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rand
HMM (hidden
Markov Model)
:
Gibbs
sampling
init
:
·
obs evidencee timestepa
·
States
= comp .
assignments
res
incorporate it
into our
model
to
all
vars S
·
State var : random
var.
2var.
fix evid. Var .,
rand. Set
encoding belief & timestep
can't
non-evid
. Var Sample
: S-P(S/c ,
r ,
: C-p(CIS , r) Sample
: W-P(WIs , r)
getting
·
evidence var
: rand. Var
change
.
gen
. Subsea. States by
new
set
encoding obse
a
timestep
at
same
looping through non-evidvar
:
Markov ASS.
of trans.
models so Represent HMM using initial
time sample
new val. Chosen Var. to &
if we
are trying to figure
out prob
Of
G
,
all
we
we have
a
distribution,
transition model
,
gen.
new sample
need to
know is G
,
knowing
G
,
and Go
bunch of
Xi
Xn)
=
samples of
Wo. our
sensor
model
P(X : /markov-blankets (v : 1)
Amatrix
Mult
:
(mxn)
(nxP)
trans.
model
ex
: weather forecasting
·
considers downstream
gives
us
a upstream
evid.
must match
Sample wo
...
Simplies
t
states of w ,
po ① US calc prob &
form
dist of Wo , (F,
b
E
F
use that to F
Calc W.
② read below
and so on
. State var Wi (the weather
(timelaspe
on day
i
Upd)
·
evidence var Fi (the forecast
on day
i)
·
Initial distP(Wol
· Sensor model P(F : Wil
* use.
Transition
model
P(Wi+ , I Wil
for
2 things needed to def HMM
:
basic
4 11
Prob of
P
,
/Po
,
Palp
,
--
like
HMM
(transition model)
above
- evidence prob aka
becomes
sensor model
: Prob
Comput
You see some evid.
exp when
more
& time t given there's
some state & time
scomplex
Shown
parts
in green
Forward Alg
:
above for HMM
B(Wi+
)dP(fi
WilB)Wil
Wi
T
time elaspe upd
:
adv. Model's
State by one timestep
Step D
B' (Wit) =
P(W :+, (wi)B(wi)
Obs
. upd :
incorporate
new ev as
Step
B(Wi +
& P(fi + 1) Witi) B' (Wit1) M
belief dist of
time i given obs
. evid.
n f
, ...,
fi
Step
② is
C
n
B(Wi)
= P(Wilf ,...,
fi) Similar
to
exact
inf.
B' (Wil
=
P(Wilf
,...,
fi - 1) W/Baye's
Shown
Particle Filtering
Parts
in
Purple HMM
use set
of Samples (particles)
to
represent belief State
· Stores n Particles (n
/XI)
·
pred .: Sample
state from
transition
model
4 X
+ + P(X + + 1 (X + )
SRPQ
·
Upd : Obs et + 1
& weight sample
based
on evid.
W
=
P(t+ + 1)X = + 1) -
Sep
②
. Fitting data
better mean getting
higher weight
· Normalize
weight across
all
particles
·
Resample
: n
times
,
sample
w/replacement
get new
particles
avoids tracking
weighted
samples
·
repeat for next time step
·
exact interference is
infeasible when
domain
of
vars
grows too large
·
Step
②
similar to likelihood weighting
from
Bayes
net
e every iteration use trans. Model
to get a new set of
samples , upd.
Way we sample by
mult
. by prob of
Seeing Curr. evid given State of sample
If your
sample
ends up on a state
w/o
Prob given
evid. bic maybe
Yoursensoris nevergoingto give a
s
can
help you
elim
that sample
bIC
I
&
:
ci
up
..
resamp
if an evid
Varabo
a
/ >
Weint
I
W
=
P(t
1)X = +
)
= 0
,
which
weight limes
, one time for each
time step
* particle filtering is not perfectly
particles : particles
: particles
:
↑ a rficus
if you have
enn,
a
Crews
particles
to
exact prob -
i S
better forComplex S
,
a bunch
of
State var (5 or 6) and a bunch
(3 , 3) W
= .
4
of evid. Var.
Machine Learning
Perception Al
· core idea : we give machines access
to data & they
learn
for themselves
Data
is often
split into training
, val.
a
al
weighs
i
e
dataset can
be divided in to
but
should
y
E
13 , do
:
set
of features ,
X (2)
Set of classes
,
poss
(a) classify
the sample using
Training Set
: used to fit model the curr· weights
Validation Set
: used to
tune hyperparam
. class predicted
byetybetheWi
(learning
rate ,
model struct , etc
. )
1
if
activationw(X)
=
Test Set
: used to test the
entire model
S
Wif(x))
·
some
types of
machine learning prob .
y
= classify
=
-1 if activationw (X)
=
WTf(X) < 0
*IRegression
: try to est. Some
numerical Val from data
(b) Compare the pred. label y to the
& bunch
of lines on
Plot ,
find
truelabe
nothinga upd
line of best fit
ex
:
feature
of houses &
finds
line of best fit to find what
Your
weights : w
f(x)
price of each house should be 3) If you went through every
* Classification
:
try
to classify
training sample wo having
to upd.
data into discrete
classes
your weights (all samples pred
.
