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The House of Mirth

The House of Mirth is a 1905 novel by American author Edith Wharton. It tells the story of Lily Bart, a well-born but impoverished woman belonging to New York City's high society around the end of the 19th century. The House of Mirth traces Lily's slow two-year social descent from privilege to a lonely existence on the margins of society. In the words of one scholar, Wharton uses Lily as an attack on "an irresponsible, grasping and morally corrupt upper class."

Before publication as a book on October 14, 1905, The House of Mirth was serialized in Scribner's Magazine beginning in January 1905. Charles Scribner wrote Wharton in November 1905 that the novel was showing "the most rapid sale of any book ever published by Scribner." By the end of December, sales had reached 140,000 copies. Wharton's royalties were valued at more than half a million dollars in today's currency. The commercial and critical success of The House of Mirth solidified Wharton's reputation as a major novelist.

Because of the novel's commercial success, some critics classified it as a genre novel. Literary reviewers and critics at the time categorized it as both a social satire and novel of manners. When describing it in her introduction to Edith Wharton's The House of Mirth: A Case Book, Carol Singley states that the novel "is a unique blend of romance, realism, and naturalism, [and thus] transcends the narrow classification of a novel of manners." The House of Mirth was Wharton's second published novel, preceded by two novellas, The Touchstone (1900) and Sanctuary (1903), and a novel, The Valley of Decision (1902).

Background, theme, and purpose

Wharton considered several titles for the novel about Lily Bart; two were germane to her purpose:

A Moment's Ornament appears in the first stanza of William Wordsworth's (1770–1850) poem, "She was a Phantom of Delight" (1804), that describes an ideal of feminine beauty:

"A moment's ornament" represents the way Wharton describes Lily's relationship to her reference group as a beautiful and well-bred socialite. Her value lasts only as long as her beauty and good-standing with the group is maintained. By centering the story around a portrait of Lily, Wharton was able to address directly the social limitations imposed upon her. These included the mores of the upper crust social class to which Lily belonged by birth, education, and breeding.

The final title Wharton chose for the novel was The House of Mirth (1905), taken from the Old Testament:

"The heart of the wise is in the house of mourning; but the heart of fools is in the house of mirth."

The House of Mirth spotlights social context as equally important to the development of the story's purpose, as the heroine. "Mirth" contrasted with "mourning" also bespeaks a moral purpose as it underscores the frivolity of a social set that not only worships money, but also uses it ostentatiously solely for its own amusement and aggrandizement. At the time the novel takes place, Old New York high society was peopled by the extraordinarily wealthy who were conditioned by the economic and social changes the Gilded Age (1870–1900) wrought. Wharton's birth around the time of the Civil War predates that period by a little less than a decade. As a member of the privileged Old New York society, she was eminently qualified to describe it authentically. She also had license to criticize the ways New York high society of the 1890s had changed without being vulnerable to accusations of envy motivated by coming from a lower social caste. She accused her peers of having lost the sense of noblesse oblige of their forebears.

Wharton revealed in her introduction to the 1936 reprint of The House of Mirth her choice of subject and her major theme:

"When I wrote House of Mirth I held, without knowing it, two trumps in my hand. One was the fact that New York society in the nineties was a field as yet unexploited by a novelist who had grown up in that little hot-house of tradition and conventions; and the other, that as yet these traditions and conventions were unassailed, and tacitly regarded as unassailable."

Wharton figured that no one had written about New York society because it offered nothing worth writing about. But that did not deter her as she thought something of value could be mined there. If only the writer could dig deeply enough below the surface, some "stuff o' the conscience" could be found. She went on to declare unabashedly that:

"[I]n spite of the fact I wrote about totally insignificant people, and 'dated' them by an elaborate stage-setting of manners, furniture and costume, the book still lives and has now attained the honour of figuring on the list of the Oxford University Press. . . . Such people always rest on an underpinning of wasted human possibilities and it seemed to me the fate of the persons embodying these possibilities ought to redeem my subject from insignificance."

The central theme of The House of Mirth is essentially the struggle between who we are and what society tells us we should be. Thus, it is considered by many to be as relevant today as it was in 1905. If its sole subject had been the excesses and lives of the rich and famous, by themselves, it is doubtful it would have remained popular for as long as it has. The House of Mirth continues to attract readers over a century after its first publication, possibly due to its timeless theme. That the life and death of Lily Bart matters to modern readers suggests that Wharton succeeded in her purpose: to critique "a society so relentlessly materialistic and self-serving that it casually destroys what is most beautiful and blameless within it."

Plot
Lily Bart is a beautiful but impoverished socialite, who is seeking a husband to secure her future. Her success is challenged by her advancing age — at twenty-nine, she has been on the "marriage market" for more than ten years — and her debts from gambling at bridge. While Lily admires the handsome and ambitious lawyer Lawrence Selden, he is too poor for her to seriously consider marrying; instead, her only prospects are the coarse and vulgar Simon Rosedale, a financier, and the wealthy but dull Percy Gryce.

Lily grew up surrounded by elegance and luxury — an atmosphere she believes she cannot live without. The loss of her father's wealth, coupled with the sudden death of her parents, left her an orphan at twenty. Lacking an inheritance or a caring protector, she adapts to life as a ward of her strait-laced aunt, Julia Peniston, from whom she receives an erratic allowance, a fashionable address, and good food, but little direction or parenting. Lily loathes the neglectful Julia and avoids her whenever possible but is forced to rely on her for her necessities and luxuries.

Lily learns that Selden and the vindictive heiress Bertha Dorset were once lovers. She also confides her money problems to Gus Trenor, a stockbroker and the husband of her childhood friend Judy, receiving from him a check for $5,000 and an investment of $4,000 in her name. Trenor tries to exploit his generosity to make a romantic move, but Lily spurns his attentions. Bertha still has feelings for Lawrence notwithstanding her recent breakup with him. When it becomes clear that Selden still has feelings for Lily, she aims to ruin the latter's budding romance with Percy by filling him in on the most salacious and scandalous rumors about Lily's card-playing and past romantic life. This effectively frightens Percy away. Lily, unaware of Bertha's machinations, blames Judy for having been the one to set the match up.

In retribution for a social snub, Lily's cousin Grace Stepney informs Julia of rumors that Lily is having an affair with Gus to obtain money so she can pay off her gambling debts. This plants seeds of doubt and discomfort in Julia who, though shocked, chooses to accept the rumors without speaking to her niece. Furthermore, Lily has soured her relationships with both Gus, who is angered by her spurning him, and Judy, because she refuses to visit her at Bellomont lest Gus confront her and reveal that she had manipulated him for financial gain.

To avoid having to spend time alone with her aunt, the Trenors, Simon Rosedale, or anyone else she considers a possible source of embarrassment or boredom, Lily begins to accept invitations from people with whom she would not ordinarily socialize. These include the Wellington Brys, who are newcomers to the New York social scene, and whose social rise is being engineered by Carry Fisher. Carry, a fallen aristocrat who supports herself by acting as a social secretary to usher newly wealthy people into fashionable society, invites Lily to social events hosted by Louisa Bry. Lily also attends the opera with Carry, Simon, and Gus. In the eyes of high society, Lily cheapens herself by voluntarily associating with her social inferiors. She returns briefly to Bellomont only to find that her peers now look at her with derision and disgust.

One of Julia's temporary servants, who is also the charwoman at Selden's apartment, sells Lily a package of torn love letters. These were written by Bertha Dorset years earlier, and they represent a chance for Lily to deal with her enemy. But instead of blackmailing Bertha into a positive relationship, Lily tries to neutralize the gossip by making herself useful to Bertha. Bertha, who is sleeping with Ned Silverton, relies on Lily to distract her husband, George. The extent to which Lily's reputation is damaged becomes obvious when she publicly appears in a way that comes across as advertising her availability for an illicit relationship. Following Carry's advice, the Wellington Brys throw a large "general entertainment" featuring a series of tableaux vivants portrayed by a dozen fashionable women in their set, including Miss Bart.

