Transcript nm_bayes1v2x - UCL Department of Geography

MSc Methods XX: YY
Dr. Mathias (Mat) Disney
UCL Geography
Office: 113, Pearson Building
Tel: 7670 0592
Email: [email protected]
www.geog.ucl.ac.uk/~mdisney
Lecture outline
• Intro to Bayes’ Theorem
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Science and scientific thinking
Probability & Bayes Theorem – why is it important?
Frequentists v Bayesian
Background, rationale
Methods: MCMC ……
Advantages / disadvantages
• Applications:
– parameter estimation, uncertainty
– Practical – basic Bayesian estimation
Reading and browsing
Bayesian methods, data analysis
• Gauch, H., 2002, Scientific Method in Practice, CUP.
• Sivia, D. S., with Skilling, J. (2008) Data Analysis, 2nd ed., OUP, Oxford.
Computational
• Numerical Methods in C (XXXX)
• Flake, W. G. (2000) Computational Beauty of Nature, MIT Press.
• Gershenfeld, N. (2002) The Nature of Mathematical Modelling,, CUP.
• Wainwright, J. and Mulligan, M. (2004) (eds) Environmental Modelling:
Finding Simplicity in Complexity, John Wiley and Sons.
Mathematical texts, inverse methods
• Tarantola (XXXX)
Kalman filters
• Welch and Bishop
• Maybeck
Reading and browsing
Papers, articles, links
P-values
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Siegfried, T. (2010) “Odds are it’s wrong”, Science News, 107(7),
http://www.sciencenews.org/view/feature/id/57091/title/Odds_Are,_Its_Wrong
Ioannidis, J. P. A. (2005) Why most published research findings are false, PLoS Medicine,
0101-0106.
Bayes
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•
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Hill, R. (2004) Multiple sudden infant deaths – coincidence or beyond coincidence, Pediatric
and Perinatal Epidemiology, 18, 320-326
(http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf)
http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/
http://yudkowsky.net/rational/bayes
Error analysis
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•
http://level1.physics.dur.ac.uk/skills/erroranalysis.php
http://instructor.physics.lsa.umich.edu/int-labs/Statistics.pdf
So how do we do science?
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Carry out experiments?
Collect observations?
Test hypotheses (models)?
Generate “understanding”?
Objective knowledge??
Induction? Deduction?
Induction and deduction
• Deduction
– Inference, by reasoning, from general to particular
– E.g. Premises: i) every mammal has a heart; ii)
every horse is a mammal.
– Conclusion: Every horse has a heart.
– Valid if the truth of premises guarantees truth of
conclusions & false otherwise.
– Conclusion is either true or false
Induction and deduction
• Induction
– Process of inferring general principles from
observation of particular cases
– E.g. Premise: every horse that has ever been
observed has a heart
– Conclusion: Every horse has a heart.
– Conclusion goes beyond information present, even
implicitly, in premises
– Conclusions have a degree of strength (weak ->
near certain).
Induction and deduction
Induction and deduction
• Example from Gauch (2003: 219) which we will
return to:
– Q1: Given a fair coin (P(H) = 0.5), what is P that 100
tosses will produce 45 heads and 55 tails?
– Q2: Given that 100 tosses 45 heads and 55 tails, what
is the P that it is a fair coin?
• Q1 is deductive: definitive answer - probability
• Q2 is inductive: no definitive answer - statistics
Aside: sound argument v fallacy
• If plants lack nitrogen, they become yellowish
– The plants are yellowish, therefore they lack N
– The plants do not lack N, so they do not become
yellowish
– The plants lack N, so they become yellowish
– The plants are not yellowish, so they do not lack N
•
•
•
•
Affirming the antecedent: p  q; p,  q ✓
Denying the consequent: p  q: ~q,  ~p ✓
Affirming the consequent: p  q: q,  p X
Denying the antecedent: p  q: ~p, ~q X
Aside: sound argument v fallacy
• Fallacies can be hard to spot in longer, more
detailed arguments:
– Fallacies of composition; ambiguity; false dilemmas;
circular reasoning; genetic fallacies (ad hominem)
• Gauch (2003) notes:
– For an argument to be accepted by any audience as
proof, audience MUST accept premises and validity
– That is: part of responsibility for rational dialogue falls to
the audience
– If audience data lacking and / or logic weak then valid
argument may be incorrectly rejected (or vice versa)
Gauch (2006): “Seven pillars of Science”
1.
