The Quick Bayes Table
This is an effort to make Bayes’ Theorem available to people without heavy math skills. It is possible that this has already been invented, because it is just a direct result of expanding something I read at Yudkowsky’s Intuitive Explanation of Bayes Theorem. In that case, excuse me for reinventing the wheel. Also, English is my second language.
When I read Yudkowsky’s Intuitive Explanation of Bayes Theorem, the notion of using decibels to measure the likelihood ratio of additional evidence struck me as extremely intuitive. But in the article, the notion was just a little footnote, and I wanted to check if this could be used to simplify the theorem.
It is harder to use logarithms than just using the Bayes Theorem the normal way, but I remembered that before modern calculators were made, mathematics carried around small tables of base 10 logarithms that saved them work in laborious multiplications and divisions, and I wondered if we could use the same in order to get quick approximations to Bayes’ Theorem.
I calculated some numbers and produced this table in order to test my idea:
|
Decibels |
Probability |
Odds |
|
-30 |
0.1% |
1:1000 |
|
-24 |
0.4% |
1:251 |
|
-20 |
1% |
1:100 |
|
-18 |
1,5% |
1:63 |
|
-15 |
3% |
1:32 |
|
-12 |
6% |
1:16 |
|
-11 |
7% |
1:12.6 |
|
-10 |
9% |
1:10 |
|
-9 |
11% |
1:7.9 |
|
-8 |
14% |
1:6.3 |
|
-7 |
17% |
1:5 |
|
-6 |
20% |
1:4 |
|
-5 |
24% |
1:3.2 |
|
-4 |
28% |
1:2.5 |
|
-3 |
33% |
1:2 |
|
-2 |
38% |
1:1.6 |
|
-1 |
44% |
1:1.3 |
|
0 |
50% |
1:1 |
|
+1 |
56% |
1.3:1 |
|
+2 |
62% |
1.6:1 |
|
+3 |
67% |
2:1 |
|
+4 |
72% |
2.5:1 |
|
+5 |
76% |
3.2:1 |
|
+6 |
80% |
4:1 |
|
+7 |
83% |
5:1 |
|
+8 |
86% |
6.3:1 |
|
+9 |
89% |
7.9:1 |
|
+10 |
91% |
10:1 |
|
+11 |
93% |
12.6:1 |
|
+12 |
94% |
16:1 |
|
+15 |
97% |
32:1 |
|
+18 |
98.5% |
63:1 |
|
+20 |
99% |
100:1 |
|
+24 |
99.6% |
251:1 |
|
+30 |
99.9% |
1000:1 |
This table’s values are approximate for easier use. The odds approximately double every 3 dB (The real odds are 1.995:1 in 3 dB) and are multiplied by 10 every 10 dB exactly.
In order to use this table, you must add the decibels results from the prior probability (Using the probability column) and the likelihood ratio (Using the ratio column) in order to get the approximated answer (Probability column of the decibel result). In case of doubt between two rows, choose the closest to 0.
For example, let’s try to solve the problem in Yudkowsky’s article:
1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
1% prior gets us −20 dB in the table. For the likelihood ratio, 80% true positive versus 9.6% false positive is about a 8:1 ratio, +9 dB in the table. Adding both results, −20 dB + 9 dB = −11dB, and that translates into a 7% as the answer. The true answer is 7.9%, so this method managed to get close to the real answer with just a simple addition.
--
Yudkowsky says that the likelihood ratio doesn’t tell the whole story about the possible results of a test, but I think we can use this method to get the rest of the story.
If you can get the positive likelihood ratio as the meaning of a positive result, then you can use the negative likelihood ratio as the meaning of the negative result just reworking the problem.
I’ll use Yudkowsky’s problem in order to explain myself. If 80% of women with breast cancer get positive mammographies, then 20% of them will get negative mammographies, and they will be false negatives. If 9.6% of women without breast cancer get positive mammographies, then 90.4% of them will get negative mammographies, true negatives.
The ratio between those two values will get us the meaning of a negative result: 20% false negative versus 90.4% true negative is between 1:4 and 1:5 ratio. We get the decibel value closest to 0, −6 dB. −20 dB − 6 dB = −26 dB. This value is between −24 dB and −30 dB, so the answer will be between 0.1% and 0.4%. The true answer is 0.2%, so it also works this way.
--
The positive likelihood ratio and the negative likelihood ratio are a good way of describing how a certain test adds additional data. We could describe the mammography test as a +9dB/-6dB test, and with only this information we know everything we need to know about the test. If the result is positive, it adds 9dB to the evidence, and if it is negative, it subtracts 6dB to it.
Simple and intuitive.
By the way, as decibels are used to measure physical quantities, not probabilities, I believe that renaming the unit would be appropriate in this case. What about DeciBayes?
- 's comment on Open Thread, May 1-15, 2012 by (9 May 2012 10:07 UTC; 3 points)
- Seeking descriptions of deciban-levels by (21 Jun 2013 17:35 UTC; 2 points)
- 's comment on How can we get more and better LW contrarians? by (19 Apr 2012 15:23 UTC; 2 points)
- 's comment on A Priori by (23 Sep 2020 12:44 UTC; 1 point)
This is cool, but I would prefer if you wrote eg 7.9:1 instead of 79:10. I think the odds always being to/from 1 would make it easier to compare them.
Ok, I changed it. It certainly seems more intituitive this way.
I don’t have the book handy, but Jaynes recommends this kind of log probability as more intuitive, and does a few examples in his book.
