, 11 April 2019 (UTC)
thread_title:  injective is not enough 

message: Hi, the section https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Discrete_distribution,_continuous_parameter_space
states that the derivative of the distribution when set to 0, "has solutions p = 0, p = 1, and p = ​49 / 80". It's not clear to me where "p = 49/80" comes from. Perhaps I'm missing something obvious, but if not, maybe a clarifying comment would be a helpful addition to the page. Thanks.  — Preceding unsigned comment added by 146.66.57.203 (talk) 05:17, 10 October 2019 (UTC)
replies: text: I just changed the verbiage to make that clearer.  Thanks, DavidMCEddy (talk) 07:32, 10 October 2019 (UTC)
replies: text:  Got it now, many thanks.  — Preceding unsigned comment added by 146.66.57.203 (talk) 15:21, 10 October 2019 (UTC)
thread_title:  Unclear conclusion on continuous parameter space 

message: I'm simply wondering the meaning of the t subscript on the random variable in these sections.  If it isn't meaningful, it should be removed for consistency, otherwise it should be explained briefly.  Cheers. Moo (talk) 19:45, 29 May 2022 (UTC)
thread_title:  Efficiency subsections X_t subscript 

message: Several methods of calculating the MLE are given (Newton-Raphson, Fisher Scoring, etc.) but no mention of the EM algorithm.  It is used so often in actually calculating the MLE that it should either be part of the Iterative Procedures list or part of the See Also. 108.18.248.126 (talk) 13:32, 5 January 2023 (UTC)
thread_title:  Adding Expectation-Maximization to Methods or Link