Actually, both pairs differ by the same factor of about 1.85256. (Same for base 100k.) (In the first case, the evidence is around 9, rather than 8, which likely accounts for the difference you got.) I'm actually a little surprised this is true - log-form is used in order to deal with things additively, rather than multiplicatively, so I wouldn't have guessed without calculation that changing the base would preserve proportions. (And I'd still want to do some math to confirm this in general.)
Note: easier to grasp than natural logarithms. Both log base e and log base 10 behave in basically the same way, and have the same arguable pitfall. However, note that this behavior is, I think, on purpose, and intended to convey the idea that 10x more probable is 10x more probable, no matter where you are on the scale of improbability. (You just have to remember where you are on the scale before making your final judgements.) Consider the opposite problem raised by the 0-1 scale, as discussed earlier in the page: 0.00001% isn't that far from 1%, same as between 10.00001% and 11%; but intuitively, clearly 0.00001% is much more different from 1% than 10.00001% is from 11%. This is the non-intuitiveness that the log-form is intended to fix, I think.
Easier to grasp perhaps, but dangerously misleading. Increasing the likelihood of an event from 10^-100 to 10^-99 is very different and much less significant than increasing it from 10^-2 (1%) to 10^-1 (10%). I hope this is covered later in this guide.
But that really gives a different magnitude to the evidence. Why not be consistent with the log base?
For example, if we were to use log base 2, the prior would be ~16.6 magnitudes strong and the evidence ~8. This means that the evidence would alter the prior by (slightly) less than half the order of magnitudes, where's in the case of log base 10 the alteration is (slightly) more than half the order of magnitudes (5 vs 2.7).
Also, imagine the absurd choice of log base 100k. The prior would remain practically intact in terms of this kind of order of magnitudes.
Wrong, they are exactly the same distances. I read the next paragraph so I get where you were going with this, but I find it confusing to start off with a blatantly wrong claim, especially when the next line compares 0.11 to 0.1 (11% to 10%) -- not to 0.100001 -- in order to describe how the significance of 0.00001 gets "lost in translation" when speaking in probabilities and not in bits.
"Extreme credences" here should likely be "infinite credences".
Even so, previous page made the exact counterpoint:
While strong evidence may not change your view of things, things, extreme evidence absolutely should make you revisit your estimate of even an infinite credence level.
I don't think these terms have been defined yet. The difference between "strength of credence" and "strength of evidence" isn't obvious to me, but it seems like it's assumed throughout the rest of the article that the reader knows what they mean.
I would expect this sentence only after another telling me that the observations were red car, honking, and punctuality. I think the next sentence should be broken apart and this should be inserted inside.
I recommend rethinking the magnet metaphor, on the grounds that it is physically wrong. If you have two magnets on either end of a ruler, and one is twice as strong as the other, then an iron ball at the center of the ruler is going to roll all the way to the larger magnet (accelerating as it goes, because inverse square law), unless I'm missing something. Perhaps a better physical metaphor would be something like rubber bands, with each bit of evidence adding another rubber band from the belief level to pins at the ends of the ruler?
Sorry, I just have a question. In the problem with pearls I can calculate
P(pearl/blue) = P(pearl AND blue) / P(blue) = 0.11 * 0.04 / (0.11 * 0.04 + 0.89 * 0.08) = 0.058
or with log-odds
P(pearl/blue) = 2^(log2(11/89) + log2(1/2)) = 2^(-4.0163018123) = 0.061
Where does the discrepancy come from?
UPD: I looked up the solution and realized that 2^(-4.016) are odds so probability is 1 / (2^4.01 + 1) = 0.058
Honestly this whole section is confusing
Actually, both pairs differ by the same factor of about 1.85256. (Same for base 100k.) (In the first case, the evidence is around 9, rather than 8, which likely accounts for the difference you got.) I'm actually a little surprised this is true - log-form is used in order to deal with things additively, rather than multiplicatively, so I wouldn't have guessed without calculation that changing the base would preserve proportions. (And I'd still want to do some math to confirm this in general.)
Note: easier to grasp than natural logarithms. Both log base e and log base 10 behave in basically the same way, and have the same arguable pitfall. However, note that this behavior is, I think, on purpose, and intended to convey the idea that 10x more probable is 10x more probable, no matter where you are on the scale of improbability. (You just have to remember where you are on the scale before making your final judgements.) Consider the opposite problem raised by the 0-1 scale, as discussed earlier in the page: 0.00001% isn't that far from 1%, same as between 10.00001% and 11%; but intuitively, clearly 0.00001% is much more different from 1% than 10.00001% is from 11%. This is the non-intuitiveness that the log-form is intended to fix, I think.
Easier to grasp perhaps, but dangerously misleading. Increasing the likelihood of an event from
10^-100to10^-99is very different and much less significant than increasing it from10^-2(1%) to10^-1(10%). I hope this is covered later in this guide.It is really confusing to apply one of the initial steps of a study as evidence to a prior which is the result (last step) of the same study.
But that really gives a different magnitude to the evidence. Why not be consistent with the log base?
For example, if we were to use log base 2, the prior would be ~16.6 magnitudes strong and the evidence ~8. This means that the evidence would alter the prior by (slightly) less than half the order of magnitudes, where's in the case of log base 10 the alteration is (slightly) more than half the order of magnitudes (5 vs 2.7).
Also, imagine the absurd choice of log base 100k. The prior would remain practically intact in terms of this kind of order of magnitudes.
Wrong, they are exactly the same distances. I read the next paragraph so I get where you were going with this, but I find it confusing to start off with a blatantly wrong claim, especially when the next line compares
0.11to0.1(11% to 10%) -- not to0.100001-- in order to describe how the significance of0.00001gets "lost in translation" when speaking in probabilities and not in bits.One of these does log( prob/ 1 - prob) the other does log( prob) ...
I get your point about orders of magnitude difference, but for me this ends up more confusing then anything.
"Extreme credences" here should likely be "infinite credences".
Even so, previous page made the exact counterpoint:
While strong evidence may not change your view of things, things, extreme evidence absolutely should make you revisit your estimate of even an infinite credence level.
Is "-1 against" the same as "+1 for"?
Expressing the first practical example entirely in terms of negative numbers seems like a poor pedagogical choice.
Phrasing as "3 bits against" and then "a further 1 bit against" may help.
Adding that the blue ones are not a great pick if you want pearls may help people understand the direction of "against".
I don't think these terms have been defined yet. The difference between "strength of credence" and "strength of evidence" isn't obvious to me, but it seems like it's assumed throughout the rest of the article that the reader knows what they mean.
It would be nice to show how to go from 99.8% to the 500:1 ratio.
I would expect this sentence only after another telling me that the observations were red car, honking, and punctuality. I think the next sentence should be broken apart and this should be inserted inside.
odds ratios?
The thing inside the log(this part) is an odds ratio, right?
I don't understand this sentence.
I recommend rethinking the magnet metaphor, on the grounds that it is physically wrong. If you have two magnets on either end of a ruler, and one is twice as strong as the other, then an iron ball at the center of the ruler is going to roll all the way to the larger magnet (accelerating as it goes, because inverse square law), unless I'm missing something. Perhaps a better physical metaphor would be something like rubber bands, with each bit of evidence adding another rubber band from the belief level to pins at the ends of the ruler?