2.
3.
4.
Realism: physical world is real;
Presuppositions: world is orderly and comprehensible;
Evidence: science demands evidence;
Logic: science uses standard, settled logic to connect
evidence and assumptions with conclusions;
5. Limits: many matters cannot usefully be examined by
science;
6. Universality: science is public and inclusive;
7. Worldview: science must contribute to a meaningful
worldview.
What’s this got to do with methods?
• Fundamental laws of probability can be derived
from statements of logic
• BUT there are different ways to apply
• Two key ways
– Frequentist
– Bayesian – after Rev. Thomas Bayes (1702-1761)
Bayes: see Gauch (2003) ch 5
• Informally, the Bayesian Q is:
– “What is the probability (P) that a hypothesis (H) is
true, given the data and any prior knowledge?”
– Weighs different hypotheses (models) in the light of
data
• The frequentist Q is:
– “How reliable is an inference procedure, by virtue of
not rejecting a true hypothesis or accepting a false
hypothesis?”
– Weighs procedures (different sets of data) in the
light of hypothesis
Probability? see S&S(1006) p9
• To Bayes, Laplace, Bernoulli…:
– P represents a ‘degree-of-belief’ or plausibility
– i.e. degree of truth, based on evidence at hand
• BUT this appears to be subjective, so P was
redefind (Fisher, Neyman, Pearson etc.) :
– P is the ‘long-run relative frequency’ with which an
event occurs, given (infinite) repeated expts.
– We can measure frequencies, so P now an objective
tool for dealing with random phenomena
Bayesian reasoning
• Bayesian view is directly related to how we do science
• Frequentist view of hypothesis testing is fundamentally
flawed (Jaynes, ch 17 for eg):
– To test H do it indirectly - invent null hypothesis Ho that denies
H, then argue against Ho
– But in practice, Ho is not (usually) a direct denial of H
– H usually a disjunction of many different hypotheses, where Ho
denies all of them while assuming things (eg normal distribution
of errors) which H neither assumes nor denies
• Jeffreys (1939, p316): “…an hypothesis that may be true is rejected
because it has failed to predict observable results that have not occurred.
This seems remarkable…on the face of it, the evidence might more
reasonably be taken as evidence for the hypothesis, not against it. The
same applies to all all the current significance tests based on P-values.”
Aside: the problem with P values
• Significance testing and P-values are widespread
• P-value < 0.05 tells you chance of fluke result is < 5% so we can
declare this result “statistically significant at the 5% level”
• BUT they tell you nothing about the effect you’re interested in (see
Siegfried (2010) for eg)
• 2 possible conclusions remain:
– i) there is a real effect
– Ii) the result is an improbable (1 in 20) fluke
• P-value does not tell you which
A P value is the probability of an observed (or more extreme) result
arising only from chance.
Credit: S. Goodman, adapted by A. Nandy
http://www.sciencenews.org/view/access/id/57253/name/feat_stati
stics_pvalue_chart.jpg
Aside: the problem with P values
• If P > 0.05 then also two conclusions:
– i) there is no real effect
– ii) test not capable of discriminating a weak effect
• All P value of < 0.05 can say is:
– There is a 5% chance of obtaining the observed (or more extreme) result if no
real effect exists i.e. if the null hypothesis is correct
• No more, no less – so doesn’t really get you very far…..
A P value is the probability of an observed (or more extreme) result
arising only from chance.
Credit: S. Goodman, adapted by A. Nandy
http://www.sciencenews.org/view/access/id/57253/name/feat_stati
stics_pvalue_chart.jpg
Bayes: see Gauch (2003) ch 5
• Prior knowledge?
