A wise man proportions his belief to the evidence David Hume Extraordinary claims require extraordinary evidence Carl Sagan Base rate neglect Outcomes Yes No Test Yes ID: 537485
Download Presentation The PPT/PDF document "Extraordinary Claims" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Extraordinary ClaimsSlide2
“A wise man… proportions his belief to the evidence.” – David HumeSlide3
“Extraordinary claims require extraordinary evidence.” – Carl SaganSlide4
Base rate neglectSlide5
Outcomes
Yes
No
Test = Yes
True Positive
False Positive
Test = No
False Negative
True NegativeSlide6
Base Rates
The
base rate
for any condition is simply the proportion of people who have the condition.
Base rate of dog owners in Hong Kong:
# of Dog Owners in Hong Kong
÷
# of Hong
KongersSlide7
Base Rate
Yes
No
Test = Yes
True Positive
False Positive
Test = No
False Negative
True NegativeSlide8
As the base rate decreases, the number of true positives decreases.Slide9
Base Rate
Yes
No
Test = Yes
True Positive
False Positive
Test = No
False Negative
True NegativeSlide10
Importance of Base Rates
Therefore, the proportion of false positives out of total positives increases:
False Positive
÷
False Positive + True PositiveSlide11
Importance of Base Rates
And the proportion of true positives out of total positives decreases:
True Positive
÷
False Positive + True PositiveSlide12
What Does It Mean?
It means that even very good tests (low false positive rate, low false negative rate) cannot reliably detect conditions with low base rates.Slide13
Example
Suppose the police have a test for telling whether someone is driving drunk:
If you are drunk, the test returns “positive” 100% of the time.
If you are not drunk, the test returns “positive” 95% of the time.
The police randomly stop 1,000 drivers and test them.Slide14
The base rate of drunk drivers is:
# of drunk drivers
÷
# of drunk drivers + # of sober driversSlide15
High Base Rate
The base rate of drunk drivers is:
# of drunk drivers =500
÷
# of drunk drivers =500 + # of sober drivers =500
Base rate = 50%Slide16
Outcomes
Drunk
Not
Drunk
Test = Yes
True Positive
False Positive
Test = No
False Negative
True NegativeSlide17
Outcomes
Drunk
Not
Drunk
Test = Yes
500
x 100%
500
x 5%
Test = No
500
x 0%
500
x 95%Slide18
Outcomes
Drunk
Not
Drunk
Test = Yes
500
25
Test = No
0
475Slide19
Good Test!
Now we have 500 true positives and only 25 false positives:
False Positive
÷
False Positive + True PositiveSlide20
Good Test!
Now we have 500 true positives and only 25 false positives:
25
÷
25 + 500
5% =Slide21
Low Base Rate
The base rate of drunk drivers is:
# of drunk drivers =5
÷
# of drunk drivers =5 + # of sober drivers =995
Base rate = 0.5%Slide22
Outcomes
Drunk
Not
Drunk
Test = Yes
True Positive
False Positive
Test = No
False Negative
True NegativeSlide23
Outcomes
Drunk
Not
Drunk
Test = Yes
5
x 100%
995 x 5%
Test = No
5
x 0%
995 x 95%Slide24
Outcomes
Drunk
Not
Drunk
Test = Yes
5
50
Test = No
0
945Slide25
Bad Test!
Now we have 5 true positives and 50 false positives:
False Positive
÷
False Positive + True PositiveSlide26
Bad Test!
Now we have 5 true positives and 50 false positives:
50
÷
50 + 5
91% =Slide27
We saw this same phenomenon when we learned that most published scientific research is false.Slide28Slide29
Low base rate of truths.Slide30
High proportion of false positives.Slide31
Base Rate Neglect Fallacy
The base rate neglect fallacy is when we ignore the base rate.
The base rate is very low, so our tests are very unreliable… but we still trust the tests.Slide32
It Matters
The Hong Kong police stop people 2 million times every year.
Only 22,500 of these cases (1%) are found to be offenses.
Even if the police have a very low false positive rate it’s clear that the base rate of offenses does not warrant these stops.Slide33
Prior probabilitiesSlide34
Probabilities
A base rate is a rate (a frequency). # of X’s out of # of Y’s. # of drunk drivers out of # of total drivers.
Sometimes a rate or a frequency doesn’t make sense. Some things only happen once, like an election. So here we talk about
probabilities
instead of rates.Slide35
Probabilities
For things that do happen a lot the probabilities are (or approximate) the frequencies:
Probability of random person being stopped by police
=
# of people stopped
by police
÷ total # of peopleSlide36
Things that Only Happened Once
The 2012 Chief Executive Election
The 1997 Handover
The great plague of Hong Kong
The second world war
The extinction of the dinosaurs
The big bangSlide37
Evidence
Drunk tests are a type of evidence. Testing positive raises the probability that you are drunk. But how much does it raise the probability?
That depends on the base rate
.Slide38
Evidence
You can also have evidence for or against one-time events. For example, you might have a footprint at the scene of a crime. This raises the probability that certain people are guilty. How much?
