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clickbait: 'Formal definitions, alternate representations, and uses of odds and odds ratios (like a 1 : 2 chance of drawing a red ball vs. green ball from a barrel).',
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text: 'Odds express relative belief: we write "the odds for X versus Y are $17 : 2$" when we think that proposition X is 17/2 = 8.5 times as likely as proposition Y.%%note: The colon denotes that we are forming a set of odds. It does not denote division, as it might in French or German.%%\n\nOdds don't say anything about how likely X or Y is in absolute terms. X might be "it will hail tomorrow" and Y might be "there will be a hurricane tomorrow." In that case, it might be the case that the odds for X versus Y are $17 : 2$, despite the fact that both X and Y are very unlikely. [1lz Bayes' rule] is an example of an important operation that makes use of relative belief.\n\nOdds can be expressed between many different propositions at once. For example, let Z be the proposition "It will rain tomorrow," the odds for X vs Y vs Z might be $(17 : 2 : 100).$ When odds are expressed between only two propositions, they can be expressed using a single [-ratio]. For example, above, the odds ratio between X and Y is 17/2, the odds ratio between X and Z is 17/100, and the odds ratio between Y and Z is 2/100 = 1/50. This asserts that X is 8.5x more likely than Y, and that Z is 50x more likely than Y. When someone says "the odds ratio of sick to healthy is 2/3", they mean that the odds of sickness vs health are $2 : 3.$\n\n[toc:]\n\n# Formal definition\n\nGiven $n$ propositions $X_1, X_2, \\ldots X_n,$ a set of odds between the propositions is a list $(x_1, x_2, \\ldots, x_n)$ of non-negative [4bc real] numbers. Each $x_i$ in the set of odds is called a "term." Two sets of odds $(x_1, x_2, \\ldots, x_n)$ and $(y_1, y_2, \\ldots, y_n)$ are called "equivalent" if there is an $\\alpha > 0$ such that $ \\alpha x_i = y_i$ for all $i$ from 1 to $n.$\n\nWhen we write a set of odds using colons, like $(x_1 : x_2 : \\ldots : x_n),$ it is understood that the '=' sign denotes this equivalence. Thus, $(3 : 6) = (9 : 18).$\n\nA set of odds with only two terms can also be written as a fraction $\\frac{x}{y},$ where it is understood that $\\frac{x}{y}$ denotes the odds $(x : y).$ These fractions are often called "odds ratios."\n\n# Example\n\nSuppose that in some forest, 40% of the trees are rotten and 60% of the trees are healthy. There are then 2 rotten trees for every 3 healthy trees, so we say that the relative *odds* of rotten trees to healthy trees is 2 : 3. If we selected a tree at random from this forest, the *probability* of getting a rotten tree would be 2/5, but the *odds* would be 2 : 3 for rotten vs. healthy trees.\n\n\n\n# Conversion between odds and probabilities\n\nConsider three propositions, $X,$ $Y,$ and $Z,$ with odds of $(3 : 2 : 6).$ These odds assert that $X$ is half as probable as $Z.$\n\nWhen the set of propositions are [1rd mutually exclusive and exhaustive], we can convert a set of odds into a set of [1rf probabilities] by [1rk normalizing] the terms so that they sum to 1. This can be done by summing all the components of the ratio, then dividing each component by the sum:\n\n$$(x_1 : x_2 : \\dots : x_n) = \\left(\\frac{x_1}{\\sum_{i=1}^n x_i} : \\frac{x_2}{\\sum_{i=1}^n x_i} : \\dots : \\frac{x_n}{\\sum_{i=1}^n x_i}\\right)$$\n\nFor example, to obtain probabilities from the odds ratio 1/3, w write:\n\n$$(1 : 3) = \\left(\\frac{1}{1+3}:\\frac{3}{1+3}\\right) = ( 0.25 : 0.75 )$$\n\nwhich corresponds to the probabilities of 25% and 75%.\n\nTo go the other direction, recall that $\\mathbb P(X) + \\mathbb P(\\neg X) = 1,$ where $\\neg X$ is the negation of $X.$ So the odds for $X$ vs $\\neg X$ are $\\mathbb P(X) : \\mathbb P(\\neg X)$ $=$ $\\mathbb P(X) : 1 - \\mathbb P(X).$ If Alexander Hamilton has a 20% probability of winning the election, his odds for winning vs losing are $(0.2 : 1 - 0.2)$ $=$ $(0.2 : 0.8)$ $=$ $(1 : 4).