660
L. Lyons
means that the hidden node i is ‘on’ (i.e. r i = 1) if
w xi x + w yi y + t i > 0,
(15.11)
or ‘off’ otherwise. Thus the boundary between events having r i on or off is a straight
line in the (x, y) plane (see Fig. 15.3c). With suitable values for the weights and
thresholds for the three hidden nodes, there will be three straight line boundaries
in the (x, y) plane shown in Fig. 15.3d. Finally, to produce the “and” of these three
conditions, the weights w jo (from the hidden node j to the output node o) and the
output threshold t o can be set as
w 1o = w 2o = w 3o = 0.4 t o = −1.0
(15.12)
to ensure that the output will be “on” only if the three hidden layers are all “on”,
i.e. that the selected input values are inside the triangular region in the (x.y) plane.
With β set at a lower level, the contour for the selected region will be smoother with
rounded corners, rather than being triangular.
It would be useful to have a similar understanding of how deep networks operate.
Tishby[50] has provided some insight on what happens in the hidden layers of a deep
neural network during the training procedure.
15.4 Parameter Determination
For a single parameter (e.g. the branching ratio for H → μ + μ − ) the parameter
range could be either a 2-sided interval or just an upper limit, at some confidence
level (typically 68% for 2-sided intervals, but usually 90% and 95% for upper
limits). For two parameters (e.g. mass and production rate for some new particle
X that decays to a top pair), their acceptable values could be those inside some 2dimensional confidence region. Alternatively an upper limit or 2-sided region for
one parameter as a function of the other could be defined; these are known as a
Raster Scan.
An upper limit on 2-variables is not a well-defined concept.
15.4.1 Upper Limits
Most recent searches for new phenomena have not found any evidence for exciting
new physics. Examples from particle physics include searches for SUSY particles,
dark matter, etc.; attempts to find substructure of quarks or leptons; looking for extra
spatial dimensions; measuring the mass of the lightest neutrino; etc. Rather than
just saying that nothing was found, it is more useful to quote an upper limit on the
sought-for effect, as this could be useful in ruling out some theories. For example in
L. Lyons
means that the hidden node i is ‘on’ (i.e. r i = 1) if
w xi x + w yi y + t i > 0,
(15.11)
or ‘off’ otherwise. Thus the boundary between events having r i on or off is a straight
line in the (x, y) plane (see Fig. 15.3c). With suitable values for the weights and
thresholds for the three hidden nodes, there will be three straight line boundaries
in the (x, y) plane shown in Fig. 15.3d. Finally, to produce the “and” of these three
conditions, the weights w jo (from the hidden node j to the output node o) and the
output threshold t o can be set as
w 1o = w 2o = w 3o = 0.4 t o = −1.0
(15.12)
to ensure that the output will be “on” only if the three hidden layers are all “on”,
i.e. that the selected input values are inside the triangular region in the (x.y) plane.
With β set at a lower level, the contour for the selected region will be smoother with
rounded corners, rather than being triangular.
It would be useful to have a similar understanding of how deep networks operate.
Tishby[50] has provided some insight on what happens in the hidden layers of a deep
neural network during the training procedure.
15.4 Parameter Determination
For a single parameter (e.g. the branching ratio for H → μ + μ − ) the parameter
range could be either a 2-sided interval or just an upper limit, at some confidence
level (typically 68% for 2-sided intervals, but usually 90% and 95% for upper
limits). For two parameters (e.g. mass and production rate for some new particle
X that decays to a top pair), their acceptable values could be those inside some 2dimensional confidence region. Alternatively an upper limit or 2-sided region for
one parameter as a function of the other could be defined; these are known as a
Raster Scan.
An upper limit on 2-variables is not a well-defined concept.
15.4.1 Upper Limits
Most recent searches for new phenomena have not found any evidence for exciting
new physics. Examples from particle physics include searches for SUSY particles,
dark matter, etc.; attempts to find substructure of quarks or leptons; looking for extra
spatial dimensions; measuring the mass of the lightest neutrino; etc. Rather than
just saying that nothing was found, it is more useful to quote an upper limit on the
sought-for effect, as this could be useful in ruling out some theories. For example in