------_=_NextPart_000_01BEB1A8.F7EAE890
Content-Type: text/plain;
charset="iso-8859-1"
Wang wrote:
> Now suppose that we were to press the button 1,000,000 times and
we
> observe that half of them resulted in H and half resulted in T
(and we
> could not detect a pattern in the sequence). If I then ask you to
say
> what the outcome of the 1,000,001st press will be, you would also
say "I
> don't know".
Yes, as long as current belief is concerned, the above three
situations (0:0, 2:2, and 500000:500000) are the same: "I don't know
(whether the next outcome will be H or T)."
However, what distinguishes ignorance and known probability is how
new
evidence will revise current belief. After a new H (or T) is
observed, the above three cases become different. I'd like to know
how this difference can be captured by BN.
Hello Pei
I have enclosed a small BN (ASCII file) that shows you how this works - at
least if I understand your concerns correctly. It is modelled using HUGIN,
and you may need to download a demo of HUGIN from www.hugin.dk
<http://www.hugin.dk> to run this example.
The enclosed network is as follows: The main node is our belief in the
probability of getting an H reading. Our problem is, of course, that we do
not know much about the machine David gave us, only that it is (after having
pressed the button) able to give us an H or a T, and that the individual
outcomes are independent. Since "I don't know" what P[H] is I have assigned
a uniform distribution (discrete, though) to the node "Belief in H", that
is, P[H] can be anything in the interval [0;1]. In the discrete world I
have chosen the probability levels [0, 0.1, 0.2, ..., 0.9, 1.0]. Since I am
indifferent to any of these probability levels I guess this describes well
that "I don't know" - whatever that may mean. Moreover, I have added 11
observation nodes, that allows me to gain more knowledge of my belief in
P[H] while I play with the machine. The observation nodes are, of course,
children of the node "Belief in H". Since I have assigned a uniform
distribution to my "Belief in H" the marginal probability of the outcome is
P[H] = P[T] = 0.5.
If you have installed the demo version of HUGIN, saved the button.net file
to a directory and double clicked on the file you are now running the HUGIN
program. You see the assigned probability distributions. If you click on
the lightning icon (Run), we may check how our belief in H changes when more
results become available.
First press the third lower icon (Expand node list) so that we easily can
see all the marginal probabilities. To make things somewhat easier, we just
want to turn Auto Propagation on: Press Alt+Enter select "Use Auto
Propagation".
Now we may run the game. Double click on the observation result you get
from David's button machine, and notice how your "Belief in H" changes when
you have the (0:0), (2:2), and (5:5) observation. Isn't this great fun?
Peter
<<button.net>>
Peter Friis Hansen
Associate Professor, Ph.D.
Department of Naval Architecture and Offshore Engineering,
Build. 101E, Technical University of Denmark,
DK-2800 Lyngby, Denmark
Tel: + 45 45 25 13 88
Fax: +45 45 88 43 25
Email: pfh@ish.dtu.dk
Web: http://www.ish.dtu.dk/pfh/
------_=_NextPart_000_01BEB1A8.F7EAE890
Content-Type: application/octet-stream;
name="button.net"
Content-Transfer-Encoding: quoted-printable
Content-Disposition: attachment;
filename="button.net"
net
{
node_size =3D (80 40);
HR_Grid_X =3D "10";
HR_Grid_Y =3D "10";
HR_Grid_GridSnap =3D "1";
HR_Grid_GridShow =3D "0";
HR_Font_Name =3D "Arial";
HR_Font_Size =3D "-12";
HR_Font_Weight =3D "400";
HR_Font_Italic =3D "0";
HR_Propagate_Auto =3D "0";
HR_Propagate_AutoSum =3D "1";
HR_Propagate_AutoNormal =3D "1";
HR_Compile_TriangMethod =3D "0";
HR_Compile_Compress =3D "0";
HR_Compile_Approximate =3D "0";
HR_Compile_ApproxEpsilon =3D "0.00001";
HR_Monitor_AutoUpdGraph =3D "0";
HR_Monitor_GraphPrecision =3D "100";
HR_Monitor_OpenGraph =3D "0";
HR_Monitor_InitStates =3D "5";
HR_Monitor_InitSD =3D "2";
HR_Color_DiscreteChance =3D "16";
HR_Color_ContinuosChance =3D "48";
HR_Color_Utility =3D "36";
HR_Color_Decision =3D "17";
HR_Groups_UserGroupsNo =3D "0";
HR_Groups_GroupNames =3D "";
HR_Groups_GroupColors =3D "";
}
node Obs_11
{
label =3D "Obs 11";
position =3D (550 240);
states =3D ("Head" "Tail");
}
node Obs_10
{
label =3D "Obs 10";
position =3D (540 180);
states =3D ("Head" "Tail");
}
node Obs_9
{
label =3D "Obs 9";
position =3D (510 130);
states =3D ("Head" "Tail");
}
node Obs_8
{
label =3D "Obs 8";
position =3D (470 80);
states =3D ("Head" "Tail");
}
node Obs_7
{
label =3D "Obs 7";
position =3D (410 30);
states =3D ("Head" "Tail");
}
node Obs_6
{
label =3D "Obs 6";
position =3D (330 0);
states =3D ("Head" "Tail");
}
node Obs_5
{
label =3D "Obs 5";
position =3D (240 0);
states =3D ("Head" "Tail");
}
node Obs_4
{
label =3D "Obs 4";
position =3D (150 10);
states =3D ("Head" "Tail");
}
node Obs_3
{
label =3D "Obs 3";
position =3D (70 40);
states =3D ("Head" "Tail");
}
node Obs_2
{
label =3D "Obs 2";
position =3D (20 90);
states =3D ("Head" "Tail");
}
node Obs_1
{
label =3D "Obs 1";
position =3D (0 160);
states =3D ("Head" "Tail");
}
node Belief
{
label =3D "Belief in H";
position =3D (270 270);
states =3D ("0" "0.1" "0.2" "0.3" "0.4" "0.5" "0.6" "0.7" "0.8" =
"0.9" "1");
}
potential (Obs_11 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_10 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_9 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_8 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_7 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_6 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_5 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_4 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_3 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_2 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Obs_1 | Belief)
{
data =3D (( 0 1 ) % 0
( 0.1 0.9 ) % 0.1
( 0.2 0.8 ) % 0.2
( 0.3 0.7 ) % 0.3
( 0.4 0.6 ) % 0.4
( 0.5 0.5 ) % 0.5
( 0.6 0.4 ) % 0.6
( 0.7 0.3 ) % 0.7
( 0.8 0.2 ) % 0.8
( 0.9 0.1 ) % 0.9
( 1 0 )); % 1
}
potential (Belief)
{
data =3D ( 0.05 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.05 );
}
------_=_NextPart_000_01BEB1A8.F7EAE890--