$\n\n# Bayes' rule\n\nOdds are exceptionally convenient when reasoning using [1lz Bayes' rule], since the [1rm prior] odds can be term-by-term multiplied by a set of [relative_likelihoods relative likelihoods] to yield the [1rp posterior] odds. (The posterior odds in turn can be normalized to yield posterior probabilities, but if performing repeated updates, it's [1zg more convenient] to multiply by all the likelihood ratios under consideration before normalizing at the end.)\n\n$$\\dfrac{\\mathbb{P}(H_i\\mid e_0)}{\\mathbb{P}(H_j\\mid e_0)} = \\dfrac{\\mathbb{P}(e_0\\mid H_i)}{\\mathbb{P}(e_0\\mid H_j)} \\cdot \\dfrac{\\mathbb{P}(H_i)}{\\mathbb{P}(H_j)}$$\n\nAs a more striking illustration, suppose we receive emails on three subjects: Business (60%), personal (30%), and spam (10%). Suppose that business, personal, and spam emails are 60%, 10%, and 90% likely respectively to contain the word "money"; and that they are respectively 20%, 80%, and 10% likely to contain the word "probability". Assume for the sake of discussion that a business email containing the word "money" [naive_bayes is thereby no more or less likely] to contain the word "probability", and similarly with personal and spam emails. Then if we see an email containing both the words "money" and "probability":\n\n$$(6 : 3 : 1) \\times (6 : 1 : 9) \\times (2 : 8 : 1) = (72 : 24 : 9) = (24 : 8: 3)$$\n\n...so the posterior odds are 24 : 8 : 3 favoring the email being a business email, or roughly 69% probability after [1rk normalizing].\n\n# Log odds\n\nThe odds $\\mathbb{P}(X) : \\mathbb{P}(\\neg X)$ can be viewed as a dimensionless scalar quantity $\\frac{\\mathbb{P}(X)}{\\mathbb{P}(\\neg X)}$ in the range $[0, +\\infty]$. If the odds of Alexander Hamilton becoming President are 0.75 to 0.25 in favor, we can also say that Andrew Jackson is 3 times as likely to become President as not. Or if the odds were 0.4 to 0.6, we could say that Alexander Hamilton was 2/3rds as likely to become President as not.\n\nThe **log odds** are the logarithm of this dimensionless positive quantity, $\\log\\left(\\frac{\\mathbb{P}(X)}{\\mathbb{P}(\\neg X)}\\right),$ e.g., $\\log_2(1:4) = \\log_2(0.25) = -2.$ Log odds fall in the [range_notation range] $[-\\infty, +\\infty]$ and are finite for probabilities inside the range $(0, 1).$\n\nWhen using a log odds form of [1lz Bayes' rule], the posterior log odds are equal to the prior log odds plus the log likelihood. This means that the change in log odds can be identified with [ the strength of the evidence]. If the probability goes from 1/3 to 4/5, our odds have gone from 1:2 to 4:1 and the log odds have shifted from -1 bits to +2 bits. So we must have seen evidence with a strength of +3 bits (a likelihood ratio of 8:1).\n\nThe convenience of this representation is what Han Solo refers to in *Star Wars* when he shouts: "Never tell me the odds!", implying that he would much prefer to be told the logarithm of the odds ratio.\n\n## Direct representation of infinite certainty\n\nIn the log odds representation, the probabilities $0$ and $1$ are represented as $-\\infty$ and $+\\infty$ respectively.\n\nThis exposes the specialness of the classical probabilities $0$ and $1,$ and the ways in which these "infinite certainties" sometimes behave qualitatively differently from all finite credences. If we don't start by being absolutely certain of a proposition, it will require infinitely strong evidence to shift our belief all the way out to infinity. If we do start out absolutely certain of a proposition, no amount of ordinary evidence no matter how great can ever shift us away from infinity.\n\nThis reasoning is part of the justification of [Cromwell's rule](https://en.wikipedia.org/wiki/Cromwell%27s_rule) which states that probabilities of exactly $0$ or $1$ should be avoided except for logical truths and falsities (and maybe [not even then](http://lesswrong.com/lw/mo/infinite_certainty/)). It also demonstrates how log odds are a good fit for measuring *strength of belief and evidence,* even if classical probabilities are a better representation of *degrees of caring* and betting odds.\n\n%%%comment: We are checking to see if users will click this button, even though we don't have the content for it yet.%%%\n%%hidden(Check my understanding):\nComing soon!\n%%',
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