User:IssaRice/Beta distribution - Revision history

IssaRice at 01:23, 5 February 2020

2020-02-05T01:23:05Z

← Older revision		Revision as of 01:23, 5 February 2020
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	Basically if we start out with a uniform prior over the bias of a coin, and see n heads and m tails, then what distribution should we have for the probability of heads? The answer is beta(n+1, m+1) (thus, beta(1,1) is equivalent to uniform(0,1)). This is basically the distribution version of [https://en.wikipedia.org/wiki/Rule_of_succession Laplace's rule of succession] (that rule only gives the expected value).		Basically if we start out with a uniform prior over the bias of a coin, and see n heads and m tails, then what distribution should we have for the probability of heads? The answer is beta(n+1, m+1) (thus, beta(1,1) is equivalent to uniform(0,1)). This is basically the distribution version of [https://en.wikipedia.org/wiki/Rule_of_succession Laplace's rule of succession] (that rule only gives the expected value of the bias of the coin).

	the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf		the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf

IssaRice at 23:49, 4 February 2020

2020-02-04T23:49:55Z

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	see also https://stats.stackexchange.com/questions/47771/what-is-the-intuition-behind-beta-distribution for a good example (using batting averages in baseball).		see also https://stats.stackexchange.com/questions/47771/what-is-the-intuition-behind-beta-distribution for a good example (using batting averages in baseball).

			an interesting property is that if we start out with a prior beta(a,b), then see n additional successes and m additional failures, our posterior becomes beta(a+n, b+m) as we would hope.

IssaRice at 23:47, 4 February 2020

2020-02-04T23:47:24Z

← Older revision		Revision as of 23:47, 4 February 2020
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	Basically if we start out with a uniform prior over the bias of a coin, and see n heads and m tails, then what distribution should we have for the probability of heads? The answer is beta(n+1, m+1). This is basically the distribution version of [https://en.wikipedia.org/wiki/Rule_of_succession Laplace's rule of succession] (that rule only gives the expected value).		Basically if we start out with a uniform prior over the bias of a coin, and see n heads and m tails, then what distribution should we have for the probability of heads? The answer is beta(n+1, m+1) (thus, beta(1,1) is equivalent to uniform(0,1)). This is basically the distribution version of [https://en.wikipedia.org/wiki/Rule_of_succession Laplace's rule of succession] (that rule only gives the expected value).

	the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf		the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf

IssaRice at 23:46, 4 February 2020

2020-02-04T23:46:51Z

← Older revision		Revision as of 23:46, 4 February 2020
Line 1:		Line 1:
	Basically if we start out with a uniform prior over the bias of a coin, and see n heads and m tails, then what distribution should we have for the probability of heads? This is basically the distribution version of [https://en.wikipedia.org/wiki/Rule_of_succession Laplace's rule of succession] (that rule only gives the expected value).		Basically if we start out with a uniform prior over the bias of a coin, and see n heads and m tails, then what distribution should we have for the probability of heads? The answer is beta(n+1, m+1). This is basically the distribution version of [https://en.wikipedia.org/wiki/Rule_of_succession Laplace's rule of succession] (that rule only gives the expected value).

	the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf		the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf

IssaRice at 23:46, 4 February 2020

2020-02-04T23:46:07Z

← Older revision		Revision as of 23:46, 4 February 2020
Line 1:		Line 1:
			Basically if we start out with a uniform prior over the bias of a coin, and see n heads and m tails, then what distribution should we have for the probability of heads? This is basically the distribution version of [https://en.wikipedia.org/wiki/Rule_of_succession Laplace's rule of succession] (that rule only gives the expected value).

	the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf		the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf

IssaRice: Created page with "the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf the one thing i don't reall..."

2020-02-04T23:43:28Z

Created page with "the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf the one thing i don't reall..."

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the derivation given here is simple enough to understand: https://web.stanford.edu/class/archive/cs/cs109/cs109.1176/lectureHandouts/15%20Beta.pdf

the one thing i don't really get is why we set <math>a=\alpha=n+1</math> and <math>b=\beta=m+1</math>. i.e. if we shift the parameters by 1, then when we look at beta(a,b), the a and b no longer track the number of successes and failures. Looking at https://stats.stackexchange.com/questions/262956/why-is-there-1-in-beta-distribution-density-function there is apparently some deep reasons why this parametrization is chosen, but i don't really understand the explanations given there.

see also https://stats.stackexchange.com/questions/47771/what-is-the-intuition-behind-beta-distribution for a good example (using batting averages in baseball).