The pièce de résistance of this highly successful event turns out to be the portrayal of Mrs. Lloyd in Sir Joshua Reynolds' famous 18th-century painting (1775–1776). The portrait shows an attractive woman suggestively clad. As the curtain opens on this last scene, the gasp of approval heard from the audience was not so much for Reynolds’ brilliant interpretation of Mrs. Lloyd as it was for the loveliness of Lily Bart herself — marking the pinnacle of her social success but also the annihilation of whatever reputation is left to her. For better or for worse, she has transitioned from a marriageable "girl" to a not-quite-reputable woman similar to Carry Fisher. Yet she does not do as Carry Fisher does and accept the loss of her respectability as the price, she must pay to maintain a position in society.

As Selden observes her in this elegantly simple tableau, he sees the real Lily Bart as if for the first time and feels the desire to be with her. He finds her alone in the ballroom toward the end of the musical interlude, as the collective praise from her admirers is subsiding. He leads her to a garden where he tells her he loves her and they kiss. Lily sighs, " 'Ah, love me, love me—but don't tell me so!' " and takes her leave. As Selden gathers his coat to leave, he is disturbed by Ned Van Alstyne's remarks, ". . . .Gad, what a show of good-looking women; but not one of 'em could touch that little cousin of mine. . . . I never knew till tonight what an outline Lily has."

Lily pleads with her aunt to help settle her debts and confesses her addiction to gambling. Julia feels taken advantage of and refuses to help her, except to cover the bill for her clothes and accessories. Feeling trapped and disgraced, she turns to thoughts of Selden as her savior and has a change of heart towards him as she looks forward to his next visit at four o'clock.

Instead, her visitor turns out to be Simon Rosedale who, so smitten by her appearance in the tableau vivant, proposes a marriage that would be mutually beneficial. Considering what Rosedale knows about her, she skillfully pleads for time to consider his offer Selden does not appear for his 4:00 appointment nor does he send word in explanation. Instead, she learns that he has departed for Havana and then on to Europe for a business trip.

To escape the rumors arising from the gossip caused by her financial dealings with Gus, and also disappointed by what she interprets as Selden's emotional withdrawal, Lily accepts Bertha's spur-of-the-moment invitation to join her and George on a Mediterranean cruise aboard their yacht; Lily is expected once again to hide Bertha's affair. Lily's decision to accept the offer proves to be her social undoing.

In order to divert the attention and suspicion of their social circle away from her, Bertha insinuates that Lily is carrying on a romantic and sexual liaison with George by instructing her not to sleep on the yacht in front of their friends at the close of a dinner the Brys held for the Duchess in Monte Carlo. Selden inadvertently helps by arranging a night's lodging, under the promise that she leave promptly in the morning. The ensuing social scandal ruins Lily's reputation and almost immediately causes her friends to abandon her and Julia to disinherit her.

Undeterred by such misfortunes, Lily fights to regain her place in high society by advising Mr. and Mrs. Gormer on their entry into the aristocracy, but when the couple learn from Bertha the "scandalous" personal background of their new secretary, they are quick to chase her out rather than risk losing their new standing. Only two friends remain for Lily: Gerty Farish (a cousin of Lawrence Selden) and Carry, who help her cope with the social ignominy of a degraded social status while continually advising Lily to marry as soon as reasonably possible.

Despite the efforts of both Carry and Farish, Lily rapidly descends through the social strata of New York City's high society. She obtains a job as personal secretary of Mrs. Hatch, a disreputable woman who very nearly succeeds in marrying a wealthy young man in Lily's former social circle. It is during this occupation she is introduced to the use of chloral hydrate, sold in drugstores, as a remedy for malaise. Selden visits her and warns her that working for Mrs Hatch is putting her in a false position, and once she realises the truth of this, Lily resigns. She then finds a job in a milliner's shop; unaccustomed to the rigors of working-class manual labor, her rate of production is low, and the quality of her workmanship is poor, exacerbated by her increased use of the drug. She is fired at the end of the New York social season, when the demand for fashionable hats has diminished.

Meanwhile, Simon reappears in her life and tries to rescue her, but Lily is unwilling to meet his terms. Simon wants Lily to use the love letters that she bought from Selden's servant to expose the love affair between Lawrence and Bertha, thus crushing a potential rival and allowing him to further ascend up the social ladder. For the sake of Selden's reputation, Lily does not act upon Rosedale's request and secretly burns the letters when she visits Selden one last time.

Lily is stopped on the street by Nettie Struther, who Lily once helped get to a sanatorium. Nettie is now married and has a baby girl. Touched by Lily's exhausted state, and eager to show her how she's profited from her help, she invites her back to her modest but lovingly looked-after tenement home, and introduces her to the baby.

Eventually, Lily receives a ten-thousand-dollar inheritance following Julia's death, which she uses to repay Gus. Distraught by her misfortunes, Lily is now crippled by drug dependence. Once she has repaid all her debts, Lily takes an overdose and dies; perhaps it is suicide, perhaps an accident. As she is dying, she hallucinates cradling Nettie's baby in her arms. That very morning, Lawrence arrives at her quarters, to finally propose marriage, but finds Lily dead. Among her belongings are bank receipts proving her dealings with Gus were honorable and that the rumors that destroyed her were false from the beginning. This realization allows him to feel sympathy for her, and he is clearly distraught at her death.

Characters
Lily Bart— Wharton paints Lily, the heroine of her novel, as a complex personality with a given name that suggests purity and a surname that implies defiance. and the foolishness that the title of the novel implies. The combination of the social pressures and conventions of her reference group and her refusal to "settle" numerous times to save herself portend a fateful destiny where she becomes complicit in her own destruction. Wharton depicts Lily as having an aesthetic purpose in life—a fine specimen to be looked at and admired. Her extraordinary beauty should have served her well to find a wealthy husband with the requisite social status that would have secured her place in upper-class New York society. However, her inner longing to become free of her society's social conventions, her sense of what is right, and her desire for love as well as money and status have thwarted her success in spite of a number of eligible admirers over the ten years she has been on the marriage market. Challenges to her success are her advancing age—she is 29 as the novel begins—the loss of her father's wealth, and the death of her parents which has left her orphaned without a caring protector, her constant efforts to "keep up with the Joneses"(4), the very modest but erratic "allowance" from her strait-laced Aunt Julia, and her gambling debts which make her the subject of vile gossip. To protect Lawrence Selden's reputation, she refuses to use damning evidence against her nemesis, Bertha Dorset, which would have recouped her ruined social standing. This leads to a tragic yet heroic ending.

Lawrence Selden—A young lawyer who, although not wealthy himself, is able to move easily within and without Old New York's elite social circles through kinship with old-line New York families. He has known Lily since her "coming out" eleven years earlier. For all this time he has been in the background of her life. He views the comings and goings of New York's high society with the detachment and the objectivity of an outsider —a characteristic that Lily not only admires but also that allows her to view those people in her surroundings in an objective, critical and a not-so-flattering way. She becomes fascinated and envies his independence from the "tribe" and the freedom that has given him. Her encounters with Selden underscore the conflict between her inner voice —her self-hood at its core— and the outer voices of her reference group. It is from Selden's description, assessment and admiration of Lily's outward characteristics that we glean those attributes that contribute to New York high society's perception and misperceptions of who she is. He can be brutally honest about Lily's superficiality and artificiality and simultaneously appreciate the sparks of freedom and spontaneity that temper these negatives. These mutual admirable qualities give way to their romantic regard for one another. He is not, however, free from the social pressure of rumor. Though he has shown Lily consistent friendship, he abandons her when she becomes the victim of appearances that put her virtue, as an unmarried woman, in question.