That depends on the prior probability
.Slide39
Priors and Posteriors
Prior probability = probability that something is true before you look at the new evidence.
Posterior probability = probability that something is true after you look at the new evidence.Slide40
Relativity
Priors and posteriors are relative to evidence. Once we look at new evidence, our posteriors become our new priors when we consider the next evidence.Slide41
Bayes’ Theorem
P (data/
hyp
.) x P(
hyp
.)
÷
P(data)
P(hypothesis/ data) =
Posterior
PriorSlide42
Prior/ Posterior Positively Correlated
P(hypothesis
/ data
) ∝ [P(data/
hyp
.) x P(
hyp
.)]Slide43
What Does It Mean
If the prior probability of some event is very low, then even very good evidence for that event does not significantly increase its probability.Slide44
“Extraordinary claims require extraordinary evidence.” – Carl SaganSlide45
Hume on miraclesSlide46
“A wise man… proportions his belief to the evidence.” – David HumeSlide47
Novel Testimony
Suppose
that we get testimony concerning something we have never experienced.
Hume imagines someone from the equatorial regions being told about frost, and snow, and ice. They have never experienced anything like that before.Slide48
It’s Strange!Slide49
Hume
thinks this person would have reason to disbelieve stories about a white powder that fell from the sky, covered everything by several inches, and then turned to water and went away
.Slide50
It’s
not that they should believe the stories are
not
true, just that they don’t have to believe they
are
true. We need more
evidence, because the prior is so low.Slide51
But
nowsuppose
someone tells us an even stranger story.
It’s
like the snow-story, in that we’ve never experienced anything like it before. But it’s even stranger, because we have
always
experienced the
opposite
before.Slide52
Miracles
For
Hume, this is the definition of a miracle. A miracle is a violation of the laws of nature. Every event or process in the world conforms to the laws of nature (for example, the laws of physics like the law of gravity)– except, if there are any, miracles. Slide53
Example
There are about 100 billion people who have lived and died in the history of humanity (and there are 7 billion more who are alive now).
As far as we know, none of the 100 billion people who have ever died and were dead for four days, later came back to life. It’s a law of nature that when you die, that’s the end, there’s no more.Slide54
Lazarus
Although there is testimony, in at least one religious book– the Christian bible– that such an event occurred at least once in history, when Jesus raised Lazarus from the dead, after he had been dead for four days.Slide55
What Should We Believe?
According to Hume, we should be wise and apportion our belief to the evidence.
Since on the one hand we have 100 billion people who died and never came back, and on the other hand we have an old legend from a book intended to make people believe its religious views, it’s most probable that
the raising of Lazarus never happened
.Slide56
Hume on Miracles
“No testimony is sufficient to establish a miracle, unless the testimony be of such a kind that its falsehood would be more miraculous than the fact which it endeavors to establish.”Slide57
Seeing and Believing
So
, for example, Hume would even say that if you
saw
someone die and come back to life, you should not believe that it really happened. Slide58
Seeing and Believing
Because
it’s always possible that what you saw was a trick, or the person was never really dead, or you were on drugs or…
Since
none of those suppositions are miraculous, you should believe them instead of believing in the miracle. They’re more likely than a violation of nature’s laws.Slide59
No more philosophy!Slide60
OK, Back to Science…
There’s a debate among scientists about Evidence Based Medicine vs. Science Based Medicine.
They sound the same, but they’re very different!Slide61
Modern Medicine
In current modern medicine the following is (one) best estimate:
37% of treatments are based on Randomized Controlled Trials
76% of treatments are based on good evidence (RCTs, observational studies)
The rest should be based on scientific theory (reasonable extension of what we know).Slide62
Evidence Based Medicine
One idea is that the 76% of tested-treatments are the “real” evidence based medicine and the rest is no better than untested alternative medicine. These are equal:
Treatments
based on scientific theory (reasonable extension of what we know
).
Untested pre-scientific or otherwise alternative treatments (e.g. homeopathy).Slide63
Difficult Tests
Some alternative treatments are difficult to test.
Homeopaths claim that their treatments are individualized. So it’s not enough to give everyone suffering from a disease the same magic water… they have to come into the shop for a personalized experience.Slide64
Can’t Placebo a Whole Shop!Slide65
False Equivalence
This means we should let the homeopaths “get away with it.” Sure, their treatments aren’t supported by science, but neither are 24% of modern treatments.Slide66
Science Based Medicine
Science based medicine, on the other hand, says we should take into account
prior probability
.
We have lots of scientific knowledge of water. Nothing about it says that chemically pure water that
in the past
contained other chemicals and
was then shaken
should behave any differently than regular chemically pure waterSlide67
Science Based Medicine
And, science based medicine says that the 24% of treatments that are not evidence based,
while they should still be tested
, are much better because of prior probability.
If science tells us why they should work, then we should believe the science even if we haven’t tested them (yet) or can’t test them.Slide68
Example
It’s immoral not to perform blood transfusions on people who have lost lots of blood.
So we can’t do a RCT on blood transfusions.
But that doesn’t mean they’re as silly as homeopathy.