The Quick Bayes Table

post by farsan · 2012-04-18T18:00:46.067Z · LW · GW · Legacy · 31 comments

Contents

31 comments

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?

31 comments

Comments sorted by top scores.

comment by ShardPhoenix · 2012-04-19T01:52:58.065Z · LW(p) · GW(p)

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.

Replies from: farsan
comment by farsan · 2012-04-20T05:54:51.630Z · LW(p) · GW(p)

Ok, I changed it. It certainly seems more intituitive this way.

comment by buybuydandavis · 2012-04-18T19:51:27.145Z · LW(p) · GW(p)

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.

comment by alexvermeer · 2012-07-12T19:47:55.528Z · LW(p) · GW(p)

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.

comment by thomblake · 2012-04-18T18:17:26.677Z · LW(p) · GW(p)

This looks correct, and seems possibly awesome.

comment by DataPacRat · 2012-04-19T02:59:32.064Z · LW(p) · GW(p)

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 .

Replies from: bogus, thomblake
comment by bogus · 2012-04-19T16:43:33.268Z · LW(p) · GW(p)

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.

Replies from: DataPacRat, thomblake
comment by DataPacRat · 2012-04-20T01:43:14.971Z · LW(p) · GW(p)

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.

comment by thomblake · 2012-04-19T17:24:31.682Z · LW(p) · GW(p)

It certainly does get called that, but I don't know how common it is.

comment by thomblake · 2012-04-19T14:39:38.264Z · LW(p) · GW(p)

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?

Replies from: Dreaded_Anomaly
comment by Dreaded_Anomaly · 2012-04-19T18:44:17.723Z · LW(p) · GW(p)

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.

comment by thomblake · 2012-04-18T19:01:00.246Z · LW(p) · GW(p)

A similar table for bits: https://gist.github.com/2415775

Replies from: farsan
comment by farsan · 2012-04-18T19:15:14.049Z · LW(p) · GW(p)

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.

Replies from: thomblake
comment by thomblake · 2012-04-18T19:16:12.774Z · LW(p) · GW(p)

Yes, decibels are easier for humans.

Replies from: JenniferRM
comment by JenniferRM · 2012-04-18T20:40:30.298Z · LW(p) · GW(p)

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.

Replies from: thomblake
comment by thomblake · 2012-04-18T21:03:32.974Z · LW(p) · GW(p)

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.

comment by Psychosmurf · 2012-04-19T07:36:11.659Z · LW(p) · GW(p)

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

Replies from: handoflixue
comment by handoflixue · 2012-04-19T22:53:12.969Z · LW(p) · GW(p)

Given the chart is for those who aren't mathematically inclined, I'd suggest this might not be a great idea.

Replies from: Psychosmurf
comment by Psychosmurf · 2012-04-20T01:31:19.237Z · LW(p) · GW(p)

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.

comment by earthwormchuck163 · 2012-04-19T02:08:39.687Z · LW(p) · GW(p)

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.

comment by David_Gerard · 2012-04-18T23:00:36.161Z · LW(p) · GW(p)

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.

comment by thomblake · 2012-04-18T18:12:58.131Z · LW(p) · GW(p)

Yudkowski

Yudkowsky

Also, fix your fonts - looks like you got some copypasta in there.

Replies from: farsan
comment by farsan · 2012-04-18T18:24:54.929Z · LW(p) · GW(p)

Ok. Corrected it, and tried to fix my fonts. I don't know if I did it right.

Replies from: thomblake
comment by thomblake · 2012-04-18T18:25:33.803Z · LW(p) · GW(p)

Looks good to me.

comment by thomblake · 2012-04-18T18:44:18.292Z · LW(p) · GW(p)

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.

Replies from: farsan
comment by farsan · 2012-04-18T18:53:01.444Z · LW(p) · GW(p)

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?

Replies from: thomblake, David_Gerard
comment by thomblake · 2012-04-18T18:59:59.816Z · LW(p) · GW(p)

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.

Replies from: farsan
comment by farsan · 2012-04-18T19:23:27.617Z · LW(p) · GW(p)

Ok, note added.

comment by David_Gerard · 2012-04-18T23:01:36.427Z · LW(p) · GW(p)

It's a commonplace in sound that 6dB = twice the signal, even if it's actually 1.995.

comment by Nnotm (NNOTM) · 2013-08-19T20:47:02.306Z · LW(p) · GW(p)

I know this is over a year old, but I still feel like this is worth pointing out:

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.

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.

comment by tut · 2012-04-25T15:56:58.005Z · LW(p) · GW(p)

DeciBayes

Surely you are shitting us ;]