Simon Rosedale—A successful and socially astute Jewish businessman—the quintessential parvenu—who has the money but not the social standing to be accepted into the circle of New York's leisure class. Building his fortune in real estate, Rosedale makes his first appearance in the story when he observes Lily leaving his apartment building after what appears to be a tryst with one of his tenants. Rosedale is interested in Lily because not only is she beautiful, but what is more important, she is also a social asset in gaining him a place in high society. She reflects that she has put herself in his power by her clumsy dress-maker fib and her refusal to allow him to take her to the station which would have given him the prestige of being seen by members of the society with whom he was aspiring to gain acceptance. As his social ascendency continues, he offers Lily marriage which would provide her a way out of her financial dilemma and her precarious social standing; she puts him off. His cleverness and business acumen serve him well to achieve a higher and higher rung on the social ladder. Lily, however, is on her way down to the point that Rosedale is no longer interested in marrying her. Despite the differences in their social standing, Rosedale by the end of the story shows compassion for Lily. He offers her a loan when he runs into her after she has lost her hat-making job—an offer she refuses.

Percy Gryce—A conservative, rich, but shy and unimaginative young eligible bachelor on whom Lily, with the support of her friend Judy Trenor, sets her sights. Percy's less than titillating personality notwithstanding, Lily works out a strategy to catch him at week-long festivities at Bellomont. Her fortuitous and successful encounter with Percy on the train to Bellomont further encourages her in pursuit of her goal. Her strategy gets interrupted, however, when Selden at week's end also appears on the scene unexpectedly. Lily then decides, on the spur of the moment, to set aside her well-thought-out tactics to pursue Percy in favor of spending some time with Selden. When, at a more rational moment, she returns to pursuing Percy, his mother-in law-to-be tells Lily at Jack Stepny's and Gwen Van Osburgh's wedding about his engagement to Evie Van Osburgh.

Bertha Dorset (Mrs. George Dorset)—A petite and pretty high-society matron whose husband George is extremely wealthy. She is first introduced catching the train to Bellomont where she boards with great fanfare and commotion. She demands that the porter find her a seat with her friends, Lily and Percy. Once at Bellomont Judy Trenor intimates to Lilly that Bertha is manipulative and also unscrupulous such that it is better to have her as a friend rather than an enemy. It is well known that Bertha is bored with her husband and seeks attention and love outside the confines of marriage. At Bellomont Bertha continues to pursue Selden in an attempt to rekindle the flame of an adulterous affair they have been carrying on but with which he has become disenamored. As Book I ends, she invites Lily to accompany her on a Mediterranean cruise to distract her husband so she can carry on an affair with Ned Silverton. Bertha understands, as a married woman, she must keep up appearances and ruthlessly impugns Lily's reputation to mask her own adultery. She spreads false rumors that besmirch Lily's virtue among their friends. Lily, as an unmarried woman without a protector, has little she can do in her own defense.

Mrs. Peniston (Julia)—Lily's wealthy, widowed Aunt –sister to Lily's father. Mrs. Peniston embodies "old school" morality and has a family pedigree that goes back to the industrious and successful Dutch families of early New York. Although she maintains an opulent residence on fashionable Fifth Avenue, she does not follow fashion or renovate constantly to maintain a chic appearance. In Lily's eyes, the Peniston home is therefore dingy. Mrs. Peniston's "Old New York" lifestyle requires keeping her drawing room neat, eating well and dressing expensively. She harbors a passive attitude and does not actively engage in life. Although she spends time in the country during the first part of the book, by the last half of the book she is a shut-in with significant heart problems. When Lily arrived in New York in financial distress after the death of her mother, Mrs. Peniston took pleasure in the public display of her generosity by agreeing to take Lily on for a year after her mother died—much to the relief of the extended family. She found, to her surprise, that she liked the volatile Lily. She therefore continued to support Lily for over a decade during Lily's fruitless search for a wealthy, socially connected husband. She indulges and passively enables Lily's habit of gallivanting with her fashionable friends, and ignores the way Lily avoids and abandons her. Although Lily is clearly Julia's favorite, displacing her previous favorite Grace Stepney, Julia never makes any verbal or written promise to provide for Lily in the long term. Although Lily and her friends believe that is "understood" that she will inherit most if not all of Julia's fortune, Julia herself never made such a statement. Indeed, her forbearance is stretched to the limit when rumors reach her that Lily gambles for money and is encouraging attention from married men who compensate her for it. As upset as Julia is by evidence of Lily's immoral behavior, she does not immediately ask Lily for details because it is easier to discredit the messenger. When Lily comes to Julia asking for money to pay various debts, including what Lily passes off as gambling debts, Julia refuses. The relationship is permanently damaged, and when Lily sails away with the Dorsets instead of cleaning up the social and financial mess she has made, Julia does not write to Lily or attempt to repair the relationship. When word reaches her that Lily has been publicly accused of having an affair with George Dorset, and when Lily continues to gallivant in Europe instead of returning home, Julia disinherits Lily in favor of the more loyal Grace Stepney. By the time Lily returns to New York, Julia has died, and nobody knows about the disinheritance until her will is read.

Judy Trenor (Mrs. Gus Trenor)—Lily's best friend and confidante— is the stereotypical high-society matron, married to Gus Trenor, a successful business man. She frequently hosts large parties and social events at their country home, Bellomont. By engaging in gossip Mrs. Trenor keeps up on the social scene. She acts as matchmaker between Lily and Percy Gryce. She uses Lily as her surrogate private secretary and spends much of her day making sure that every detail of her events is done to perfection. This includes poring over lists to decide which guests are the most desirable to invite, which have been "stolen" by another conflicting event, and which unmarried men and women should be set up together. She invites Selden to Bellomont on anonymous advice to keep Mrs. George Dorset entertained. Judy suffers from frequent headaches and avoids noisy environments such as the opera. She must sometimes cancel minor engagements on short notice, and prefers to stay at Bellomont when her head hurts.

Gus Trenor—Judy Trenor's husband—a massive man with a heavy carnivorous head and a very red complexion. He is a successful stock market speculator and an advocate of Simon Rosedale's acceptance in high society circles although he considers him a bounder. He is also a notorious flirt and looks for attention in relationships with women outside of his marriage. Gus becomes enamored with Lily, a frequent guest at his wife's weekend social events. He uses his financial investment skills and a large sum of his own money in a risky investment for Lily which she agrees to. The proceeds from this speculation will help her pay her gambling debts and other expenses necessary to keep up appearances. The investment pays off for Lily financially, as Gus intends that it should, but the friendship turns sour when Lily is unwilling to exchange romantic attention for money the way Gus believes she tacitly agrees to do.

Carry Fisher (Mrs. Fisher)—A small, fiery and dramatic divorcée. She is perceived as carrying "a general air of embodying a 'spicy paragraph';"(70) and according to Mrs. Trenor, ". . .most of her alimony is paid by other women's husbands." (91) She sponges money from Gus Trenor to cover her bills much to his wife's chagrin. Although Gus accepts romantic favors from Mrs. Fisher in exchange for paying her bills and investing her money in the stock market, he considers her a "battered wire-puller"(94) in comparison to the fresh and unsullied Miss Bart. Carry is also known for bringing newcomers into high society such as Rosedale and the Wellington Brys, who had managed the miracle of making money in a falling market. After Lily has been expelled from the upper class by Bertha, Carry is one of the few people who still show compassion toward her, offering Lily support and job opportunities. Carry is an example of a woman who finds ways to earn money and to succeed in society despite being divorced and somewhat disreputable. Her presence in the story refutes the notion that Lily has no choice except to self-destruct.

