Talk:Likelihood function

message: Article does not communicate well. 
A relatively simple matter turns out to be difficult to understand.
—Preceding unsigned comment added by 80.212.104.206 (talk) 13:34, 18 September 2010 (UTC)
thread_title: Unclassified comments

message: Can someone tell me what the arrow notation is suppose to mean? --Huggie (talk) 11:30, 3 April 2010 (UTC)
replies: text: I can't figure this out either. Did you ever find an answer? Jackmjackm (talk) 20:00, 23 September 2023 (UTC)
replies: text: The arrows indicate mapping, so  \theta \mapsto f(x|\theta)  is saying that for each parameter value \theta, we can define a function f(x|\theta) giving the probability (or probability density) of the data x given \theta. I.e., we map \theta to a function f(\cdot|\theta). I hope this is somewhat helpful and not too circular! So many suspicious toenails (talk) 16:16, 26 September 2023 (UTC)
thread_title:  The arrow 

message: I added the context tag because the article starts throwing mathematical functions and jargon around from the very beginning with no explanation of what the letters and symbols mean. Rompe 04:40, 15 July 2006 (UTC)
replies: text: The tag proposes making it more accessible to a general audience.  A vernacular usage makes likelihood synonymous with probability, but that is not what is meant here.  I doubt this topic can be made readily comprehensible to those not familiar at the very least with probability theory.  So I question the appropriateness of the "context" tag.  The article starts with the words "In statistics,...".  That's enough to tell the general reader that it's not about criminology, church decoration, sports tactics, chemistry, fiction writing, etc.  If not such preceeding words were there, I'd agree with the "context" tag. Michael Hardy 23:55, 16 July 2006 (UTC)
thread_title:  Context tag 

message: Which came first? the common use as in "in all likelihood this will not occur" or the mathematical function?
:See History of probability.  "Probable and probability and their cognates in other modern languages derive from medieval learned Latin probabilis ... .  The mathematical sense of the term is from 1718. ... The English adjective likely is of Germanic origin, most likely from Old Norse likligr (Old English had geliclic with the same sense), originally meaning "having the appearance of being strong or able" "having the similar appearance or qualities, with a meaning of "probably" recorded from the late 14th century. Similarly, the derived noun likelihood had a meaning of "similarity, resemblance" but took on a meaning of "probability" from the mid 15th century."  Mathematical formalizations of probability came later, starting primarily around roughly 1600.  Ronald Fisher is credited with popularizing "likelihood" in its modern sense beginning around 1912, according to the Wikipedia article on him.  DavidMCEddy (talk) 15:47, 21 March 2016 (UTC)
thread_title:  Which came first 

message: An earlier version of this page said "In a sense, likelihood works backwards from probability: given B, we use the conditional probability Pr(A|B) to reason about A, and, given A, we use the likelihood function L(A|B) to reason about B. ".  This makes sense;  i.e. it says it's backwards, and it is.  
The current version uses L(B|A) instead, i.e. it says: "In a sense, likelihood works backwards from probability: given B, we use the conditional probability Pr(A|B) to reason about A, and, given A, we use the likelihood function L(B|A) to reason about B. "  This does not make sense.  It says it's backwards, but it talks as if Pr and L are interchangeable.
How about switching back to the earlier version, and providing a concrete example to help clarify it?  Possible example: Given that a die is fair, we use the probability of getting 10 sixes in a row given that the die is fair to reason about getting 10 sixes in a row;  or given that we got 10 sixes in a row, we use the likelihood of getting 10 sixes in a row given that the die is fair to reason about whether the die is fair.  (Or should it say "the likelihood that the die is fair given that 10 sixes occur in a row"?  What exactly is the definition of "likelihood" used in this sort of verbal context, anyway?) --Coppertwig 20:28, 24 August 2007 (UTC)
replies: text: And the preamble has the variable and the parameter confused in "Equivalently, the likelihood may be written to emphasize that it is the probability of observing sample  given ,..." 109.250.93.11 (talk) 16:32, 17 November 2022 (UTC)
thread_title:  Backwards 

message: I believe it is a good habit for mathematical articles on Wikipedia, to start with a simple heuristical explanation of the concept, before diving into details and formalism. In this case I think it should be made clearer that the likelihood is simply the pdf regarded as a function of the parameter rather than of the data. 
Perhaps the fact that while the pdf is a deterministic function, the likelihood is considered a random function, should also be adressed.
Thomas Tvileren (talk) 07:30, 17 April 2009 (UTC)
thread_title:  Needs a simpler introduction? 

message: For a bernoulli trial, is there a significant meaning for the median of the likelihood function?  —Preceding unsigned comment added by Fulldecent (talk • contribs) 16:30, 13 August 2009 (UTC)
replies: text: The Bernoulli trial has a probability distribution function fP defined by fP(0) = 1−P and fP(1) = P. This means that the likelihood function is  Lx defined by L0(P) = 1−P and L1(P) = P for 0≤P≤1. For x=0 the maximum likelihood estimate of P is 0; the median is 1−1/√2 = 0.29; and the mean value is 1/3=0.33. For x=1 the maximum likelihood estimate of P is 1; the median is 1/√2 = 0.71; and the mean value is 2/3=0.67. These are point estimates for P. Some likelihood functions have a well defined maximum likelihood value but no median. Other likelihood functions have median but no mean value. See for example the German tank problem#Likelihood function.  Bo Jacoby (talk) 22:27, 3 September 2009 (UTC)
thread_title:  Median 

message: The definition given here is the opposite that given by D'Agostini, Bayesian Reasoning in Data Analysis (2003). From
pp. 34-35: "The possible values x which may be observed are classified in belief by f(x|\mu).  This function is traditionally called `likelihood' and summarizes all previous knowledge on that kind of measurement..."  In other words, it is the probability of an effect x given a parameter (cause) \mu.  The definition given in this entry, proportional to the probability of a cause given the effect (f(\mu|x)) seems more useful, as the concept is more important, but is it possible that there is more than one definition in use in the literature?
LiamH (talk) 02:10, 4 October 2009 (UTC)
thread_title:  Probability of causes and not probability of effects? 

