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Think Bayes - Bayesian Statistics Made Simple.txt
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think O
bayes O
bayesian O
statistics O
made O
simple O
version O
1.0.9 O
think O
bayes O
bayesian O
statistics O
made O
simple O
version O
1.0.9 O
allen O
b. O
downey O
green O
tea O
press O
needham O
, O
massachusetts O
copyright O
© O
2012 O
allen O
b. O
downey O
. O
green O
tea O
press O
9 O
washburn O
ave O
needham O
ma O
02492 O
permission O
is O
granted O
to O
copy O
, O
distribute O
, O
and/or O
modify O
this O
document O
under O
the O
terms O
of O
the O
creative O
commons O
attribution-noncommercial O
3.0 O
unported O
license O
, O
which O
is O
available O
at O
http O
: O
//creativecommons.org/ O
licenses/by-nc/3.0/ O
. O
preface O
0.1 O
my O
theory O
, O
which O
is O
mine O
the O
premise O
of O
this O
book O
, O
and O
the O
other O
books O
in O
the O
think O
x O
series O
, O
is O
that O
if O
you O
know O
how O
to O
program O
, O
you O
can O
use O
that O
skill O
to O
learn O
other O
topics O
. O
most O
books O
on O
bayesian O
statistics O
use O
mathematical O
notation O
and O
present O
ideas O
in O
terms O
of O
mathematical O
concepts O
like O
calculus O
. O
this O
book O
uses O
python O
code O
instead O
of O
math O
, O
and O
discrete O
approximations O
instead O
of O
con- O
tinuous O
mathematics O
. O
as O
a O
result O
, O
what O
would O
be O
an O
integral O
in O
a O
math O
book O
becomes O
a O
summation O
, O
and O
most O
operations B
on O
probability B
distributions O
are O
simple O
loops O
. O
i O
think O
this O
presentation O
is O
easier O
to O
understand O
, O
at O
least O
for O
people O
with O
pro- O
gramming O
skills O
. O
it O
is O
also O
more O
general O
, O
because O
when O
we O
make O
modeling B
decisions O
, O
we O
can O
choose O
the O
most O
appropriate O
model O
without O
worrying O
too O
much O
about O
whether O
the O
model O
lends O
itself O
to O
conventional O
analysis O
. O
also O
, O
it O
provides O
a O
smooth O
development O
path O
from O
simple O
examples O
to O
real- O
world O
problems O
. O
chapter O
3 O
is O
a O
good O
example O
. O
it O
starts O
with O
a O
simple O
ex- O
ample O
involving O
dice B
, O
one O
of O
the O
staples O
of O
basic O
probability B
. O
from O
there O
it O
proceeds O
in O
small O
steps O
to O
the O
locomotive B
problem I
, O
which O
i O
borrowed O
from O
mosteller O
’ O
s O
fifty O
challenging O
problems O
in O
probability B
with O
solutions O
, O
and O
from O
there O
to O
the O
german O
tank O
problem O
, O
a O
famously O
successful O
application O
of O
bayesian O
methods O
during O
world O
war O
ii O
. O
0.2 O
modeling B
and O
approximation O
most O
chapters O
in O
this O
book O
are O
motivated O
by O
a O
real-world O
problem O
, O
so O
they O
involve O
some O
degree O
of O
modeling O
. O
before O
we O
can O
apply O
bayesian O
methods O
( O
or O
any O
other O
analysis O
) O
, O
we O
have O
to O
make O
decisions O
about O
which O
parts O
of O
the O
vi O
chapter O
0. O
preface O
real-world O
system O
to O
include O
in O
the O
model O
and O
which O
details O
we O
can O
abstract O
away O
. O
for O
example O
, O
in O
chapter O
7 O
, O
the O
motivating O
problem O
is O
to O
predict O
the O
winner O
of O
a O
hockey B
game O
. O
i O
model O
goal-scoring O
as O
a O
poisson O
process B
, O
which O
implies O
that O
a O
goal O
is O
equally O
likely O
at O
any O
point O
in O
the O
game O
. O
that O
is O
not O
exactly O
true O
, O
but O
it O
is O
probably O
a O
good O
enough O
model O
for O
most O
purposes O
. O
in O
chapter O
12 O
the O
motivating O
problem O
is O
interpreting O
sat O
scores O
( O
the O
sat O
is O
a O
standardized B
test I
used O
for O
college O
admissions O
in O
the O
united O
states O
) O
. O
i O
start O
with O
a O
simple O
model O
that O
assumes O