Ned Silverton—A young man, whose first intention was to live on proofreading and write an epic, but ended up living off his friends. Ned's romantic relationship at the Bellomont house party is with Carry Fisher. Six months later, Ned accompanies Lily and the Dorsets on their Mediterranean cruise. He has an affair with Mrs Dorset, who manages to keep it concealed from most of society. Ned's increasing gambling addiction consumes not only his resources but that of his sisters, who live in poverty while he travels with Bertha Dorset. After the fling with Bertha ends, Ned participates in a scheme to help a purportedly wealthy but disreputable woman to marry the younger brother of Gwen and Evie Van Osburgh. This conspiracy, in which Lily is implicated, helps ensure Lily's downfall.

Evie Van Osburgh—A young, innocent, dull, and conservative, stay-at-home kind of a girl, heiress to a substantial fortune. Judy Trenor paired her sister, Gwen, with Percy Gryce at the Sunday-night supper at Bellomont. Evie ends up getting engaged within six weeks of their stay at Bellomont to Percy Gryce due to Bertha Dorset's match-making skills.

Gerty Farish—Selden's cousin. She is a kind, generous woman who occupies herself with charity work, but Lily despises her because of her less than glamorous appearance. In Book Two, Gerty becomes one of Lily's only friends, giving her a place to stay and taking care of her when everyone else abandons her. Lily does not wish to continue living with Gerty or combining resources because she is unwilling to lower herself to the standard of living Gerty can afford.

Jack Stepney and Gwen (Van Osburgh) Stepney—A very wealthy couple—guests at Bellomont just before celebrating their wedding at the Van Osburgh's estate six weeks later. They belong to Old New York's high society, although their money comes from Gwen's side. Prior to his marriage, the nearly bankrupt Jack has business dealings with Simon Rosedale and has tried to afford him entrée into New York's high society. After marrying Gwen, Jack becomes plump, conservative, and complacent. Jack is Lily's cousin so he agrees to shelter her for the night after Bertha kicks her off her yacht for ostensibly carrying on romantically with Bertha's husband.

Grace Stepney—Lily's middle-aged cousin lives in a boarding house and has spent most of her life waiting on Julia Peniston. Until Lily arrived, Grace was Julia's favorite: she shares Julia's conservatism and sense of propriety, and she is willing to help out during the fall cleaning. Grace attracts and remembers all manner of gossip related to high society in general and to Lily in particular, but does not generally gossip herself. Lily treats Grace very poorly, regarding her as insignificant. When Lily prevails on her aunt Julia to exclude Grace from a family dinner party, Grace retaliates. She relays to Aunt Julia the talk about Lily's attention to Gus Trenor in exchange for money that Lily used to pay gambling debts, and stops protecting Lily from the otherwise predictable consequences of Lily's actions. When Lily comes to Grace after the reading of the will, Grace does not have the money to give Lily the loan she is asking for because the assets Grace inherited from Julia are still tied up in probate. When Lily asks Grace to borrow money against Aunt Julia's estate and lend or give it to Lily, Grace refuses.

Critical reception
In the contemporary book review "New York Society Held up to Scorn in three New Books" (15 October 1905) The New York Times critic said that The House of Mirth is "a novel of remarkable power" and that "its varied elements are harmoniously blended, and [that] the discriminating reader who has completed the whole story in a protracted sitting, or two, must rise from it with the conviction that there are no parts of it which do not properly and essentially belong to the whole. Its descriptive passages have verity and charm, it has the saving grace of humor, its multitude of personages, as we have said, all have the semblance of life."

The publication of the novel prompted letters to the editor of the "New York Times Saturday Review of Books" which argued the merits of the story, saying that the novel was a faithful and true portrait of the New York City gentry, while detractors said that it impugned the character of the city's social élite as a heartless and materialist leisure class.

Adaptations
The novel The House of Mirth (1905) has been adapted to radio, the stage and the cinema.

 * The Play of the novel The House of Mirth (1906), by Edith Wharton and Clyde Fitch.
 * La Maison du Brouillard (1918), directed by Albert Capellani, featured Katherine Harris Barrymore as Lily Bart; a French silent film.
 * The House of Mirth was presented on radio's Theatre Guild on the Air December 14, 1952. The one-hour adaptation starred Joan Fontaine and Franchot Tone.
 * The House of Mirth (1956), directed by John Drew Barrymore. Matinee Theatre: Season 2, Episode 56. (4 December 1956)
 * The House of Mirth (1981), directed by Adrian Hall. A television film for the Public Broadcasting System in the U.S.
 * Edith Wharton's The House of Mirth (1995) adapted for the stage by Dawn Keeler. The play was performed by the Cambridge Theatre Company at the Theatre Royal in Winchester, England. After the initial production, the play toured England for nine weeks. This modern adaptation offers a late-twentieth-century interpretation of Lily Bart's story that emphasizes freedom, relationships, and tragedy.
 * The House of Mirth (2000), directed by Terence Davies, featured Gillian Anderson as Lily Bart.
 * Composer Garth Baxter has written the opera Lily based upon The House of Mirth, with a libretto by Lisa VanAuken. Lily is a two-act opera in the romantic tradition with full orchestra. Arias from the opera have been featured in two albums: Katherine Keem Sings Songs and Arias by Garth Baxter, from Centaur Records, and ASK THE MOON, music for voice and piano by Garth Baxter from Navona Records.

Reviews

 * Kornasky, L. (2014). "Edith Wharton in context". Studies in American Naturalism, 9(1), 107.
 * Preston, C. (2004). "The critical reception of Edith Wharton". The Yearbook of English Studies, 34(1) Nineteenth-Century Travel Writing,336-338.Accessed: 01-04-2016 03:52 UTC.DOI: 10.2307/3509561
 * Singley, C., & Moseley, A. (2007). "Wharton and Cather". American Literary Scholarship, 2007(1), 139-168.
 * Wharton, Edith. (1905). "Mr. Sturgis's Belchamber", Bookman, 21(May), 309-310.
 * Wharton, Edith. (1905). "Mr. Sturgis's Belchamber", Bookman, 21(May), 309-310.

title: The House of Mirth


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Canta (magazine)

Canta (stylised CANTA) is the official magazine of the University of Canterbury Students' Association (UCSA) of the University of Canterbury, established in 1930.

The magazine generally features a light-hearted tone, with an emphasis on short, image-heavy articles.

Canta is a member of the Aotearoa Student Press Association (ASPA) and the New Zealand Press Council.

Previous issues are archived and freely available in electronic form on the website of the National Library of New Zealand.

Publication information
Canta is freely available around The University of Canterbury campus every second Monday in term time. It is 48 pages long. The Managing Editor is Liam Donnelly and Print Editor is Liam Stretch. There are over 60 regular contributors.

