Only one of the following two stories
is true.

Oakland First Fridays: Maya and Elsa
were with me at work yesterday, August 2^{nd}, and overheard
me making plans to go to Oakland's First Friday – a street/gallery
food/art/music/people-watching festival. Maya asked whether I was
going to take them. I said, “No, you will be with your mom
tonight.”.

Elsa - “Are we with you next Friday?”

I - “Yes.”

Elsa - “Can we go with you to Oakland
First Friday next week?”

Black Ravens and Pink Flamingoes:

Maya, Elsa and I were walking along the
bay in Redwood Shores with Heather, a colleague. It was close to dusk
and we saw a number of birds: a small number of red-winged
blackbirds, crows, moor hens and cormorants, all mostly black; and a
few killdeer, snowy egrets, a mourning dove nesting in a flowerpot at
the entrance to Heather and Chris' home, scrub jays, great blue
herons and clapper rails, all non-black. We also saw nine black
ravens. I casually said, “Hmm, looks like all ravens are black.”,
which seemed to go un-noticed at the time. Soon after, we saw another
raven, which was also black. Heather said, “Oh, that raven is
black, looks like you are right that all ravens are black.”

A short while later, Maya pointed out a
pink flamingo.

Maya said, “Look Dad, a pink
flamingo! Looks like you are even more right that all ravens are
black!”

“What in the world do pink flamingoes
have to do with black ravens?”

“Daa-ad! They do, I read it in my
'Great Philosophers' book and searched for it

online!
You said that 'All ravens are black.' That is logically equivalent to
it's contrapositive, 'All non-black objects are non-ravens.' When
Heather saw an additional raven and it was black, she pointed out
(and you did not disagree) that it was evidence in favor of your
original proposition that 'All ravens are black.' The contrapositive
I just mentioned is logically equivalent. So when we see a non-black
object (it is pink) and it is not a raven (since it is a flamingo),
you should agree that it is evidence in favor of 'All ravens are
black.'!”

“Oh god! Can't you just read some
normal book about tribes of cats or that girl Parsnip or something
who shot her little sister with an arrow? I suppose you are right!”

“Dad, okay, it seems to make sense
logically, but I don't know if it is true in some statistical sense,
all that “SQL, SQL, standard error and p-value” stuff you keep
talking about now.”

“Let's take a crack at it. Let's
denote

Proposition 1: “All ravens are
black.” For the sake of simplicity let's just say that we are
talking about birds and that we are trying to see if (the set of
individual birds that are) ravens have some special property that
distinguishes them from all birds (as individuals, not species).
Then, practically speaking, the contrapositive and logical equivalent
of Prop. 1 is

Proposition 2: “All non-black birds
are non-ravens.”

Prior to Heather's observation, the
evidence we had looked like

All birds of all colors |
Black |
non-Black |
All colors |

Ravens |
9 |
0 |
9 |

non-Ravens |
50 |
8 |
58 |

All birds |
59 |
8 |
67 |

Then she saw a black raven, and our
evidence changed to

All birds of all colors |
Black |
non-Black |
All colors |

Ravens |
10 |
0 |
10 |

non-Ravens |
50 |
8 |
58 |

All birds |
60 |
8 |
68 |

And we agreed that this helped verify
that prop 1 is true, in the sense that it allowed us to be more
confident that Prop. 1 is true.

Then you saw a pink flamingo, and our
evidence became

All birds of all colors |
Black |
non-Black |
All colors |

Ravens |
10 |
0 |
10 |

non-Ravens |
50 |
9 |
59 |

All birds |
60 |
9 |
69 |

And you are saying that this also
allows us to be more confident that Prop. 1 is true. Let's forget
about Prop.2 for a while. So for verifying that “All ravens are
black.”, a pink flamingo is worth as much as a black raven. I can't
think with concrete numbers, so let's use some abcedra:

All birds of all colors |
Black |
non-Black |
All colors |

Ravens |
a |
0 |
a |

non-Ravens |
b |
c |
b+c |

All birds |
a+b |
c |
a+b+c |

On the basis of this evidence, the
probability of any bird being non-black is

p = c/(a+b+c).