message: since P(x|theta) is describing sets of data points (as if a vector), shouldn't it be put in bold?
theta represents a vector (or set) of parameters, and x represents a vector of data points from a sample.
I might be wrong about this, thought it would be worth mentioning
SuperChocolate (talk) 14:49, 18 September 2014 (UTC)
thread_title:  putting x and theta in bold 

message: This section seems to trace the etymology and usage of the word "likelihood", which generally seems irrelevant to the specific concept of likelihood functions. Recommend heavily truncating or removing this section.  — Preceding unsigned comment added by Denziloe (talk • contribs) 09:55, 7 September 2017 (UTC)
replies: text: Yes, I much agree. The Wikipedia article is about the likelihood function in mathematical statistics. That function is not what Peirce was referring to in his papers, when he discussed likelihood. Nor is a detailed etymology of the word relevant to the function. I have removed most of the section, and added an additional citation for Fisher.  BetterMath (talk) 08:20, 9 November 2017 (UTC)
thread_title:  "Historical remarks" section 

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thread_title:  External links modified 

message: Can you please help me understand why you deleted the following:  
A more detailed discussion of the history of likelihood in statistics is given by the following sources.
	 
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??? These all seem relevant to this article and useful for a reader interested in the history of the subject.  Thanks, DavidMCEddy (talk) 20:04, 28 February 2019 (UTC)
replies: text: It is not helpful to list all of these as a quasi section-ending bibliography. They should either be included as individual footnotes for specific points, or added to the general "further reading" section of the article. Some of these references appear to be of very narrow scope (like Stigler 1978 and Pratt 1976), others of broader scope but better suited for different articles (Stigler 1986 and 1999 should rather be included in History of statistics). --bender235 (talk) 20:58, 28 February 2019 (UTC)
replies: text: OK.  Thanks.  DavidMCEddy (talk) 21:33, 28 February 2019 (UTC)
thread_title:  Why deleting "A more detailed discussion of history of likelihood ..."? 

message: Your addition of "Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof" reminded me that George E. P. Box got PhDs for at least two grad students in the late 1970s / early 1980s for what I called, "psyching out the robustifiers":  At that time some of the leaders in "robust statistics" were publishing papers claiming certain procedures were "distribution free" and "robust to outliers".  These grad students showed that if you assumed some long-tailed distribution(s) like double-exponential or Cauchy distribution(s), then a standard likelihood ratio or Bayesian procedure would recommend exactly or approximatly what the "robustifiers" were recommending.  
After showing that this "robust" or "distribution free" procedure was essentially likelihood ratio or Bayes, Box would then ask the "robustifiers" to just be honest and tell us what distributions they were assuming.  But, of course, they couldn't, because they had not thought of the procedure in those terms.  
This was crudely similar to one of the famous Bayesians of that period saying something like, "There are two types of statisticians:  Bayesians and one who don't know they are Bayesians, also called "frequentists".  DavidMCEddy (talk) 06:28, 14 March 2019 (UTC)
thread_title:  Inverse logic 

message: Thanks very much for your contributions to this article.  
However, I wonder why did you convert my ambiguous singular vs. plural construction for "parameter(s)" to unambiguous plural "parameters" in the first sentence, from:  
: In statistics, a likelihood function (often simply a likelihood) is function \mathcal{L}(\theta, y) of parameter(s) 
to:  
: In statistics, a likelihood function (often simply a likelihood) is function \mathcal{L}(\theta, y) of parameters
?  The standard estimation for the Poisson and binomial distributions, for example, include only one parameter -- singular.  
Thanks DavidMCEddy (talk) 02:45, 9 April 2019 (UTC)
replies: text: I know there are likelihood functions were the parameter vector is a singleton, but I don't think writing "parameters" suggests that this special case doesn't exist. Personally, I don't like writing singular and plural "simultaneously" with parentheses. --bender235 (talk) 04:26, 9 April 2019 (UTC)
thread_title:  parameter(s) singular or plural?   

message: Should we discuss in this articles the assumptions that are usually made about the properties of the likelihood function in the context of maximum likelihood estimation? I'm not talking about the properties of the MLE (asymptotic normality, etc.) but properties specifically of the likelihood function in the context of MLE that are usually labelled (and barely discussed in applied work) as "regularity conditions." It shouldn't be more than a paragraph or two based on Chanda (1954) and Mäkeläinen et al. (1981). Any thoughts? --bender235 (talk) 02:33, 25 July 2019 (UTC)
replies: text: This sounds worthwhile. I would learn something. Attic Salt (talk) 22:21, 25 July 2019 (UTC)
replies: text: I gave it a first try. Feel free to comment. --bender235 (talk) 00:34, 7 August 2019 (UTC)
thread_title:  Properties of the likelihood function for MLE 

message: I find this section a bit confusing:  
What is n?  I do not see a definition anywhere.
"s_{n}^{-1} is well-defined in an open neighborhood about 0 with probability going to one" -- I wonder what this notion of probability might mean.
DavidRideout (talk) 19:14, 8 September 2021 (UTC)
thread_title:  Likelihood equations