There are a variety of sections which typically appear in every issue. These include:
 * Campus News (including Fake News)
 * The President's Piece
 * Comedian's Corner
 * Flat Famous
 * Gig Guide
 * Lucky Dip (blind date)
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Talk:Mohammedia

Opening heading
What! No mention of Al Jazeera? Wetman 07:23, 5 Aug 2004 (UTC) fedalasurf pour voir la plage des sablesttes —Preceding unsigned comment added by 213.90.198.3 (talk) 15:13, 12 March 2009 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: page moved. Vegaswikian (talk) 22:27, 13 August 2011 (UTC)

Mohammédia → Mohammedia – per WP:ENGLISH, WP:Place, WP:DIACRITICS and WP:UE Encyclopedia britannica uses Mohammedia

Google books and scholar return higher hits (in English) for Mohammedia than Mohammédia, the latter spelling is just typically French and does not accurately describe the native spelling. The adding of the é is of superfluous value even for the French language. Not to mention that all of this is about a place whose native name is not in Latin alphabet Tachfin (talk) 16:28, 6 August 2011 (UTC)
 * Supprot, Mohammédia is French form. Takabeg (talk) 03:15, 13 August 2011 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

title: Talk:Mohammedia


text:
User:Nick Nolan

Hi, this is just my personal page, nothing much to see. The preferred methods of contacting me are through email at malglico@gmail.com or on my talk page.

title: User:Nick Nolan


text:
Talk:Maximum likelihood estimation

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 27 August 2021 and 19 December 2021. Further details are available on the course page. Student editor(s): Jiang1725. Peer reviewers: EyeOfTheUniverse, 5740 Grant L, Jimyzhu.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 03:39, 17 January 2022 (UTC)

removal
I removed this from the article, until it can be made more NPOV and more encyclopedic. Currently reads more like a list of observations than true encyclopedic content and needs more explanation. --Lexor|Talk 07:25, 5 Aug 2004 (UTC)

 * Maximum likelihood is one of the main methods used by frequentist (i.e. non-Bayesian) statisticians. Bayesian arguments against the ML and other point estimation methods are that

 * all the information contained in the data is in the likelihood function, so why use just the maximum?Bayesian methods use ALL of the likelihood function and this is why they are optimal.

 * ML methods have good asymptotic properties (consistency and attainment of the Cramer-Rao lower bound) but there is nothing to recommend them for analysis of small samples

 * the method doesn't work so well with distributions that have many modes or unusual shapes. Apart from the practical difficulties of getting stuck in local modes, there is the difficulty of interpreting the output, which consists of a point estimate plus standard error. Suppose you have a distribution for a quantity that can only take positive values, and your ML estimate for the mean comes out at 1.0 with a standard error of 3? Bayesian methods gives you the entire posterior distribution as output, so you can make sense of it and then decide what summaries are appropriate.

Style
This whole article reads like an 80's highschool textbook. As a matter of fact, a lot wikipedia's articles on difficult to understand subjects read like they're out of an 80's highschool textbook, making them only useful to people who have already know the subject back to front, making the entire wikipedia project a failure. —Preceding unsigned comment added by 4.232.174.170 (talk • contribs)

 * Clarity issues in a statistics article (a subject that is less than clear) should not be used to make the inference that "the entire wikipedia project a failure" Rschulz 23:56, 1 Mar 2005 (UTC)

 * This article was never intended to be accessible to secondary school students generally (although certainly there are those among them who would understand it). I would consider this article successful if mathematicians who know probability theory but do not know statistics can understand it.  And I think by that standard it is fairly successful, although more examples and more theory could certainly be added.  If someone can make this particular topic comprehensible to most people who know high-school mathematics, through first-year calculus or perhaps through the prerequisites to first-year calculus, I would consider that a substantially greater achievement.  But that would tak more work. Michael Hardy 00:40, 10 Jan 2005 (UTC)

 * I would consider this article a failure if only "mathematicians who know probability theory" can understand it. I got to the article via a link from Constellation diagram, and learned nothing from it that I didn't already know from the other article—not even what it is used for.  121a0012 05:41, 14 June 2006 (UTC)
 * I think there is kind of a style struggle in mathematics whereas some people prefer math text to say what it means and discuss itself introspectively, and others prefer a terse no nonsense style. I find the discussion based approach to be healthier.  One of the problems though is it isnt neccessarily encyclopedic to use that tone.  I mean most important math can be stated in one or two sentence.  Doesnt mean that the sentence will be approachable.  But it will be factually complete saying all there is to say about the subject.  So then an encyclopedia editor struggles with the fact that you have to say more to make it approachable, but at the same time you can say less and say all there is to say.  It is a problem unique to mathematics. Jeremiahrounds 13:01, 20 June 2007 (UTC)
 * But most mathematicians who know probability theory do not know this material, so it should not be considered a failure if they understand it. That is not to say it should not be made accessible to a broader audience.  But that will take more work, so be patient. Michael Hardy 17:42, 14 June 2006 (UTC)

 * I totally agree with the original complaint that this along with many other wikipedia math articles are too heavy going. For that reason I replaced the first paragraph with something more digestible.  I see no reason to dive into using math symbols right in the first paragraph.  --Julian Brown 02:50, 30 August 2007 (UTC)

 * As a university student learning statistics, I think this article needs improvement. It would be good if a graph of the likelihood of different parameter values for p was added (with the maximum pointed out) to the example. This addition would require adding some specific data to the example. Also, the example should be separated from the discussion about MLE, to make sure people understand that the binominal distribution is only used for this case. The reasons why it is good to take the log of likelihood are not discussed. Further the discussion about what makes a good estimator (and how MLE is related to other estimators) could be expanded. Rschulz 23:56, 1 Mar 2005 (UTC)

 * The "left as an exercise to the reader" part is definitely gratuitous and needs to go. I came to this page to learn about the subject, not for homework problems.

 * a user in CA Good article, don't be so hard on the author(s), of course could be better, but most of us have day jobs, but I would change the notation, as this was confusing " The value (lower-case) x/n observed in a particular case is an estimate; the random variable (Capital) X/n is an estimator." seems to conflict with the excellent example at the end for finding maximum likelihood x/n in a bionomial distribution of x voters in a sample of n (without replacement). Now, next question, can anybody explain the Viterbi algorithm to a high-schooler? 01 March 2004

 * I don't see the conflict. The lower-case x in the example at the end is not a random variable, but an observed realization. Michael Hardy 22:50, 2 Mar 2005 (UTC)

 * To chime in on this discussion: reading this article felt too much like a high school text book, and too little like an encyclopedia. Though I agree that this article should be (in part) accessible to laymen, elaborating to this amount of detail makes it hard for non-laymen to get to what they want to know.  I believe that laymen are more than adequately catered to in the premier section of this article.  The rest of the article should highlight important properties and results.  Editors, remember, this is an encyclopedia, a reference book, not a text book, and not Wikiversity.  The article can be shortened significantly by: (1) Trimming the huge (seemingly repetitive) list of "It has widespread applications in various fields, including:..." list. (2) Assuming the reader already has a basic knowledge of what statistics is or at least that he has read/understood the statistics article of Wikipedia. (3) Cutting the hand-holding mathematics in the "Examples" section, which should be moved to Wikiversity. Solace1  —Preceding unsigned comment added by 207.237.81.59 (talk) 13:34, 11 November 2008 (UTC)

Difference between likelihood function and probability density function
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)

where is the spanish version?

Is it possible to get the text shown as it should read by people who don't know latex code instead of symbols such as x_1 for subindexes or x^1 for superindexes, etc. I understand those but not other symbols which are used in this article, and in any case they hamper reading. Thanks! Xinelo 14:47, 21 September 2006 (UTC)

 * Maybe it was a temporary problem; they look OK to me now. Michael Hardy 15:06, 21 September 2006 (UTC)

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

Thank You!!
This article is fantastic. It is more understandable than my class notes and has helped me greatly for my class. I also really appreciated the use of examples. Many many thanks to the people who wrote it!. Poyan 8:09, 6 December 2006 —The preceding unsigned comment was added by 128.100.36.147 (talk) 13:09, 6 December 2006

Sloppiness
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))
 * 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)

Is this true
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 undefined 05:00, 15 May 2007 (UTC)
 * 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 undefined 06:37, 15 May 2007 (UTC)
 * It was true but not relevant, so I dropped it when I expanded this section. RVS (talk) 00:17, 25 December 2008 (UTC)

Some more applications please
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)indiaman1

I think the article is not "well written" "MLE is a popular statistical method" - I think such comments should be left to established textbooks. In Germany it could be Fahrmeir (et al.), may be others as well. These professors with the 3rd + x editions of basic statistics. There should be mentioning of competitive measurements (confidence intervals, potential misuse of the p-value (deciding after the fact which of the confidence levels to use etc.)) For a non-english (non-american) it would be interesting who are considered the leading statistics profs in the UK etc. [Gaschroeder]   -- Gaschroeder (talk) 14:47, 3 October 2010 (UTC) -- Gaschroeder (talk) 14:51, 3 October 2010 (UTC)

