I can follow the calculation of diseasitis - that's standard math that I learned in school. What I have a problem to follow is how you get to the "absolute propability" of 3 / (3 + 4). I think the "3+4" are the 3 parts red water and 4 parts blue water, but where does the other 3 come from?
Wait ... is that again the 3 parts red? So 3 Parts of 7 parts in all?
Hm ... I think I have solved my question ;-)
The inverse of multiplication is division. To the mathematically steadfast this is completely obvious but I wager this is exactly the point where most non-mathematically inclined people will become confused and give up or will simply read on without absorbing the whole message. Maybe make this mathematical step more clearly?
"Likely" refers to probability, and yet the point of this essay is to explain probability. Therefore, the use of "likely" is, in a sense, circular reasoning. After all, what does "likely" mean? It's not explained here. It suggests an outcome frequency of sorts and so this statement and others like it is an attempt to arrive at an outcome frequency (equivalent to the proportions of red and blue water that make it down through) by referring to another outcome frequency; thus the circularity.
Better to stick with the proportions themselves by explaining that, however much red water makes it down through, there will be three times as much of it as there is blue water that makes it down through. Say that some fraction, f, of the blue water molecules makes it down through; then for every 100 molecules of water, f x 80 blue molecules make it down through and 3f x 20 red molecules make it down through, making for proportions of 60f red to 80f blue. Scaling down those proportions by dividing both by f, we get 60:80, which can be further scaled down to 3:4.
Note that the factor of 3, i.e. the "likelihood ratio" (by which the initial proportions of 20:80 are multiplied) is explicit in the previous paragraph. (It's in the statement, "3f x 20 red molecules make it down through".) Putting it another way, the previous paragraph makes it clear that multiplying by 3 will give the same final proportions ("posterior odds") as will, in taking a frequency approach, multiplying 20 by 0.9 and 80 by 0.3, since the latter proportions can be scaled by dividing each by 0.3: (0.9/0.3 x 20):(0.3/0.3 x 80) = (3 x 20):1 x 80 = 3:4.
I don't understand how the waterfall concept helps illustrate the "odds form": the amount of each type of water reaching the pool is still expressed as a probability rather than jointly being expressed as the likelihood ratio. The fact that these likelihoods don't matter -- only their ratio -- was the the critical conceptual blockage for me.
Ah, insightful! I hadn't seen forms of Bayes' Rule other than the probability form before today, and this is very helpful (well, perhaps I had seen them but it hasn't "hit me" until now).
I like that this is emphasized. To further emphasize, I think a formula should be added as a block level element underneath.
I think it'd be clearer to have two different headers. The way it's set up right now, I didn't initially see that this one article is talking about two different (but related) approaches.
Wording seem less clear then it could be here, what does it mean to say it “produces better problem-solving.” What about something like:
. . . that participants arrive at the correct answer more often when the problems is presented in terms of frequencies, 20 patients, rather then probabilities, 20% of patients.”
I liked this explanation. In particular, the obvious hard way vs sneaky easy way contrast caught my attention.
Perhaps that could even serve as an introductory motivating sentence? (e.g. "In this post we'll explore an obvious hard way and also a sneaky easy way to do calculations using Bayes's Rule.")
(At a higher level, do we want readers to be able to flag portions of a page with a variety of labels, such as, unclear, appears to be factually incorrect, contradictory, etc?)
Agree. Could be replaced with “similar” or “similar in form”. The sentence could also be change to say something like “This problem is just like . . .”
Title says "Relative Odds" and then the article uses "relative likelihood" to describe the same concept. That's confusing.
I can follow the calculation of diseasitis - that's standard math that I learned in school. What I have a problem to follow is how you get to the "absolute propability" of 3 / (3 + 4). I think the "3+4" are the 3 parts red water and 4 parts blue water, but where does the other 3 come from? Wait ... is that again the 3 parts red? So 3 Parts of 7 parts in all? Hm ... I think I have solved my question ;-)
The inverse of multiplication is division. To the mathematically steadfast this is completely obvious but I wager this is exactly the point where most non-mathematically inclined people will become confused and give up or will simply read on without absorbing the whole message. Maybe make this mathematical step more clearly?