– What is known beyond the particular experiment at
hand, which may be substantial or negligible
• We all have priors: assumptions, experience,
other pieces of evidence
• Bayes approach explicitly requires you to
assign a probability to your prior (somehow)
• Bayesian view - probability as degree of belief
rather than a frequency of occurrence (in the
long run…)
Bayes’ Theorem
• The “chief rule involved in the process of
learning from experience” (Jefferys, 1983)
• Formally:
P ( H | D) µ P ( D | H ) ´ P ( H )
• P(H|D) = Posterior i.e. probability of hypothesis
(model) H being true, given data D
• P(D|H) = Likelihood i.e probability of data D
being observed if H is true
• P(H) = Prior i.e. probability of hypothesis being
true before measurement of D
Bayes’ Theorem
• Importance? P(H|D) appears on the left of BT
• It solves the inverse (inductive) problem –
probability of a hypothesis given some data
• This is how we do science in practice!
• We don’t have access to infinite repetitions of
expts (the ‘long run frequency’ view)
Bayes’ Theorem
P ( Hypoth. | Data, I ) µ P ( Data | Hypoth., I ) ´ P ( Hypoth. | I )
• I is ‘background information’ as there is ‘no such thing as
absolute probability’ (see S & S p 5)
• P(rain today) will depend on clouds this morning, whether
we saw forecast etc. etc. – I is usually left out but ….
• Power of Bayes’ Theorem
– Relates the quantity of interest i.e. P of H being true given D, to
that which we might estimate in practice i.e. P of observing D,
given H is correct
Bayes’ Theorem & marginalisation
• To go from to  to = we need to divide by P(D|I)
P ( D | H, I ) ´ P ( H, I )
P ( H | D, I ) =
P(D | I )
• Where P(D|I) is known as the ‘Evidence’
• Normalisation constant which can be left out for parameter
estimation as independent of H
• But is required in model selection for e.g. where data
amount may be critical
Bayes’ Theorem & marginalisation
• Generally, using X for Hypothesis, and Y for Data
prob (Y | X, I ) ´ prob ( X, I )
prob ( X | Y, I ) =
prob(Y | I )
prob ( X | I ) =
+¥
ò prob ( X,Y | I ) dY
-¥
• Where prob(X|I) is the marginalisation equation
• But if Y is a proposition, how can we integrate over it?
Bayes’ Theorem & marginalisation
•
•
•
•
Suppose instead of Y and Y (not Y) we have a set of alternative possibilities:
Y1, Y2, …. YM = {Yk}
Eg M candidates for an election, Y1 = prob. candidate 1 will win, Y2 cand. 2
will win etc.
Prob that X is true e.g. that unemployment will fall in 1 year, irrespective of
who wins (Y) is
M
As long as
prob ( X | I ) = å prob ( X,Yk | I )
k=1
M
å prob ( X,Y | I ) =1
k
k=1
•
i.e. the various probabilities {Yk} are exhaustive and mutually exclusive, so
that if one Yk is true, all others are false but one must be true
Bayes’ Theorem & marginalisation
•
•
•
•
As M gets larger, we approach prob ( X | I ) =
ò prob ( X,Y | I ) dY
-¥
Eg we could consider an arbitrarily large number of propositions about the
range in which my weight WMD could lie
Choose contiguous intervals and large enough range (M ∞), we will have a
mutually exclusive, exhaustive set of possibilities
So Y represents parameter of interest (WMD in this case) and integrand
prob(X,Y|I) is now a distribution – probability density function (pdf)
pdf ( X,Y = y | I ) = lim
prob ( X, y £ Y < y + d y | I )
dy
d y®0
•
+¥
And prob. that Y lies in finite range between y1 and y2 (and X is also true) is
prob ( X, y1 £ Y < y2 | I ) =
y2
ò p df ( X,Y | I ) dY
y1
Eg Laplace and the mass of Saturn
• Laplace (1749-1827) estimated MSaturn from orbital data
• i.e. posterior prob(M|{data},I) where I is background knowledge of
orbital mechanics etc.
• Shaded area under posterior pdf shows degree of belief that m1 ≤
MSaturn < m2 (he was right to within < 0.7%)
• How do we interpret this pdf in terms of frequencies?
