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Maybe it's just my liberal arts education but discussions like this and comments like "hypothesis H1’ ... that yields a high probability for the evidence P(E|H1

by pc86 4y ago

Maybe it's just my liberal arts education but discussions like this and comments like "hypothesis H1’ ... that yields a high probability for the evidence P(E|H1’)" or "evidence strongly supports H1/H1’ vis-à-vis H0" mean absolutely nothing to me without a concrete, real-world example of what H1, H1', and H0 are.

time_to_smile 4y ago
It's not the presence of a liberal arts education, it's the absence of a mathematical one (and technically mathematics is one of the 'liberal arts'). I have a liberal arts background, but also a statistics background and reading this is no problem for me. Reading and thinking mathematically is a literacy the same way reading sheet music is a type of literacy. Just like there are degrees of literacy for reading and writing same with these disciplines. Some people can't read sheet music at all. I can name the notes when reading sheet music, and know where they are on a piano, but, unlike many musician friends, I can't "hear the music" when I read. Similarly it takes time and practice to "read mathematics". To someone with no practice the notation appears as scribbles. A more experienced mathematician will pause and work through what's happening. At a certain point, you start to experience the same "hearing the music" with mathematical writing.
defen 4y ago
> Formulate a vague hypothesis H1 so broad that it is not extremely unlikely (thus has non-negligible prior, P(H1) > 0), then in the updating step sneakily introduce a much more specific hypothesis H1’ that is far less likely a priori, but that yields a high probability for the evidence P(E|H1’). H0 is "the lottery is fair" (null hypothesis). H1 is "somebody rigged the lottery". It's not outrageously unlikely, it's happened many times before and it's something people can easily understand, and it's not specific about the details. So maybe someone would say "Without knowing more, I think there's a 1% chance that happened". H1’ is "The lottery is rigged by a cult that worships the multiples of {9}" - This is the sneaky step. If it weren't such an obviously outrageous example, you might not notice that this specific hypothesis is very unlikely. However, H1’ perfectly predicts the evidence that we did observe! So basically, if you plug in the numbers, without adjusting for the fact that H1’ does not have a 1% chance of being true, you turn "I think there's a 1% chance someone rigged the lottery" into "The lottery was almost certainly rigged - 99.9997% chance!"
williamcotton 4y ago
He defines those as concrete real-world examples! Event {E}: on October 1, 2022, the PCSO Grand Lotto in the Philippines, which draws six numbers from {1} to {55} at random, managed to draw the numbers {9, 18, 27, 36, 45, 54} (though the balls were actually drawn in the order {9, 45,36, 27, 18, 54}). In other words, they drew exactly six multiples of nine from {1} to {55}. In addition, a total of {433} tickets were bought with this winning combination, whose owners then had to split the {236} million peso jackpot (about {4} million USD) among themselves. Null hypothesis {H_0}: The lottery is run in a completely fair and random fashion. Alternative hypothesis {H_1}: The lottery is rigged by some corrupt officials for their personal gain. Alternative hypothesis {H'_1}: The lottery is rigged by a cult that worships the multiples of {9}, and views October 1 as their holiest day. On this day, they will manipulate the lottery to only select those balls that are multiples of {9}. So then with substitution: Remark 1: The contrast between alternative hypothesis that "the lottery is rigged by some corrupt officials for their personal gain" and alternative hypothesis that "the lottery is rigged by a cult that worships the multiples of {9}, and views October 1 as their holiest day. On this day, they will manipulate the lottery to only select those balls that are multiples of {9}" illustrates a common demagogical rhetorical technique when an advocate is trying to convince an audience of an alternative hypothesis, namely to use suggestive language (“`I’m just asking questions here”) rather than precise statements in order to leave the alternative hypothesis deliberately vague. In particular, the advocate may take advantage of the freedom to use a broad formulation of the hypothesis (such as "the lottery is rigged by some corrupt officials for their personal gain") in order to maximize the audience’s prior odds of the hypothesis, simultaneously with a very specific formulation of the hypothesis (such as "the lottery is rigged by a cult that worships the multiples of {9}") in order to maximize the probability of the actual event of "they drew exactly six multiples of nine from {1} to {55}" occurring under this hypothesis. (A related technique is to be deliberately vague about the hypothesized competency of some suspicious actor, so that this actor could be portrayed as being extraordinarily competent when convenient to do so, while simultaneously being portrayed as extraordinarily incompetent when that instead is the more useful hypothesis.) This can lead to wildly inaccurate Bayesian updates of this vague alternative hypothesis, and so precise formulation of such hypothesis is important if one is to approach a topic from anything remotely resembling a scientific approach. [EDIT: as pointed out to me by a reader, this technique is a Bayesian analogue of the motte and bailey fallacy.]