I'm failing to grasp how the probability conversion works and so some further explanation may be needed
has to be 18:42. 42 is the sum of 18 and 24 ( these are the proportions of water).
"Likely" refers to probability, and yet the point of this essay is to explain probability. Therefore, the use of "likely" is, in a sense, circular reasoning. After all, what does "likely" mean? It's not explained here. It suggests an outcome frequency of sorts and so this statement and others like it is an attempt to arrive at an outcome frequency (equivalent to the proportions of red and blue water that make it down through) by referring to another outcome frequency; thus the circularity.
Better to stick with the proportions themselves by explaining that, however much red water makes it down through, there will be three times as much of it as there is blue water that makes it down through. Say that some fraction, f, of the blue water molecules makes it down through; then for every 100 molecules of water, f x 80 blue molecules make it down through and 3f x 20 red molecules make it down through, making for proportions of 60f red to 80f blue. Scaling down those proportions by dividing both by f, we get 60:80, which can be further scaled down to 3:4.
Note that the factor of 3, i.e. the "likelihood ratio" (by which the initial proportions of 20:80 are multiplied) is explicit in the previous paragraph. (It's in the statement, "3f x 20 red molecules make it down through".) Putting it another way, the previous paragraph makes it clear that multiplying by 3 will give the same final proportions ("posterior odds") as will, in taking a frequency approach, multiplying 20 by 0.9 and 80 by 0.3, since the latter proportions can be scaled by dividing each by 0.3: (0.9/0.3 x 20):(0.3/0.3 x 80) = (3 x 20):1 x 80 = 3:4.
I don't understand how the waterfall concept helps illustrate the "odds form": the amount of each type of water reaching the pool is still expressed as a probability rather than jointly being expressed as the likelihood ratio. The fact that these likelihoods don't matter -- only their ratio -- was the the critical conceptual blockage for me.
How did it convert to 3/7th is unclear.
Answer of interest.
Question of interest.
90% of the red water makes it to the shared pool. 30% of the blue water makes it to the shared pool.
Ah, insightful! I hadn't seen forms of Bayes' Rule other than the probability form before today, and this is very helpful (well, perhaps I had seen them but it hasn't "hit me" until now).
I like that this is emphasized. To further emphasize, I think a formula should be added as a block level element underneath.
I think it'd be clearer to have two different headers. The way it's set up right now, I didn't initially see that this one article is talking about two different (but related) approaches.
It should be clarified that “the bottom” here refers to the pool.
This sentence should be written above the previous paragraph: 18/24 is 3/4, not 3/7.
The banner reading "Your proposal has been submitted" lingers into subsequent editing processes, causing confusion.
This page is screwed.
A panel within the first diagram reads:
Impossible to see where this comes from. Revise to read:
Wording seem less clear then it could be here, what does it mean to say it “produces better problem-solving.” What about something like:
I liked this explanation. In particular, the obvious hard way vs sneaky easy way contrast caught my attention.
Perhaps that could even serve as an introductory motivating sentence? (e.g. "In this post we'll explore an obvious hard way and also a sneaky easy way to do calculations using Bayes's Rule.")
This text is out of sync with the graphic -- the pic actually shows black tongue depressors.
Do we want citation needed norms on Arbital?
(At a higher level, do we want readers to be able to flag portions of a page with a variety of labels, such as, unclear, appears to be factually incorrect, contradictory, etc?)
I think isomorphic is too advanced vocabulary to be assumed for Math 1. Would this be a good opportunity to use a popover with the definition?
Agree. Could be replaced with “similar” or “similar in form”. The sentence could also be change to say something like “This problem is just like . . .”