– Some ensemble of universes all constant other than MSaturn? Distribution of
MSaturn in repeated experiments?
– But data consist of orbital periods, and these multiple expts. didn’t happen
Best estimate of M
Degree of
certainty of M
The posterior pdf expresses
ALL our best understanding
of the problem
Example: is this a fair coin?
Heads I win, tails you lose?
• H? HT? HTTTTHTHHTT?? What do we mean fair?
• Consider range of contiguous propositions (hypotheses)
about range in which coin bias-weighting, H might lie
• If H = 0, double tail; H = 1, double head; H = 0.5 is fair
• E.g. 0.0 ≤ H1 < 0.01; 0.01 ≤ H2 < 0.02; 0.02 ≤ H3 < 0.03 etc.
Example: is this a fair coin?
• If we assign high P to a given H (or range of Hs), relative to
all others, we are confident of estimate of ‘fairness’
• If all H are equally likely, then we are ignorant
• This is summarised by conditional (posterior) pdf
prob(H|{data},I)
• So, we need prior prob(H,I) – if we know nothing let’s use
flat (uniform) prior i.e.
ì 1 0 £ H £1
prob ( H | I ) = í
î 0 otherwise
P ( Hypoth. | Data, I ) µ P ( Data | Hypoth., I ) ´ P ( Hypoth. | I )
Example: is this a fair coin?
• Now need likelihood i.e. prob({data}|H,I)
• Measure of chance of obtaining {data} we have actually
observed if bias-weighting H was known
• Assume that each toss is independent event (part of I)
• Then prob(R heads in N tosses) is given by binomial
theorem i.e.
prob ({data} | H, I ) µ H (1- H )
R
N-R
– H is chance of head and there are R of them, then there must be NR tails (chance 1-H).
P ( Hypoth. | Data, I ) µ P ( Data | Hypoth., I ) ´ P ( Hypoth. | I )
Example: is this a fair coin?
• How does prob(H|{data},I) evolve?
TTT
HT
HTTTH
prob ({data} | H, I ) µ H (1- H )
R
N-R
ì 1 0 £ H £1
prob ( H | I ) = í
î 0 otherwise
Gaussian prior μ = 0.5, σ = 0.05
• How does prob(H|{data},I) evolve?
H0 (mean) not always at peak
Particularly when N small
T
prob ({data} | H, I ) µ H (1- H )
R
N-R
ì 1 0 £ H £1
prob ( H | I ) = í
î 0 otherwise
Summary
• The posterior pdf summarises our knowledge, based on
{data} and prior
– Note{data} in this case actually np.random.binomial(N, p)
• Weak prior shifted easily
ì 1 0 £ H £1
prob ( H | I ) = í
î 0 otherwise
• Stronger Gaussian prior (rightly) requires a lot more data to
be convinced
( H-m )
prob ( H | I ) = e 2s
• See S & S for other priors….
• Bayes’ Theorem encapsulates the learning process
2
2
P ( Hypoth. | Data, I ) µ P ( Data | Hypoth., I ) ´ P ( Hypoth. | I )
Summary
• Takes a lot of coin tosses to estimate H to within 0.2-0.3
• If we toss 10 times and get 10 T, this might be strong
evidence for bias
• But if we toss 100 times and get 45H 55T, difference still
10 BUT much more uncertain
• Gaussian: Although H(0.5) ~ 250000 H(0.25), 1000 tosses
gets posterior to within 0.02
P ( Hypoth. | Data, I ) µ P ( Data | Hypoth., I ) ´ P ( Hypoth. | I )
Reliability and uncertainty
• Can we summarise PDF prob(H|{data},I) concisely (mean, error)?
• Best estimate Xo of parameter X is given by condition
dP
d 2P
=0
<0
dX Xo
dX 2 Xo
• Also want measure of reliability (spread of pdf around Xo)
f¢ a
f ¢¢ a
• Use Taylor series expansion f ( x ) = f ( a) + ( ) ( x - a) + ( ) ( x - a) 2 +...
1!
2!
