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€? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? ð›Ð 2 " !q likelihood é ü©ñÒMbP? :bÛÅ ;bÛÅ ð›Ð 2 " ñœ prior é ü©ñÒMbP? :bÛÅ :bÛÅ €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? €? ð›Ð 5 d TXET0RGI 7 d Notebook0 Notebook0 Notebook0 Author: John Bechhoefer (johnb 'at' sfu.ca)
Date: March 15, 2009
v. 1.0
Illustration of formation of beliefs using Bayesian inference.
The program simulates coin flipping and is inspired by an
example from the book of D. S. Sivia and J. Skilling, "Data
Analysis: A Bayesian Tutorial" 2nd ed.
(Oxford University Press, Oxford, 2006).
Reset: unbiased corresponds to a
uniform prior while Reset: biased corresponds to a level of
bias equivalent to having seen 80 heads out of 100 flips.
As you make individual flips, beliefs about the coin gradually
build up. (The flips are done using a fair coin.) Note how it
takes many more flips to "overcome" a biased prior than with
a uniform prior. But with enough data, beliefs do change!
If you are impatient, you can do 100 flips at once using
the Flip 100 button. R H
Geneva , xU 9ù x H H d dg(ü H H Ø( d h F ÿ H H H H $ $ RDuÉ ¼ R MPSR WMT1 *íÿÿ ˆÿ îÿÿ L$uÉ €ÿÿ È(uÉ // Platform=Macintosh, IGORVersion=6.100, architecture=Intel
Silent 101 // use | as bitwise or -- not comment.
DefaultFont "Geneva"
MoveWindow/P 19,52,525,663
MoveWindow/C 2,637,1206,794
CoinFlipsGraph()
OpenNotebook/N=Notebook0/W=(5,44,341,376)/J=54897 "Notebook0"
Window CoinFlipsGraph() : Graph
PauseUpdate; Silent 1 // building window...
Display /W=(341,44,1271,623) posterior
ModifyGraph width=792,height=432
ModifyGraph mode=7
ModifyGraph lSize=2
ModifyGraph hbFill=5
ModifyGraph usePlusRGB=1
ModifyGraph plusRGB=(65535,32768,32768)
ModifyGraph nticks=2
ModifyGraph fSize=18
ModifyGraph axOffset(bottom)=0.25
ModifyGraph manTick(left)={0,5,0,0},manMinor(left)={4,0}
Label left "\\Z24Probabilty density"
Label bottom "\\Z24Probability of Heads"
SetAxis left -0.1,12
TextBox/C/N=mytext0/F=0/A=MC/X=-3.01/Y=45.83 ""
ControlBar 48
Button button0,pos={14,8},size={60,30},proc=ButtonProc,title="Flip 1",fSize=18
Button button1,pos={269,8},size={150,30},proc=ButtonProc_1,title="Reset: unbiased"
Button button1,fSize=18
Button button2,pos={444,8},size={130,30},proc=ButtonProc_2,title="Reset: Biased"
Button button2,fSize=18
ValDisplay valdisp0,pos={774,9},size={127,25},title="N flips ",fSize=18
ValDisplay valdisp0,limits={0,0,0},barmisc={0,1000},value= #"N"
ValDisplay valdisp1,pos={624,10},size={135,25},title="N heads ",fSize=18
ValDisplay valdisp1,limits={0,0,0},barmisc={0,1000},value= #"Nh"
Button button3,pos={110,7},size={90,30},proc=ButtonProc_3,title="Flip 100"
Button button3,fSize=18
EndMacro
#pragma rtGlobals=1 // Use modern global access method.
