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text: '[summary: "Likelihood", when speaking of Bayesian reasoning, denotes *the probability of an observation, supposing some hypothesis to be correct.*\n\nSuppose our piece of evidence $e$ is that "Mr. Boddy was shot." One of our suspects is Miss Scarlett, and we denote by $H_S$ the hypothesis that Miss Scarlett shot Mr. Boddy. Suppose that if Miss Scarlett *were* the killer, we'd have predicted in advance a 20% probability she would use a gun, and an 80% chance she'd use some other weapon.\n\nThen the *likelihood* from the evidence, to Miss Scarlett being the killer, is 0.20. Using [1rj conditional probability notation], $\\mathbb P(e \\mid H_S) = 0.20.$\n\nThis doesn't mean Miss Scarlett has a 20% chance of being the killer; it means that if she is the killer, our observation had a probability of 20%.\n\nRelative likelihoods are a key ingredient for [1ly Bayesian reasoning] and one of the quantities plugged into [1lz Bayes's Rule].]\n\nConsider a piece of evidence $e,$ such as "Mr. Boddy was shot." We might have a number of different hypotheses that explain this evidence, including $H_S$ = "Miss Scarlett killed him", $H_M$ = "Colonel Mustard killed him", and so on.\n\nEach of those hypotheses assigns a different probability to the evidence. For example, imagine that _if_ Miss Scarlett _were_ the killer, there's a 20% chance she would use a gun, and an 80% chance she'd use some other weapon. In this case, the "Miss Scarlett" hypothesis assigns a *likelihood* of 20% to $e.$\n\nWhen reasoning about different hypotheses using a [-probability_distribution probability distribution] $\\mathbb P$, the likelihood of evidence $e$ given hypothesis $H_i$ is often written using the [1rj conditional probability] $\\mathbb P(e \\mid H_i).$ When reporting likelihoods of many different hypotheses at once, it is common to use a [-likelihood_function,] sometimes written [51n $\\mathcal L_e(H_i)$].\n\n[1rq Relative likelihoods] measure the degree of support that a piece of evidence $e$ provides for different hypotheses. For example, let's say that if Colonel Mustard were the killer, there's a 40% chance he would use a gun. Then the absolute likelihoods of $H_S$ and $H_M$ are 20% and 40%, for _relative_ likelihoods of (1 : 2). This says that the evidence $e$ supports $H_M$ twice as much as it supports $H_S,$ and that the amount of support would have been the same if the absolute likelihoods were 2% and 4% instead.\n\nAccording to [1lz Bayes' rule], relative likelihoods are the appropriate tool for measuring the [22x strength of a given piece evidence]. Relative likelihoods are one of two key constituents of belief in [bayesian_reasoning Bayesian reasoning], the other being [1rm prior probabilities].\n\nWhile absolute likelihoods aren't necessary when updating beliefs by Bayes' rule, they are useful when checking for [227 confusion]. For example, say you have a coin and only two hypotheses about how it works: $H_{0.3}$ = "the coin is random and comes up heads 30% of the time", and $H_{0.9}$ = "the coin is random and comes up heads 90% of the time." Now let's say you toss the coin 100 times, and observe the data HTHTHTHTHTHTHTHT... (alternating heads and tails). The _relative_ likelihoods strongly favor $H_{0.3},$ because it was less wrong. However, the _absolute_ likelihood of $H_{0.3}$ will be much lower than expected, and this deficit is a hint that $H_{0.3}$ isn't right. (For more on this idea, see [227].)',
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Summary: '"Likelihood", when speaking of Bayesian reasoning, denotes *the probability of an observation, supposing some hypothesis to be correct.*\n\nSuppose our piece of evidence $e$ is that "Mr. Boddy was shot." One of our suspects is Miss Scarlett, and we denote by $H_S$ the hypothesis that Miss Scarlett shot Mr. Boddy. Suppose that if Miss Scarlett *were* the killer, we'd have predicted in advance a 20% probability she would use a gun, and an 80% chance she'd use some other weapon.\n\nThen the *likelihood* from the evidence, to Miss Scarlett being the killer, is 0.20. Using [1rj conditional probability notation], $\\mathbb P(e \\mid H_S) = 0.20.$\n\nThis doesn't mean Miss Scarlett has a 20% chance of being the killer; it means that if she is the killer, our observation had a probability of 20%.\n\nRelative likelihoods are a key ingredient for [1ly Bayesian reasoning] and one of the quantities plugged into [1lz Bayes's Rule].'
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