Non-independent variables
I have added a section on non-independent variables. I hope this proves useful to someone :) Velocidex (talk) 04:44, 19 March 2008 (UTC)
 * 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)

For generality this section really needs something said about mixed discrete-continuous distributions. Melcombe (talk) 10:45, 19 March 2008 (UTC)

Mathematical precision
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)

Typography
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)

link has pw
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)

applications list
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)

about which distribution
(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)
 * 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)

Consistency
Why is it listed that "The MLE is asymptotically unbiased, i.e., its bias tends to zero as the sample size increases to infinity" under "Under certain (fairly weak) regularity conditions" when one of the conditions listed is "The maximum likelihood estimator is consistent." Isn't being consistent the same as being asymptotically unbiased? cancan101 (talk) 00:11, 3 March 2009 (UTC)
 * Technically they are not the same thing; consistent includes the spread of the distriburion reducing, while the othef only needs the mean to approach the right value. Your point however is broadly correct, as consistency almost implies asymptotically unbiased (one exclusion being that the mean stll need not exist for consistency). I think what is meant is really that unbiasedness usually holds under similar conditions to those needed for consistency. The level of technical detail in the article does not naturally incline to including more detail. I think it is meant to be a general warning not to expect that maximum likelihood will always "work".

Article title
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)

 * I agree - why the change? The article even starts "Maximum likelihood estimation (MLE) ..." 128.61.125.131 (talk) 14:55, 28 January 2010 (UTC)

 * 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)

 * 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.  …  st pasha  »  18:42, 29 January 2010 (UTC)

Error in pdf formula?
Shouldn't this:
 * The joint probability density function of these $$n$$ random variables is then given by:
 * $$f(x_1,\ldots,x_n)=\frac{1}{2\pi\sqrt{\text{det}(\Sigma)}} \exp\left( -\frac{1}{2} \left[x_1-\mu_1,\ldots,x_n-\mu_n\right]\Sigma^{-1}    \left[x_1-\mu_1,\ldots,x_n-\mu_n\right]^T       \right)$$

be this?:
 * The joint probability density function of these $$n$$ random variables is then given by:
 * $$f(x_1,\ldots,x_n)=\frac{1}{(2\pi)^n\sqrt{\text{det}(\Sigma)}} \exp\left( -\frac{1}{2} \left[x_1-\mu_1,\ldots,x_n-\mu_n\right]\Sigma^{-1}    \left[x_1-\mu_1,\ldots,x_n-\mu_n\right]^T       \right)$$

as per Multivariate_normal_distribution? Aaron McDaid (talk - contribs) 22:10, 2 April 2009 (UTC)

 * I've changed it to this:
 * $$ f(x_1,\ldots,x_n)=\frac{1}{(2\pi)^{n/2} \sqrt{\text{det}(\Sigma)}} \exp\left( -\frac{1}{2} \left[x_1-\mu_1,\ldots,x_n-\mu_n\right]\Sigma^{-1} \left[x_1-\mu_1,\ldots,x_n-\mu_n\right]^T \right)$$
 * (Alternatively, one could put (2π)n under the radical.) Michael Hardy (talk) 01:42, 3 April 2009 (UTC)
 * (Alternatively, one could put (2π)n under the radical.) Michael Hardy (talk) 01:42, 3 April 2009 (UTC)
 * (Alternatively, one could put (2π)n under the radical.) Michael Hardy (talk) 01:42, 3 April 2009 (UTC)

Weak consistency or strong consistency?
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)

Insufficient content
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). ...  st pasha  » talk » 10:31, 2 September 2009 (UTC)

(Dis?)motivational example
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.  …  st pasha  »  07:50, 24 January 2010 (UTC)

“Convergence of maximum likelihood estimator” gone
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: Sentence: removed.  …  st pasha  »  08:11, 24 January 2010 (UTC)
 * 1) It silently assumes that θ is a scalar, although the article states otherwise;
 * 2) It concludes the “proof” with Pr[Sn(θ)>0] → 0, while in fact this probability is equal to 1 for all n.
 * 3) The general idea to prove a theorem without stating it, and without specifying the assumptions needed for it, is faulty.

Uniqueness?
This article made a lot of statements about "the" mle, when this seem contrary to the literature. Old examples of Cramér show the importance of including local minima (and the problems of restricing attention to only maxima).

Second, this article stated a lot of nice properties of the mle, and included interpretations that the (?) MLE method was the best. But many other estimators have those properties, which may be proved under wider conditions---see the maximum probability estimator of Wolfowitz (no relation!!), and Le Cam on the superior properties of Bayes methods and a lot of examples where the MLE has trouble. I tried to soften uniqueness claims ("the"), where this was possible. No doubt others can improve my edits. Thanks! Kiefer.Wolfowitz (talk) 00:36, 25 January 2010 (UTC)

 * I'm not sure what it is that you're trying to say. The MLE is one of the generic estimation techniques, applicable to a wide range of problems. There are typically other methods which can be applied to same problems, and some of those methods are as good as MLE. But they can't be better than MLE (at least up to the second-order), because MLE is already efficient — at least as long as we're talking about the MSE loss function.
 * The maximum probability estimator — what is it? As for the “Bayes estimator” — there is no such thing. There is a bayesian school of thought for estimation, and within that school many estimators can be constructed. Neither of those estimator is “superior” to MLE, because they work with different model: they require that additional prior knowledge about the value of the parameter was available.
 * I'm not saying that MLE is like “best from the best” or anything like that, but the article should be kept encyclopedic, and any comparison to other estimation methods be done in a separate section (or maybe even a separate article).  …  st pasha  »  08:18, 25 January 2010 (UTC)

 * On the first point, the article does have "For certain problems the maximum likelihood estimates may not be unique, or even may not exist" fairly early on. There does seem scope to add some new sections to cover the non-uniqueness case from both a theoretical-asymptotic viewpoint and a practical one. Another thing possibly not mentioned fully enough is the case where estimates are on the boundary of the parameter space (asymptotic properties). Melcombe (talk) 10:48, 25 January 2010 (UTC)

Proposed restructure
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)

 * 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)

 * 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.  …  st pasha  »  18:52, 26 January 2010 (UTC)

 * 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)

Discrete case
So, does anybody know if there is a justification for the use of MLE in case when either the parameters or the data belong to a discrete set?  …  st pasha  »  19:48, 26 January 2010 (UTC)

 * MLE can be proved to be have good properties for one-parameter exponential families, e.g. binomial, negative binomial, Poisson, etc. A good reference is the work of Erling Andersen, who especially emphasized the importance of conditional maximum likelihood estimation, which avoids the inconsistency of the MLE in many cases. (For a counter-example to conditional mle, see Deb. Basu's selected writings (Ghosh, ed.).) MLE has trouble with multiple parameters, and the bestiary of likelihoods (partial, profile, etc.) have worse properties than MLE with one parameter. Kiefer.Wolfowitz (talk) 22:49, 26 January 2010 (UTC)

 * Cox's Principles of Statistical Inference ISBN 0-521-68576-2 has (page 143) discussion of a case where the mean of a normal distribution is known to be an integer... but this more in an "inference" context than "estimation". Page 144 in the same book has the example of a mixture of two normals being a case where standard asymptotic theory doesn't apply, which relates to another earlier question. Melcombe (talk) 10:32, 27 January 2010 (UTC)