• Use L = loge[prob(H|{data},I)] - varies much more slowly with X
• Expand about X-Xo = 0 so
1 d2L
L = L ( X0 ) +
2 dX 2
( X - X0 )
2
+...
X0
• First term is constant, second term linear (X-Xo) not important as we
are expanding about maximum. So, ignoring higher order terms….
Reliability and uncertainty
• We find
é1 d 2L
prob ( X | { data}, I ) » Aexp ê
2
êë 2 dX
( X - X0 )
X0
2
ù
ú
úû
• Where A is a normalisation constant. So what is this function??
• It is pdf of Gaussian (normal) distribution i.e.
é ( x - m)2 ù
1
ú
prob ( x | m, s ) =
exp ê2
s 2p
êë 2s úû
• Where μ, σ are maximum and width (sd)
• Comparing, we see μ at Xo and
æ d2L
s = - çç 2
è dX
-1 2
ö
÷
÷
X0 ø
• So X = Xo ±σ
http://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svg
Reliability and uncertainty
• From the coin example prob ( H | { data}, I ) µ H R (1- H )
• So L = loge éë prob ( H | { data}, I )ùû = const + R loge ( H ) + ( N - R) loge (1- H )
• Therefore
dL
R ( N - R)
EXERCISE: verify
=
=0
dH H H 0 (1- H 0 )
these expressions
N-R
0
yourself
• So Ho = R/N, and then
d 2L
dH 2
=H0
R ( N - R)
N
=
H 2 (1- H ) 2
H 0 (1- H 0 )
s=
H 0 (1- H 0 )
N
• Ho tends to a constant, therefore so does Ho(1-Ho), so σ  1/√ N
• So can express key properties of pdf using Ho and σ
• NB largest uncertainty (σmax ) when Ho = 0.5 i.e. easier to identify
highly-biased coin than to be confident it is fair
Reliability and uncertainty
•
•
•
•
Asymmetric pdf? Ho still best estimate
But preal more likely one side of Ho than another
So what does ‘error bar’ mean then?
Confidence intervals (CI)
95%CI
X
– shortest interval enclosing X% of area under pdf, say 95%
• Assuming posterior pdf normalised
(total area = 1) then need X1, X2
X
such that prob ( X1 £ X < X2 | {data}, I ) = ò prob ( X | {data}, I ) dX » 0.95
X
• The region X1 ≤ X < X2 is the shortest 95% CI
• For normalised pdf, weighted average given by X = ò Xprob ( X | { data}, I ) dX
• Multimodal pdf?
2
1
– As pdf gets more complex, single estimates of mean not relevant
– Just show posterior pdf, then you can decide…..
A more complex example: mean of Gaussian
• For N data,
é ( x - m)2 ù
1
prob ( xk | m, s ) =
exp ê- k 2 ú
2s
s 2p
êë
úû
• Given data {xk}, what is best estimate of μ and error, σ?
N
• Likelihood?
prob ({ xk } | m, s , I ) = Õ prob ( xk | m, s , I )
k=1
• Simple uniform prior?
ìï A =1 m - m
( max min ) mmin £ m £ mmax
prob ( m | s , I ) = prob ( m | I ) = í
ïî
0
otherwise
• Log(Posterior),L
N
L = loge éë prob ( m | { xk }, s , I )ùû = const - å
k=1
( xk - m )
2s 2
2
A more complex example: mean of Gaussian
• For best estimate μo
• So
•
•
•
•
•
N
N
k=1
k=1
å xk = å m 0 = N m 0
N
dL
x -m
=å k 2 0 =0
dm u0 k=1 s
and best estimate is simple mean i.e.
N
d2L
1
N
= -å 2 = - 2
2
dm u0
s
k=1 s
Confidence depends on σ i.e.
s
And so m = m0 ±
N
Here μmin = -2 μmax = 15
If we make larger?
Weighting of error for each point?
1 N
m 0 = å xk
N k=1
Common errors: ignored prior
• After Stirzaker (1994) and Gauch (2003):
• Blood test for rare disease occurring by chance in
1:100,000. Test is quite accurate:
– Will tell if you have disease 95% of time i.e. p = 0.95
– BUT also gives false positive 0.5% of the time i.e. p = 0.005
• Q: if test says you have disease, what is the probability this
diagnosis is correct?
