Apr 18, 2012

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?