 * Perhaps the most important condition for the success of MLE in irregular cases is the following:
 * * MLE was used on this irregular problem by David Cox!
 * Cox's recent Principles mentions a practical example where the mle has substantial bias, making it dangerous in application, for optimal design of experiments for logistic regression, citing Rolf Sundberg (EM method) and Vågero, I believe. (Cox and Sunberg are usually fond of MLE.) Kiefer.Wolfowitz (talk) 14:17, 27 January 2010 (UTC)

 * On discrete parameters, the case where the mean of a normal distribution is known to be an integer, is also used for two Examples by Kendall&Stuart (Advanced Th. of Statistics, Vol 2, 3rd Edition, Examples 18.21-2). They indicate that the second provides an example where the MLE is not consistent. Melcombe (talk) 14:37, 1 February 2010 (UTC)

I object
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)

Who is Pfanzagl?
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)

 * The inadequate reference has been improved, now reading "Pages 207-208 in ". Kiefer.Wolfowitz (talk) 11:37, 29 June 2010 (UTC)
 * 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)

Vandalism: Administrators, consider raising the protection level
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)

Revert
I'm reverting the last edit by Kiefer.Wolfowitz due to several reasons: (1) For many purposes, the maximum-likelihood estimator has poor theoretical properties --- not explained in the following text (what poor properties, which purposes?), and this sentence didn't fit into the paragraph anyway, (2) ... mathematical statisticians consider a related estimator that finds critical points of the log-likelihood function. -- not true, since right after you find the MLE estimator you compute the information matrix (−Hessian of the objective function), and verify that this matrix is positive definite, which means that you have the maximum; (3) When evaluated on a sequence of observations from the true distribution, a subsequence of the estimator-evaluations ... --- saying the same as before, only in far more complicated language; (4) ..., for some (unspecified) sample size n --- stated as such the property becomes false: for a given n (even unspecified) you cannot estimate with arbitrary precision, however for arbitrary precision you can find the right sample size n, as was stated initially (and why n suddenly bold?); (5) if the limiting likelihood function ℓ(θ --- again not true. Identification condition is necessary and sufficient for ℓ to have the global maximum (assuming the function ℓ is well-defined), for example see Lemma 2.2 in Handbook of Econometrics 36.  //  st pasha  » 18:43, 16 March 2011 (UTC)

 * I am on the road and so I cannot reference everything. I shall reply more fully later. Considering zero-score estimators is standard: See Ferguson's article,
 * which I believe is cited in the enthusiastic book of Pawitan. Kiefer.Wolfowitz  (Discussion) 08:17, 17 March 2011 (UTC)
 * which I believe is cited in the enthusiastic book of Pawitan. Kiefer.Wolfowitz  (Discussion) 08:17, 17 March 2011 (UTC)

 * BTW, it is possible to link to Handbook of Economics chapters:

Are there other ways of estimating?
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)
 * 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)
 * 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)

Kolmogorov in the introduction
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)
 * 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)

Subsequence of the sequence converges in probability
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)
 * 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)

Error in the article: Consistency
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)

Problem in "Higher-order properties" section
I believe that the bias calculation has a typo. The tensor $$J_{j,ik}$$ should actually be $$J_{i,jk}$$. The cited source is equation (20) from Cox and Snell (1968) (http://www.jstor.org/stable/2984505). This problem is there as well.

I checked two other sources, which comply with the correction I propose:

 * Equation (16) from Shenton, L. R., & Wallington, P. A. (1962). The Bias of Moment Estimators with an Application to the Negative Binomial Distribution. Biometrika, 49(1/2), 193. http://doi.org/10.2307/2333481
 * 7th equation in page 29 from Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80(1), 27–38. http://doi.org/10.1093/biomet/80.1.27

I'm not an expert on this topic, but rather an advanced user. If somebody confirms this, I would make the changes. — Preceding unsigned comment added by Fbalzarotti (talk • contribs) 21:55, 28 February 2016 (UTC)

 * Be bold (but not reckless): You've done some good homework.  Make the change.  I don't have time to check the math nor the sources you cite.
 * I suggest include a single footnote citing all three sources with a brief comment listing first Shelton and Wallington (1962) and Firth (1993) then noting that Cox and Snell (1968) write $$J_{j,ik}$$ where the other sources write $$J_{i,jk}$$, and you believe the other sources (because ...???)
 * Do you have an example where using $$J_{i,jk}$$ would give a different answer from using $$J_{j,ik}$$? If you do, then you might do a simulation to confirm -- and it would be worth mentioning that example in the text and your simulation in a footnote.
 * For the normal distribution, $$K_{ijk}$$ = 0 = $$J_{j,ik}$$, for all $$i$$, $$j$$, and $$k$$. For Poisson regression and logistic regression, $$J_{j,ik}$$ = 0;  all the bias comes from $$K_{ijk}$$.  These three examples don't expose the difference.
 * (I'm not an expert in this either.)  DavidMCEddy (talk)

 * Thanks for the advice, I'll be bold and go ahead. I believe the others authors and not Cox because I went through the calculations myself, I'm quite confident given that I found the other articles confirming them.
 * As for adding an example, I'm not sure. The situation with which I'm working is a bit odd.

 * I'm using a set $$\{n_1, ..., n_m\}$$ of random variables following a multinomial distribution with parameters $$\{p_1, ..., p_m\}$$ where $$ p_m = 1-\sum _{i=0}^{m-1}p_i$$ and $$ N = \sum _{i=0}^{m}n_i$$. When working with maximum likelihood estimators for the parameters $$p_i$$ the necessary tensors for the bias estimation are $$K_{ijk}=2N[\delta_{ijk}\,p_i^{-2}-p_{m}^{-2}]$$ and $$J_{j,ik} = N\,(1-2p_m)\,p_m^{-2}$$, where the indices $$\{i,j,k\}$$ run from 1 to m-1. They're both symmetric in $$\{i,j,k\}$$ and the problem doesn't show. So far, so good.

 * I'm actually working in a situation in which the parameters $$\{p_1, ..., p_m\}$$ depend on a set of variables $$\{x_1, ..., x_k\}$$. I'm using a maximum likelihood estimator for these variables $$ x_i $$ and I need to calculate an approximation of their bias. In this context the result for $$ J_{j,ik} $$ is
 * $$ J_{j,ik} = N E\left[

\left(\sum_{u=1}^m \frac{\partial p_u}{\partial x_j} \right) \left(\sum_{u=1}^m -\frac{1}{p_u} \frac{\partial p_u}{\partial x_i} \frac{\partial p_u}{\partial x_k} + \frac{\partial^2 p_u}{\partial x_i \partial x_k} \right) + \sum_{u=1}^m \frac{\partial p_u}{\partial x_j}  \left( -\frac{1}{p_u} \frac{\partial p_u}{\partial x_i} \frac{\partial p_u}{\partial x_k} + \frac{\partial^2 p_u}{\partial x_i \partial x_k} \right) \, \left(\frac{1}{p_u}-2\right) \right] $$
 * which is not symmetric. By interchanging $$j$$ by $$i$$ the result completely changes. In general, the functional dependence $$ \bar{p}(\bar{x}) $$ makes this tensor non-zero and non-symmetric, which are the cases with which I'm working.
 * I have calculated and simulated all this, but I'm afraid that posting this whole thing as an example may not be exactly informative, but quite disruptive. Fbalzarotti (talk) 15:42, 29 February 2016 (UTC)

 * I'm impressed.
 * After you get your work published -- at least as a tech report on the web -- I might like to see a line added saying something like what I wrote above: "For the multivariate normal distribution, $$K_{ijk}$$ = 0 = $$J_{j,ik}$$, for all $$i$$, $$j$$, and $$k$$.  For Poisson regression and logistic regression, $$J_{j,ik}$$ = 0;  all the bias comes from $$K_{ijk}$$.  For an example in which $$J_{j,ik}$$ is not zero, see" your tech report.
 * Conflict of interest policy "strongly discourages" "contributing to Wikipedia about yourself, family, friends, clients, employers, or your financial or other relationships." For examples like this, they suggest you use the  template on a talk page like this:  Provide the suggested change with a link to your paper, noting that it's your work.  Then hope that someone else will make the actual change.
 * And good luck with your research. DavidMCEddy (talk) 17:40, 29 February 2016 (UTC)

Move to hyphenated title
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)
 * 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)
 * Oh I see, thank you. Tayste (edits) 20:09, 20 June 2016 (UTC)

Maximum a posteriori estimation
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)

 * 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)

According to the french page, this is completely wrong (and also present in the introduction, and terribly misleading since it's a direct contradiction

https://fr.wikipedia.org/wiki/Maximum_de_vraisemblance#Histoire

It roughly says that it was a misconception, and fisher disproved it in 1922.