EDIT: He introduces it in Chapter 4, pg 91, and uses it in chapter 4 and a little in chapter 5.
I made an Anki deck from this table, with the following card layouts:
dB <-> percent
percent <-> odds
dB <-> odds
You can find it here: Quick-Bayes-Table.anki.
Note: I rounded some of the values even further, for simplicity.
This looks correct, and seems possibly awesome.
I calculated a similar table for my own purposes, one version of which can be seen at http://www.orionsarm.com/eg-article/4d5bf17b86585 , another at http://www.lojban.org/tiki/bei%27e .
Isn’t this unit actually called the deciban, with the base unit being known as either the Ban or the Hartley? The WP article on Bayes factors also gives a table of qualitative interpretations, sourced from Harold Jeffreys. Note that your tables seem to give five decibans (1.6 bits of evidence) as “beyond a reasonable doubt”, whereas Jeffreys would merely describe such a degree of evidence as “substantial”. “Strong” evidence requires more than 3.3 bits, or 10 decibans.
Decibans, deciHartleys, and decidits are, indeed, close to being the ‘official’ names for this unit, insofar as any exist.
As for the ‘beyond a reasonable doubt’ line, some time ago, I went looking for numbers, and the best I found was a specialized survey which asked how certain people would have to be in order to convict someone as being guilty ‘beyond a reasonable doubt’. The answers varied depending on the crime, from 75% for petty larceny to 95% for murder; so I assumed that the higher numbers were because of the significant punishments involved, and that people wanted to be extra sure that if they voted to convict, that the person really deserved it; and thus that the ‘real’ meaning of ‘beyond a reasonable doubt’ was the lower number, 75%.
A bit of Googling turns up http://www.law.northwestern.edu/faculty/fulltime/diamond/papers/conflictBetweenPrecisionAndFlexibility.pdf , which appears to be what I was looking at.
It certainly does get called that, but I don’t know how common it is.
I do like “Jaynes” as the name of the unit.
Though… are there other examples of where we have a unit that mathematically is unitless and just represents a multiplication factor, other than Decibels?
Radians and degrees (measures of angle) are common dimensionless units.
A similar table for bits: https://gist.github.com/2415775
Good one.
I chose the decibel scale instead of using bits because bits were a bit awkward when the probabilities were close to 50%. From 0 bits to 1 the probability jumps 16.666%, and the odds doubles, but with decibels the first jump is about 6%, and doubles the odds around 3 decibels, and multiplies them by 10 in exactly 10 decibels.
Yes, decibels are easier for humans.
I’m pretty sure I’m human, but I like bits better because they have a natural interpretation as the number of answers you’ve received to well crafted yes/no questions, which is something that a 10 year old can understand pretty easily.
I meant the math is easier. The same reason you multiply the log by 10 when using decibels—that way, you can talk about 11 decibels instead of 1.1, which would confuse and frighten people.
The table isn’t legible with LessWrong’s dark theme.
You can shorten the table by about half if you eliminate the negative logarithms by using the laws of logarithms.
For example, −20 decibles in terms of probability is just 100% - (The probability corresponding to +20 decibles), and the odds ratios simply occur in reverse order. That is 20 db = 100:1 and −20 db = 1:100
Given the chart is for those who aren’t mathematically inclined, I’d suggest this might not be a great idea.
I’d give most people who even attempt to assign probabilities to their beliefs in the first place enough credit that they can perform very simple arithmetical operations.
I am putting a printout of this chart on my desk until I have it internalized. No more fumbling around trying to do numerical updates in conversation in real time.
I like this a lot. Enough that I might even actually use it at all. Not sure I’d memorise the table, though pinning some percentage points to decibels should be reasonably memorable.
(I’ve been messing with decibels of sound lots lately. I’m thinking of 30dB as certainly the sort of range I can resolve to 3dB steps and even 1.5dB steps, ’cos hearing can do that. So it looks like it might work because it maps it easily onto a domain I’m used to. I realise probability isn’t sound and I expect my probability resolution to end up different …)
I wonder if this is a good way to get better resolution on one’s probability sense.
Yudkowsky
Also, fix your fonts—looks like you got some copypasta in there.
Ok. Corrected it, and tried to fix my fonts. I don’t know if I did it right.
Looks good to me.
Your odds ratios, and thus your decibels, are imprecise. I don’t know if that was approximation on purpose to simplify calculation, or what?
For example, 1% is an odds ratio of 1:99, which is
10 * log(1/99) =~ -19.96.Exactly, I used approximations on purpose, but the real approximated value in this case is the 1%. The ratio that actually gets −20 dB is 1:100.
I felt that getting approximated but round results was worth the imprecision. If I used values like −19.96 on the table, then people without the patience to handle maths wouldn’t be able to use it as well.
Should I explain about the imprecisions of this table better in the article?
It seems like the obvious thing to do, but it’s worth having a tiny note that percent values are approximate, just because they look exact.
Ok, note added.
It’s a commonplace in sound that 6dB = twice the signal, even if it’s actually 1.995.
I know this is over a year old, but I still feel like this is worth pointing out:
You weren’t using the likelihood ratio, which is one value, 8.33… in this case. You were using the numbers you use to get the likelihood ratio.
But the same likelihood ratio would also occur if you had 8% and 0.96%, and then the “negative likelihood ratio” would be about 0.93 instead of 0.22.
You simply need three numbers. Two won’t suffice.
Surely you are shitting us ;]