Pixels are features in
an corr. ) , then
terminate. Else ,
repeat
img. Of
a # &
classes
,
what Step
we're trying to Pred. (0-9) *
weights
def .
the line drawn
Clustering
: try to group Similar eX
: 2D data
data into clusters
naturally >0 Perception
is pos
·
Types
of
learning :
< perception
is neg.
3 supervised
:
training data
x 3
40 - 1 = - 1
has labels ,
e. g. classification & if the
res .
happens to be incour
ex
:
or
digits)
then we
upd. Our weight by
:
training data adding the true class *
feat.
has no
labels ,
e. g.
Clustering
4[-i]s
& don't
know exactly what
You're looking for, but
Want to see
if there is
Neural
Networks :
naturally see
what Struct
appear
·
motivation :
most
prob
are non-linear
type of
·
Common neural
network
Naive Bayes
classification
Class is the multilayer
· Goal
: Create a
model that can pred
Perception a lg.
a label
y given
features ,
where we instead of
having binary
assume all features are ind.
affected - 1 ,
I choices
after each
by label
Pred .
label y
Y feat.
node ,
we use a non-linear
ex
: Spam
filter
L activation
func.
·
y is in Espam , Ham
F. F ... Fr &
NodesStill
dot prod .
their
·
Fi in Eo , 13 is
whether word
i inputs w/ their own weight
appears in the
email rectors
· Label email based on
the higher
of these use gradient descent to
two prob:
Upd. all weights-
P(y
= nam/F ,
= f
,.. ., Fn
=
fn) backpropagation
X
G
P(y = Spam /Fi
= f ,... ,
Fn
=
fn) *
When
applying
assume that there is any
type
non-linearity , 00
of relationship blu
words allowing us to
We can
pred.
get dist
for y by multiplying proby given each more
complex classes X
of the
feature var.
=>
if we
come up
we a neural
· Generalized
: network
that is complex
spred (f
, ... ,
fn)
= argmaxP(y
=
y(F
= f , ...
Fu
=
fn)
enough ,
we can come up w/
a line that inc.
all circles
argmaxF
and no crosses
create multles
a
Class
we're
of
some
1
Planning
Xi ~
to classify
I
↓ Maximum
Likelihood EstimationLimes
each t a
a
· How to est. CPTs since we don't actually
feat.
know them
(input) ; do X 3
for each node ↑
parameter
est
. W/MLE inputs/feat . I weight
· find Prob. (CPT
val) O
= PC.
) Such
that we
maximize the likelihood of observing ist layer (made of a bunch
of nodes
Our observation ,
P
(observations (0)
eage is
network is still linear A CPT blu we're
· Ans. is actually fairly intuitive. Y S mult. # by features
& fle ,
given data (F, Y)
P(y
=
y)
=
MLE(OI(F
, y))
F. Fz ... FN
=
(H
ex. w(y
=
y) ( +o + al
eX)
Sigmond Func.
PFMEexw(y=
aka activation
·
look at
a ton of emails
that
have
already been classified for
us ,
& find each edge
(CPT),
What is
prob given that
email
isspam,ithassome word
ina
word in
it
perception
Binary
Perception
·idea:
linearlyseparates datain
a
a
(def
. by a set of weights
· if data
is linearly separable ,
the
alg. Will perfectly classify the data
· to find boundaries -w
WTf(x)
= 0
that don't have
to cross
the origin ,
incorp. a "bias" feat.
that always has val
of
Gi
· creating perception model
:
wehave a bunchofdataa
features on some O
kind F
of graph a
. Where
each
feat
. is either pos. Or Olinear
neg ., so
given just
feat. We
want to *
perception figures
out what line
build
model that can
you can draw to
take these feat.
Perfectly
div. 2
give usanest for
beclasses
one
train
valid e
test Logistics Reg. Conti inference by enum.
·
idea
: instea of simply using
WiX (binary Perception) & LPCTPC, S)
Classifying
,
applysigmoid
func.
on wiX
Prior
sampling
·
results always
blw 0
and I
:
can pred.
Prob .
Unlike
binary
4
Al
:
perceptron
rand
.
gen. Samples
·ifwecan
getasetof weights
samples discard sampstate
every
inconsistent wh
logistic
regression form can comb, of evid.
give us
a good
classification ass Calc. Prob.
for a set of data pts
.