that O
all O
sat O
questions O
are O
equally O
diffi- O
cult O
, O
but O
in O
fact O
the O
designers O
of O
the O
sat O
deliberately O
include O
some O
questions O
that O
are O
relatively O
easy O
and O
some O
that O
are O
relatively O
hard O
. O
i O
present O
a O
second O
model O
that O
accounts O
for O
this O
aspect O
of O
the O
design O
, O
and O
show O
that O
it O
doesn O
’ O
t O
have O
a O
big O
effect O
on O
the O
results O
after O
all O
. O
i O
think O
it O
is O
important O
to O
include O
modeling B
as O
an O
explicit O
part O
of O
problem O
solving O
because O
it O
reminds O
us O
to O
think O
about O
modeling B
errors O
( O
that O
is O
, O
errors O
due O
to O
simplifications O
and O
assumptions O
of O
the O
model O
) O
. O
many O
of O
the O
methods O
in O
this O
book O
are O
based O
on O
discrete O
distributions O
, O
which O
makes O
some O
people O
worry O
about O
numerical O
errors O
. O
but O
for O
real-world O
prob- O
lems O
, O
numerical O
errors O
are O
almost O
always O
smaller O
than O
modeling B
errors O
. O
furthermore O
, O
the O
discrete O
approach O
often O
allows O
better O
modeling B
decisions O
, O
and O
i O
would O
rather O
have O
an O
approximate O
solution O
to O
a O
good O
model O
than O
an O
exact O
solution O
to O
a O
bad O
model O
. O
on O
the O
other O
hand O
, O
continuous O
methods O
sometimes O
yield O
performance O
advantages—for O
example O
by O
replacing O
a O
linear- O
or O
quadratic-time O
compu- O
tation O
with O
a O
constant-time O
solution O
. O
so O
i O
recommend O
a O
general O
process B
with O
these O
steps O
: O
1. O
while O
you O
are O
exploring O
a O
problem O
, O
start O
with O
simple O
models O
and O
im- O
plement O
them O
in O
code O
that O
is O
clear O
, O
readable O
, O
and O
demonstrably O
correct O
. O
focus O
your O
attention O
on O
good O
modeling B
decisions O
, O
not O
optimization B
. O
2. O
once O
you O
have O
a O
simple O
model O
working O
, O
identify O
the O
biggest O
sources O
of O
error B
. O
you O
might O
need O
to O
increase O
the O
number O
of O
values O
in O
a O
discrete O
approximation O
, O
or O
increase O
the O
number O
of O
iterations O
in O
a O
monte O
carlo O
simulation B
, O
or O
add O
details O
to O
the O
model O
. O
3. O
if O
the O
performance O
of O
your O
solution O
is O
good O
enough O
for O
your O
applica- O
tion O
, O
you O
might O
not O
have O
to O
do O
any O
optimization B
. O
but O
if O
you O
do O
, O
there O
are O
two O
approaches O
to O
consider O
. O
you O
can O
review O
your O
code O
and O
look O
0.3. O
working O
with O
the O
code O
vii O
for O
optimizations O
; O
for O
example O
, O
if O
you O
cache B
previously O
computed O
re- O
sults O
you O
might O
be O
able O
to O
avoid O
redundant O
computation O
. O
or O
you O
can O
look O
for O
analytic O
methods O
that O
yield O
computational O
shortcuts O
. O
one O
benefit O
of O
this O
process B
is O
that O
steps O
1 O
and O
2 O
tend O
to O
be O
fast O
, O
so O
you O
can O
explore O
several O
alternative O
models O
before O
investing O
heavily O
in O
any O
of O
them O
. O
another O
benefit O
is O
that O
if O
you O
get O
to O
step O
3 O
, O
you O
will O
be O
starting O
with O
a O
ref- O
erence O
implementation B
that O
is O
likely O
to O
be O
correct O
, O
which O
you O
can O
use O
for O
regression B
testing I
( O
that O
is O
, O
checking O
that O
the O
optimized O
code O
yields O
the O
same O
results O
, O
at O
least O
approximately O
) O
. O
0.3 O
working O
with O
the O
code O
the O
code O
and O
sound O
samples O
used O
in O
this O
book O
are O
available O
from O
https O
: O
// O
github.com/allendowney/thinkbayes O
. O
git O
is O
a O
version O
control O
system O
that O
allows O
you O
to O
keep O
track O
of O
the O
files O
that O
make O
up O
a O
project O
. O
a O
collection O
of O
files O
under O
git O
’ O
s O
control O
is O
called O