function MCinitialize()
SetAxis left -0.1,12
variable /g N=0,Nh=0 // Nh = number of heads; N = number of flips
string /g flip="" // will be "H" when you flip heads, "T" when you flip tails
textbox/C/N=mytext0 flip
make /o/n=1001 prior,likelihood=0,posterior=0
SetScale/I x 0,1,"", prior, likelihood,posterior
prior = 1 // normalized uniform distribution
posterior = prior
end
function MCinitialize_biased()
SetAxis left -0.1,12
variable /g N=0,Nh=0 // Nh = number of heads; N = number of flips
string /g flip="Biased prior + "
textbox/C/N=mytext0 flip
make /o/n=1001 prior,likelihood=0,posterior=0
SetScale/I x 0,1,"", prior, likelihood,posterior
prior = StatsBinomialPDF(80,x,100) // biased distribution for initial condition
variable a = area(prior); prior /=a; // normalize it
posterior = prior
end
function MCflip(Nf) // flips a coin Nf times (for now, Nf not implemented, =1)
variable Nf
Nvar Nh,N // Nh = number of heads; N = number of flips
Svar flip
make /o/n=(Nf) fliptemp=round(enoise(0.5)+0.5) // 1 for heads, 0 for tails
variable deltaNh = sum(fliptemp)
killwaves fliptemp
if (Nf == 1) // makes HTHH... display string when flipping once
if (deltaNh == 1)
flip +="H"
else
flip += "T"
endif
else
flip = "" // get rid of display when doing big flips
SetAxis /A left // turn off fixed scaling
endif
Nh += deltaNh // update number of heads
N += Nf // update number of flips
wave prior, likelihood,posterior
likelihood = StatsBinomialPDF(deltaNh,x,Nf) // the prob. dist IF you know Ph
posterior = likelihood*prior // Bayesian update
variable A = area(posterior)
posterior /= A // normalize
prior = posterior // update prior for next flip(s)
textbox/C/N=mytext0 "\Z24" + flip
end
Function ButtonProc_1(ctrlName) : ButtonControl
String ctrlName
MCinitialize()
End
Function ButtonProc_2(ctrlName) : ButtonControl
String ctrlName
MCinitialize_biased()
End
Function ButtonProc(ctrlName) : ButtonControl
String ctrlName
MCflip(1)
End
Function ButtonProc_3(ctrlName) : ButtonControl
String ctrlName
MCflip(100)
End
Window CoinFlipsGraph() : Graph
PauseUpdate; Silent 1 // building window...
Display /W=(341,44,1271,623) posterior
ModifyGraph width=792,height=432
ModifyGraph mode=7
ModifyGraph lSize=2
ModifyGraph hbFill=5
ModifyGraph usePlusRGB=1
ModifyGraph plusRGB=(65535,32768,32768)
ModifyGraph nticks=2
ModifyGraph fSize=18
ModifyGraph axOffset(bottom)=0.25
ModifyGraph manTick(left)={0,5,0,0},manMinor(left)={4,0}
Label left "\\Z24Probabilty density"
Label bottom "\\Z24Probability of Heads"
SetAxis left -0.1,12
TextBox/C/N=mytext0/F=0/A=MC/X=-3.01/Y=45.83 ""
ControlBar 48
Button button0,pos={14,8},size={60,30},proc=ButtonProc,title="Flip 1",fSize=18
Button button1,pos={269,8},size={150,30},proc=ButtonProc_1,title="Reset: unbiased"
Button button1,fSize=18
Button button2,pos={444,8},size={130,30},proc=ButtonProc_2,title="Reset: Biased"
Button button2,fSize=18
ValDisplay valdisp0,pos={774,9},size={127,25},title="N flips ",fSize=18
ValDisplay valdisp0,limits={0,0,0},barmisc={0,1000},value= #"N"
ValDisplay valdisp1,pos={624,10},size={135,25},title="N heads ",fSize=18
ValDisplay valdisp1,limits={0,0,0},barmisc={0,1000},value= #"Nh"
Button button3,pos={110,7},size={90,30},proc=ButtonProc_3,title="Flip 100"
Button button3,fSize=18
EndMacro