– 80% of health experts questioned gave the wrong answer (Gauch,
2003: 211)
– Use 2-hypothesis form of Bayes’ Theorem
Common errors: ignored prior
• Back to our disease test
P ( H1| D) P ( D | H1) P ( H1)
=
´
P ( H 2 | D) P ( D | H 2 ) P ( H 2 )
P ( H1| D)
p(correct)p(have)
=
=
P ( H 2 | D) p(correct)p(have) + p correct p have
(
) (
( 0.95 ´ 0.00001)
» 1: 500
( 0.95´ 0.00001) + ( 0.005´ 0.99999)
)
• Correct diagnosis only 1 time in 500 - 499 false
+ve!
• For a disease as rare as this, the false positive
rate (1:200) makes test essentially useless
What went wrong?
• Knowledge of general population gives prior
odds diseased:healthy 1:100000
• Knowledge of +ve test gives likelihood odds 95:5
• Mistake is to base conclusion on likelihood odds
• Prior odds completely dominate
– 0.005 x 0.99999 ~ 0.005 >> 0.95 x 0.00001 ~1x10-6
The tragic case of Sally Clark
• Two cot-deaths (SIDS), 1 year apart, aged 11 weeks and
8 weeks. Mother Sally Clark charged with double
murder, tried and convicted in 1999
– Statistical evidence was misunderstood, “expert” testimony was
wrong, and a fundamental logical fallacy was introduced
• What happened?
• We can use Bayes’ Theorem to decide between 2
hypotheses
– H1 = Sally Clark committed double murder
– H2 = Two children DID die of SIDS
•
•
http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayestheorem/
http://yudkowsky.net/rational/bayes
44
The tragic case of Sally Clark
P ( H1| D) P ( D | H1) P ( H1)
=
´
P ( H 2 | D) P ( D | H 2 ) P ( H 2 )
prob. of H1 or H2
given data D
Likelihoods i.e. prob. of
getting data D IF H1 is
true, or if H2 is true
Very important PRIOR probability i.e.
previous best guess
• Data? We observe there are 2 dead children
• We need to decide which of H1 or H2 are more
plausible, given D (and prior expectations)
• i.e. want ratio P(H1|D) / P(H2|D) i.e. odds of H1 being
true compared to H2, GIVEN data and prior
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The tragic case of Sally Clark
• ERROR 1: events NOT independent
• P(1 child dying of SIDS)? ~ 1:1300, but for affluent nonsmoking, mother > 26yrs ~ 1:8500.
• Prof. Sir Roy Meadows (expert witness)
– P(2 deaths)? 1:8500*8500 ~ 1:73 million.
– This was KEY to her conviction & is demonstrably wrong
– ~650000 births a year in UK, so at 1:73M a double cot death is a 1
in 100 year event. BUT 1 or 2 occur every year – how come?? No
one checked …
– NOT independent P(2nd death | 1st death) 5-10 higher i.e. 1:100 to
200, so P(H2) actually 1:1300*5/1300 ~ 1:300000
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The tragic case of Sally Clark
• ERROR 2: “Prosecutor’s Fallacy”
– 1:300000 still VERY rare, so she’s unlikely to be innocent, right??
• Meadows “Law”: ‘one cot death is a tragedy, two cot deaths is suspicious and,
until the contrary is proved, three cot deaths is murder’
– WRONG: Fallacy to mistake chance of a rare event as chance that
defendant is innocent
• In large samples, even rare events occur quite frequently someone wins the lottery (1:14M) nearly every week
• 650000 births a year, expect 2-3 double cot deaths…..
• AND we are ignoring rarity of double murder (H1)
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The tragic case of Sally Clark
• ERROR 3: ignoring odds of alternative (also very rare)
– Single child murder v. rare (~30 cases a year) BUT generally significant
family/social problems i.e. NOT like the Clarks.