· May have to compute
· How do we compute
Use
large # of samples
radient descent for
unlikely scenarios
·
see also
:
multiclass logistic
· eX
: gen. the following
regression ,usingsoftmaxfuna
Samples
C ,
iS ,
r ,
w
Gradient Ascent
!
Descent
:
S ,
S ,
Lr ,
w
·
given acc. measure, C P(W
=
w)
in order
to come up
S
L
S
RP(w
=
w)
=
Wbetternetworkaa
or
dec. our loss
wh
· Goal
: Want to find Param .
that maximizes obje
· For i
= 1 ,
2 , ...,
n (in topolog
fund.
or minimizes
loss
order (
func.
Sample X : from
·
if
closed-form
formula for P(X : /
Parents (Xi))
global
Optimum does
not exist ,
· Return
(X ,, Xz
,
...,
Xn)
can use gradient ascent/
descent
· Observation
: gradient is dir.
of steepest inc ... by repeatedly Rejection
Samp.
following the gradient ,
we can
X
idea
: we can
imm. Stop
chase
maxima/minima
gen.
Samples as soon
as
· Gradient ascent
: will not
they
become inconst. w/
rand.
Initialize w accept our
evid.
While
w not converged do
:
any still
discard most
w = w
GVwf(w) sample
samples but takes
end
that has
less time
to gen.
·
Gradient descent
:
inconsist.
those
& rand. Initialize w evid ;
·
Al9 :
While w not
converged do :
same for input
: evidence
w = w
GVwf
(w)
likelihood
e, ..., 2
end
& Gibbs
For i,from
Traing Neural
Networks
Parents
leat
· set
the weights to some
initial
Val.
reject
: return
& no sample
is
· input training data , run
forward pass
to generate val
gen.
in this
at all modes ,
calc. loss func.
cycle
On final output return (X 1, Xz, ..., Xn)
·
run
backwards pass-calc
. the we want P(CIr , w -
gradient of loss
w/ respect
to can
throw
away
each of weights
samples
w/ ir orTw
·
use gradient
descent
to upd.
all weights
C ,
iS ,
r ,
w
· repeat w/ more
data
i
Bayes Net
:
S ,
S ,
r ,
w
·
acyclic
graph ·
use a
dirrob
.
table to
L
C
S
encode relationships blw var. S R
Prob. of an assignment
P(X , Xz , ...,
Xn)
P(X : /Parents (Xi)
& L
W S eX : Alarm (A) goes off if there's a
:
burglary (B) or
earthquake (E) ,
res .
in John (5) or Mary (M)
calling
B
E Likelihood weighting
P(2)
logistic
fun
· CPTs
:PCBI
,
PLE
· idea : Rejection sampling
· takes
in some
input from
may reject a
lot of
to 0 and maps from
0 to P(5IA)
,
PCMIAlyv
Y
M
samples
if evid. is
Uses : · P(
j ,
m)
= Unlikely
the
if used as activation func. p(
. P(
. P(
.
P(+ j) + a)
Var. While we
which you can apply imm. P(
after a linear layer weight
sample by
don't
for get
bis ,Sa
& pred. the class
of something prob.
of evid
Ind in Bayes
Net
:
given
Parents
include bias
val
·
ex:
if ourneural
networkis
as
1) each
node is cond , ind.
Of all samples are
used!
to o , sigmoid
func will map all its ancestor
nodes (non-
· downstream
var. inf.
to a val from O to 1 descendants) in
the graph
, by upstream
var ,
but
if given-o
, it will given all of its parents upstream
var.
likely be class of O
- each
node is cond. ind.
Of
want
to see all
3 + 1000 ,
it will likely
be all other var. given
its evid .,
not just
class
of 1 b/C very
big Markov blanket (consisting that
which inf.
anything
in middle like of its
parents ,
children ,
downstream
vars.
O will be interpreted as Children's other Parents)
·
Al9 :
Probability like 0. 5 saying
take
al
at
input
: evidence
i t 's 50/50 e, ..., 2
Variable
Elim
i inv. the
var. 3
Forivid.
Logistic
Regression :
1
Clim.
hidden var.
X by
you're
hw(X)
=
(1 I
e-wax)
joining (multiplying
trying
var.
uses smaller +g + )
all
factors inv. X
to elim -Xi
= Obs.
we want to fine
tune our
factors
summing
out 4 value ,
weights , such
that when we
than Factors
: unnorm.
Prob
Set W
= W
pass in W
X ,
which is a single infer,
proportional
to actual
P(X
Parents (Xi)
layer
in NN ,
into sigmoid func
. by
enum. Prob , but doesn't sum
to
return (X,
, Xz, ...,
Xn)
,
w
we get
val from o to
< P(
+/
ike a
prob
dist-
shoulda
network w/
just I layer
more unnec. terms outside