En 1912, un malentendu a laissé croire que le critère absolu pouvait être interprété comme un estimateur bayésien avec une loi a priori uniforme^2. Fisher réfute cette interprétation en 1921^2 — Preceding unsigned comment added by 143.121.239.90 (talk) 13:14, 5 March 2018 (UTC)

Section "Principles"
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 x1, x2, …, xn 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 { f(· (where θ is a vector of parameters for this family), called the parametric model, so that θ0). 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 f(x1, …, xn, 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)

Subsection "Asymptotic normality"
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)

Furthermore, the subsection begins with this claim: "In a wide range of situations, maximum likelihood parameter estimates exhibit asymptotic normality". The claim is false: the only situations are those in which the data is simulated. Thus it appears that the subsection was written under a misconception.

For the above reasons, I have removed almost all of the subsection. BetterMath (talk) 20:30, 29 January 2018 (UTC)

The level of pedantry leveled here is absurd. By this reasoning, asymptotic consistency should be removed as well, along with second-order efficiency, ... and pretty much everything on this page. Assuming there exists a true model is completely standard within the context of parameter estimation. If you want to argue about model selection or model uncertainty, then go somewhere else for that. (204.111.241.221 (talk) 21:37, 28 July 2018 (UTC))

revising the lede 2019-04-10
On 2019-04-10 the lead paragraph of this article began as follows:

 * In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model such that having obtained a given set of observed data was most probable. Specifically, this is done by expressing the joint probability density function of the observed data $$\mathbf{y}$$ as a likelihood function $$\mathcal{L}(\theta;\mathbf{y})$$ of the parameter vector $$\theta$$ and finding its maximum value over a parameter space $$\Theta$$.

I have problems with this:
 * The first "sentence" is difficult to parse.
 * The likelihood is not always the probability density, e.g., with Censoring (statistics).

I therefore propose the following:

 * In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model so the observed data is most probable. Specifically, this is done by finding the value of the parameter (or parameter vector) $$\theta$$ that maximizes the likelihood function $$\mathcal{L}(\theta;\mathbf{y})$$, which is the joint probability (or probability density) of the observed data $$\mathbf{y}$$, over a parameter space $$\Theta$$.

I've made this change, hoping that it's clearer, more concise and more compelling, while still being accurate. DavidMCEddy (talk) 01:49, 11 April 2019 (UTC)

 * I like your changes, except for "...parameter (or parameter vector)...". I would prefer if we could avoid using parentheses in the lede. --bender235 (talk) 01:55, 11 April 2019 (UTC)

 * OK: Then how about "parameter" without the "(or parameter vector)"?
 * Most people who have studied this would understand that a "single value" may not be a single number but a point in a vector space.
 * Change this if you wish. I think it's clearer for people who have not studied this extensively with the paranthetical "(or parameter vector)" than without, but I can live with this deletion.
 * If you really insist on "parameter vector", I could live with that also, recognizing that the real line is a vector space of one dimension. DavidMCEddy (talk) 02:51, 11 April 2019 (UTC)

injective is not enough
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)

 * Correct. No objection from my side. --bender235 (talk) 03:54, 11 April 2019 (UTC)

Possibly unnecessary algebraic complexity
The section on Continuous distribution, continuous parameter space contains the following pair of formulae:

 * $$f(x_1,\ldots,x_n \mid \mu,\sigma^2) = \prod_{i=1}^{n} f( x_{i}\mid \mu, \sigma^2) = \left( \frac{1}{2\pi\sigma^2} \right)^{n/2} \exp\left( -\frac{ \sum_{i=1}^{n}(x_i-\mu)^2}{2\sigma^2}\right),$$

or more conveniently,

 * $$f(x_1,\ldots,x_n \mid \mu,\sigma^2) = \left( \frac{1}{2\pi\sigma^2} \right)^{n/2} \exp\left(-\frac{ \sum_{i=1}^{n}(x_i-\bar{x})^2+n(\bar{x}-\mu)^2}{2\sigma^2}\right),$$

As far as I can see, in working through the derivations that follow, the second formula (described as "convenient") doesn't actually add anything. Am I missing the illumination that this second formula is supposed to provide? Attic Salt (talk) 15:11, 2 July 2019 (UTC)

 * It's "convenient", because it isolates the MLEs for the mean, $$\hat{\mu} = \bar{x}$$, and standard deviation, $$\hat{\sigma} = \sum_{i=1}^{n}(x_i-\bar{x})^2 / n$$.
 * If you want to add this, I would support that. However, I'm not going to take the time to do it myself.  DavidMCEddy (talk) 12:49, 3 July 2019 (UTC)

 * Yes, I see that, but this doesn't affect the derivations of the MLE that follows. The MLE is algebraically easier to obtain without that expansion. Attic Salt (talk) 12:55, 3 July 2019 (UTC)
 * I went ahead an simplified the derivation. Attic Salt (talk) 17:10, 11 July 2019 (UTC)

Unclear conclusion on continuous parameter space
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)

 * I just changed the verbiage to make that clearer. Thanks, DavidMCEddy (talk) 07:32, 10 October 2019 (UTC)

 * Got it now, many thanks. — Preceding unsigned comment added by 146.66.57.203 (talk) 15:21, 10 October 2019 (UTC)

typo in asymptotic behavior?
In the section "consistency", look at the expression after the sentence "it can also be shown that the maximum likelihood estimator converges in distribution to a normal distribution". I think that the $$\sqrt{n}$$ should not be there, or at least it depends on the definition of the loglikelihood that enters the information matrix $$I$$. If $$I$$ is based on the log-likelihood of one datum, then we need the $$n$$. If $$I$$ is based on the log-likelihood of $$n$$ data, then the $$n$$ should not be explicit in the formula. Indeed, the german page does not report the $$n$$, and I think that it is reported here in the english one because whoever wrote it took the formula from the cited source without realizing that in the source the log-likelihood was defined for one datum only — Preceding unsigned comment added by 176.242.186.224 (talk) 28 May 2020 (UTC)

Efficiency subsections X_t subscript
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)

Adding Expectation-Maximization to Methods or Link
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)

title: Talk:Maximum likelihood estimation


text:
Talk:Jason Robertson (activist)

Requested move 16 May 2021

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: Moved  (t · c)  buidhe  22:15, 23 May 2021 (UTC)

Jason Robertson → Jason Robertson (activist) – no clear WP:PRIMARYTOPIC Joeykai (talk) 15:52, 16 May 2021 (UTC)
 * Support and DAB at base name.--Ortizesp (talk) 15:02, 17 May 2021 (UTC)
 * Strong support the hockey player has way more views (17,352) than this one (181) even the rugby union player gets more (219)[|Jason_Robertson_(ice_hockey)|Jason_Robertson_(rugby_union)].  Crouch, Swale  ( talk ) 08:53, 18 May 2021 (UTC)
 * Strong Support per Couch, Swale. -Drdisque (talk) 14:47, 18 May 2021 (UTC)!

title: Talk:Jason Robertson (activist)