– P(1 murder) ~ 30:650000 i.e. 1:21700
– Double MUCH rarer, BUT P(2nd|1st murder) ~ 200 x more likely given first,
so P(H1|D) ~ (1/21700* 200/21700) ~ 1:2.4M
• So, two very rare events, but double murder ~ 10 x rarer than
double SIDS
• So P(H1|D) / P(H2|D)?
– P (murder) : P (cot death) ~ 1:10 i.e. 10 x more likely to be double SIDS
– Says nothing about guilt & innocence, just relative probability
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The tragic case of Sally Clark
• Sally Clark acquitted in 2003 after 2nd appeal (but not on
statistical fallacies) after 3 yrs in prison, died of alcohol
poisoning in 2007
– Meadows “Law” redux: triple murder v triple SIDS?
• In fact, P(triple murder | 2 previous) : P(triple SIDS| 2 previous) ~ ((21700 x
123) x 10) / ((1300 x 228) x 50) = 1.8:1
• So P(triple murder) > P(SIDS) but not by much
• Meadows’ ‘Law’ should be:
– ‘when three sudden deaths have occurred in the same family, statistics give no
strong indication one way or the other as to whether the deaths are more or less
likely to be SIDS than homicides’
From: Hill, R. (2004) Multiple sudden infant deaths – coincidence or beyond coincidence, Pediatric and
Perinatal Epidemiology, 18, 320-326 (http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf)
49
Common errors: reversed conditional
• After Stewart (1996) & Gauch (2003: 212):
– Boy? Girl? Assume P(B) = P(G) = 0.5 and independent
– For a family with 2 children, what is P that other is a girl, given that
one is a girl?
• 4 possible combinations, each P(0.25): BB, BG, GB, GG
• Can’t be BB, and in only 1 of 3 remaining is GG possible
• So P(B):P(G) now 2:1
– Using Bayes’ Theorem: X = at least 1 G, Y = GG
– P(X) = ¾ and so P(X ÇY ) P(X) = 14 43 =1 3
Stewart, I. (1996) The Interrogator’s Fallacy, Sci. Am., 275(3), 172-175.
Common errors: reversed conditional
• Easy to forget that order does matter with
conditional Ps
– As P ( X ÇY ) = P (Y Ç X ) and P ( X ÈY ) = P (Y È X )but
– P ( X |Y ) ¹ P (Y | X ) as this is cause & effect
– Gauch (2003) notes use of “when” in incorrectly phrasing Q: For a family
with 2 children, what is P that other is a girl, when one is a girl?
– P(X when Y) not defined
– It is not P(X|Y), nor is it P(Y|X) or even P(X AND Y)
Stewart, I. (1996) The Interrogator’s Fallacy, Sci. Am., 275(3), 172-175.
Common errors: reversed conditional
• Relates to Prosecutor’s Fallacy again
– Stewart uses DNA match example
– What is P(match) i.e. prob. suspect’s DNA sample matches that from crime
scene, given they are innocent?
– But this is wrong question – SHOULD ask:
– What is P(innocent) i.e. prob. suspect is innocent, given a DNA match?
• Note Bayesian approach – we can’t calculate likelihood of innocence (1st case), but we can
estimate likelihood of DNA match, given priors
• Evidence: DNA match of all markers P(match|innocent) = 1:1000000.
• BUT question jury must answer is P(innocent|match). Priors?
– Genetic history and structure of population of possible perpetrators
– Typically means evidence about as strong as you get from match using half genetic
markers, but ignoring population structure
– Evidence combines mulitplicatively, so strength goes up as ~ (no. markers)1/2
Stewart, I. (1996) The Interrogator’s Fallacy, Sci. Am., 275(3), 172-175.
Common errors: reversed conditional
• If P(innocent|match) ~ 1:1000000 then P(match|innocent) ~ 1:1000
• Other priors? Strong local ethnic identity? Many common ancestors
within 1-200 yrs (isolated rural areas maybe)?
• P(match|innocent) >> 1:1000, maybe 1:100
• Says nothing about innocence, but a jury must consider whether the
DNA evidence establishes guilt beyond reasonable doubt
Stewart, I. (1996) The Interrogator’s Fallacy, Sci. Am., 275(3), 172-175.