How much evidence is there to reject
the hypothesis that some ravens are non-black? The specific null
hypothesis in this case is

H : “Some raven-birds are
non-black.”,

since we are hypothesizing that
raven-birds are just any other individual bird, some of which are
non-black.

Now, if the above H were true, and we
make 'a' observations of birds which happen to be ravens, the
probability P of getting 0 non-black is the product of the
probabilities that each observed bird in this set of ravens is
non-black, i.e. P = (1-p)*(1-p)*... a times = (1-p)^{a}.

(Or you can use B[p,a](i) =
C(a,i)*(1-p)^(a-i)*p^i for i = 0.)

This is the P-value, the probability of
the consequence of the null hypothesis in this experiment, that we
can use to reject the null hypothesis with 1-P degree of confidence.
Recall that the smaller that P is, the more confidence we have in
rejecting the Null Hypothesis, or “accepting the validity of
proposition 1”.

So now we can ask about the relative
merits of observing a black raven or a pink flamingo. Which
additional observation reduces the P-value more? Let's use
calclueless, since I am not secretive enough to do discrete math.
What that means is we want to compare the marginal change (partial
derivative) in the P-value when we make an additional observation of
a black raven a → a + 1 vs. when we see a pink flamingo c → c +
1.

Since ln(P) = a*ln(1-p),

(d/d a) ln(P) = ln(1-p) and some work
shows that the change in the P-value by observing a black raven,
adding 1 to a:

(d/d a)P = P * ln(1-p),

which is less than 0, meaning that
observing a black raven does reduce the P-value and increases the
confidence in “All ravens are black.”!

Similarly,

(d/d c)p = (1-p)/(a + b+ c) ( > 0)

(d/d p)P = -a*P/(1 – p) (< 0) .

Simplifying, the change in the P-value
by observing a pink flamingo, adding 1 to c:

(d/d c)P = -a*P/(a+b+c),

which is less than 0, meaning that
**observing a pink flamingo also reduces the P-value and increases
the confidence in “All ravens are black.”**!

The question that now remains is wether
pink flamingoes are more valuable evidence than black ravens, i.e.
which change decreases the P-value more:

|(d/d c)P| ?> |(d/d a)P|, which is
equivalent, since P > 0, to

|(d/d c)ln(P)| ?> |(d/d a)ln(P)|

working through algebra

a/(a+b+c) ?> -ln(1-p).

Simplifying,

the condition we are looking for is

**e^(a/(a+b+c)) ?> 1+ c/(a+b)**

For a = 10, b = 50
and c = 8, it turns out that this is just marginally true!

**A pink flamingo is just as valuable as a black raven in verifying that all ravens are black!**

See the
plot below

Since most birds
are actually non-black, had we seen already a very large number of
non-black non-ravens:

All birds of all colors |
Black |
non-Black |
All colors |

Ravens |
10 |
0 |
10 |

non-Ravens |
50 |
90 |
140 |

All birds |
60 |
90 |
150 |

Then the
incremental value of seeing a pink flamingo would have been much less
than that of seeing a black raven, for two reasons, one since we
would have had a large proportion of non-black birds, the expected
proportion of non-black ravens would have been correspondingly
higher, making it all the more unlikely to see *no*
non-black ravens amongst the additional ravens. Second, as the graph
above shows, the incremental value of each non-black non-raven when
we've already seen a lot does very little to increase our confidence
that all ravens are black. For example, in the Rann of Kutch “*Another
pink flamingo, ho hum!*”

So,
Maya, does it now make statistical sense that your pink flamingo
sighting was just as important as Heather's black raven sighting for
verifying that all ravens are black?”

“Yes,
Dad, I think I want to be a doctor when I grow up.”

The
next day, we went out for lunch, and suddenly Elsa piped up, “Dad!
More black ravens! I just saw an orange chicken!”

“Can
they fly?”