by pc86 4y ago

time_to_smile 4y ago

It's not the presence of a liberal arts education, it's the absence of a mathematical one (and technically mathematics is one of the 'liberal arts'). I have a liberal arts background, but also a statistics background and reading this is no problem for me. Reading and thinking mathematically is a literacy the same way reading sheet music is a type of literacy. Just like there are degrees of literacy for reading and writing same with these disciplines. Some people can't read sheet music at all. I can name the notes when reading sheet music, and know where they are on a piano, but, unlike many musician friends, I can't "hear the music" when I read. Similarly it takes time and practice to "read mathematics". To someone with no practice the notation appears as scribbles. A more experienced mathematician will pause and work through what's happening. At a certain point, you start to experience the same "hearing the music" with mathematical writing.

defen 4y ago

> Formulate a vague hypothesis H1 so broad that it is not extremely unlikely (thus has non-negligible prior, P(H1) > 0), then in the updating step sneakily introduce a much more specific hypothesis H1’ that is far less likely a priori, but that yields a high probability for the evidence P(E|H1’). H0 is "the lottery is fair" (null hypothesis). H1 is "somebody rigged the lottery". It's not outrageously unlikely, it's happened many times before and it's something people can easily understand, and it's not specific about the details. So maybe someone would say "Without knowing more, I think there's a 1% chance that happened". H1’ is "The lottery is rigged by a cult that worships the multiples of {9}" - This is the sneaky step. If it weren't such an obviously outrageous example, you might not notice that this specific hypothesis is very unlikely. However, H1’ perfectly predicts the evidence that we did observe! So basically, if you plug in the numbers, without adjusting for the fact that H1’ does not have a 1% chance of being true, you turn "I think there's a 1% chance someone rigged the lottery" into "The lottery was almost certainly rigged - 99.9997% chance!"

williamcotton 4y ago

He defines those as concrete real-world examples! Event {E}: on October 1, 2022, the PCSO Grand Lotto in the Philippines, which draws six numbers from {1} to {55} at random, managed to draw the numbers {9, 18, 27, 36, 45, 54} (though the balls were actually drawn in the order {9, 45,36, 27, 18, 54}). In other words, they drew exactly six multiples of nine from {1} to {55}. In addition, a total of {433} tickets were bought with this winning combination, whose owners then had to split the {236} million peso jackpot (about {4} million USD) among themselves. Null hypothesis {H_0}: The lottery is run in a completely fair and random fashion. Alternative hypothesis {H_1}: The lottery is rigged by some corrupt officials for their personal gain. Alternative hypothesis {H'_1}: The lottery is rigged by a cult that worships the multiples of {9}, and views October 1 as their holiest day. On this day, they will manipulate the lottery to only select those balls that are multiples of {9}. So then with substitution: Remark 1: The contrast between alternative hypothesis that "the lottery is rigged by some corrupt officials for their personal gain" and alternative hypothesis that "the lottery is rigged by a cult that worships the multiples of {9}, and views October 1 as their holiest day. On this day, they will manipulate the lottery to only select those balls that are multiples of {9}" illustrates a common demagogical rhetorical technique when an advocate is trying to convince an audience of an alternative hypothesis, namely to use suggestive language (“`I’m just asking questions here”) rather than precise statements in order to leave the alternative hypothesis deliberately vague. In particular, the advocate may take advantage of the freedom to use a broad formulation of the hypothesis (such as "the lottery is rigged by some corrupt officials for their personal gain") in order to maximize the audience’s prior odds of the hypothesis, simultaneously with a very specific formulation of the hypothesis (such as "the lottery is rigged by a cult that worships the multiples of {9}") in order to maximize the probability of the actual event of "they drew exactly six multiples of nine from {1} to {55}" occurring under this hypothesis. (A related technique is to be deliberately vague about the hypothesized competency of some suspicious actor, so that this actor could be portrayed as being extraordinarily competent when convenient to do so, while simultaneously being portrayed as extraordinarily incompetent when that instead is the more useful hypothesis.) This can lead to wildly inaccurate Bayesian updates of this vague alternative hypothesis, and so precise formulation of such hypothesis is important if one is to approach a topic from anything remotely resembling a scientific approach. [EDIT: as pointed out to me by a reader, this technique is a Bayesian analogue of the motte and bailey fallacy.]