Probability theory - Revision history

Benrbray: /* Probability Distributions */

2021-07-12T17:25:57Z

Probability Distributions

← Older revision		Revision as of 18:25, 12 July 2021
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	* When $X$ and $Y$ are random variables, $ P(X\|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y\|X) \) is the conditional probability distribution of $Y$ given $X$.		* When $X$ and $Y$ are random variables, $ P(X\|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y\|X) \) is the conditional probability distribution of $Y$ given $X$.
	* $P(A,B,C\|X,Y,Z)$ may or may not be equivalent to $P(B,A,C\|Y,Z,X)$, depending on who you ask.		* $P(A,B,C\|X,Y,Z)$ may or may not be equivalent to $P(B,A,C\|Y,Z,X)$, depending on who you ask.
	* Sometimes we use $ P(X_1, X_2, X_3 $ to mean the probability of observing a sequence events $X_1$, $X_2$, $X_3$ in that particular order.		* Sometimes we use $ P(X_1, X_2, X_3) $ to mean the probability of observing a sequence events $X_1$, $X_2$, and $X_3$ in that particular order, in contrast to $ P(X_2, X_1, X_3) $.
	* In Bayesian statistics, $ P(X\|Y;\theta) $ or $ P_\theta(X\|Y) $ means "the probability of X, given Y, parameterized by $\theta$".		* In Bayesian statistics, $ P(X\|Y;\theta) $ or $ P_\theta(X\|Y) $ means "the probability of X, given Y, parameterized by $\theta$".

Benrbray: /* Probability Distributions */

2021-07-12T17:24:51Z

Probability Distributions

← Older revision		Revision as of 18:24, 12 July 2021
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	* When $X$ and $Y$ are random variables, $ P(X\|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y\|X) \) is the conditional probability distribution of $Y$ given $X$.		* When $X$ and $Y$ are random variables, $ P(X\|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y\|X) \) is the conditional probability distribution of $Y$ given $X$.
	* $P(A,B,C\|X,Y,Z)$ may or may not be equivalent to $P(B,A,C\|Y,Z,X)$, depending on who you ask.		* $P(A,B,C\|X,Y,Z)$ may or may not be equivalent to $P(B,A,C\|Y,Z,X)$, depending on who you ask.
			* Sometimes we use $ P(X_1, X_2, X_3 $ to mean the probability of observing a sequence events $X_1$, $X_2$, $X_3$ in that particular order.
	* In Bayesian statistics, $ P(X\|Y;\theta) $ or $ P_\theta(X\|Y) $ means "the probability of X, given Y, parameterized by $\theta$".		* In Bayesian statistics, $ P(X\|Y;\theta) $ or $ P_\theta(X\|Y) $ means "the probability of X, given Y, parameterized by $\theta$".

Benrbray: tag with unpleasantness and ambiguities categories

2021-07-12T17:06:29Z

tag with unpleasantness and ambiguities categories

← Older revision		Revision as of 18:06, 12 July 2021
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			[[Category:Unpleasantness]]
			[[Category:Ambiguities]]

	Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$), which appear to accept almost any expression as an argument!		Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$), which appear to accept almost any expression as an argument!

Benrbray: Created page with "Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or \(\math..."

2021-07-12T17:05:35Z

Created page with "Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or \(\math..."

New page

Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$), which appear to accept almost any expression as an argument!

Unlike other areas of mathematics, it is the names of variables that matter, rather than the order in which they appear.

==Random Variables==

Variable names matter!

==Probability Distributions==

The magic function $ P $ (sometimes written $\mathbb{P}$ or $\mathrm{Pr}$) can accept almost any expression as an argument! Consider:

* When $X$ is a random variable, $ P(X) $ is its probability distribution.
* When $X$ is an event, $ P(X) $ is the probability of that event with respect to some implicit underlying distribution
* When $X$ and $Y$ are random variables, $ P(X|Y) $ is the conditional probability distribution of $X$ given $Y$, while ( P(Y|X) \) is the conditional probability distribution of $Y$ given $X$.
* $P(A,B,C|X,Y,Z)$ may or may not be equivalent to $P(B,A,C|Y,Z,X)$, depending on who you ask.
* In Bayesian statistics, $ P(X|Y;\theta) $ or $ P_\theta(X|Y) $ means "the probability of X, given Y, parameterized by $\theta$".

==Expected Value==

See Carpenter 2020, [https://statmodeling.stat.columbia.edu/2020/02/05/abuse-of-expectation-notation/ "Abuse of Expectation Notation"]

← Older revision		Revision as of 18:25, 12 July 2021
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	* When \(X\) and \(Y\) are random variables, \( P(X\|Y) \) is the conditional probability distribution of \(X\) given \(Y\), while ( P(Y\|X) \) is the conditional probability distribution of \(Y\) given \(X\).		* When \(X\) and \(Y\) are random variables, \( P(X\|Y) \) is the conditional probability distribution of \(X\) given \(Y\), while ( P(Y\|X) \) is the conditional probability distribution of \(Y\) given \(X\).
	* \(P(A,B,C\|X,Y,Z)\) may or may not be equivalent to \(P(B,A,C\|Y,Z,X)\), depending on who you ask.		* \(P(A,B,C\|X,Y,Z)\) may or may not be equivalent to \(P(B,A,C\|Y,Z,X)\), depending on who you ask.
	* Sometimes we use \( P(X_1, X_2, X_3 \) to mean the probability of observing a sequence events $X_1$, $X_2$, $X_3$ in that particular order.		* Sometimes we use \( P(X_1, X_2, X_3) \) to mean the probability of observing a sequence events \(X_1\), \(X_2\), and \(X_3\) in that particular order, in contrast to \( P(X_2, X_1, X_3) \).
	* In Bayesian statistics, \( P(X\|Y;\theta) \) or \( P_\theta(X\|Y) \) means "the probability of X, given Y, parameterized by \(\theta\)".		* In Bayesian statistics, \( P(X\|Y;\theta) \) or \( P_\theta(X\|Y) \) means "the probability of X, given Y, parameterized by \(\theta\)".

← Older revision		Revision as of 18:24, 12 July 2021
Line 18:		Line 18:
	* When \(X\) and \(Y\) are random variables, \( P(X\|Y) \) is the conditional probability distribution of \(X\) given \(Y\), while ( P(Y\|X) \) is the conditional probability distribution of \(Y\) given \(X\).		* When \(X\) and \(Y\) are random variables, \( P(X\|Y) \) is the conditional probability distribution of \(X\) given \(Y\), while ( P(Y\|X) \) is the conditional probability distribution of \(Y\) given \(X\).
	* \(P(A,B,C\|X,Y,Z)\) may or may not be equivalent to \(P(B,A,C\|Y,Z,X)\), depending on who you ask.		* \(P(A,B,C\|X,Y,Z)\) may or may not be equivalent to \(P(B,A,C\|Y,Z,X)\), depending on who you ask.
			* Sometimes we use \( P(X_1, X_2, X_3 \) to mean the probability of observing a sequence events $X_1$, $X_2$, $X_3$ in that particular order.
	* In Bayesian statistics, \( P(X\|Y;\theta) \) or \( P_\theta(X\|Y) \) means "the probability of X, given Y, parameterized by \(\theta\)".		* In Bayesian statistics, \( P(X\|Y;\theta) \) or \( P_\theta(X\|Y) \) means "the probability of X, given Y, parameterized by \(\theta\)".

Probability theory - Revision history

Benrbray: /* Probability Distributions */

Benrbray: /* Probability Distributions */

Benrbray: tag with unpleasantness and ambiguities categories

Benrbray: Created page with "Probability theory makes a number of pragmatic abuses of notation. There are a lot of symbols, such as the magic function \( P \) (sometimes written \(\mathbb{P}\) or \(\math..."