00:00:00
to talk about the estimation
00:00:01
maximization or em algorithm okay and
00:00:05
I'm going to illustrate this with an
00:00:07
example um most of the content in this
00:00:10
video was taken from the internet so you
00:00:12
can find the sources I think are are uh
00:00:15
sided whenever they were used and you
00:00:18
can go deeper into into this uh by
00:00:22
visiting those sources now I'm going to
00:00:25
illustrate this with an example let's
00:00:28
say you have two
00:00:31
cases and each of these cases or
00:00:34
conditions have a success or failure so
00:00:37
for example you have uh two marketing
00:00:40
campaigns in one city and you have uh
00:00:45
buyer outcome or something and then you
00:00:47
want to determine whether that outcome
00:00:49
came from campaign a or campaign B right
00:00:52
and the outcome is whether campaign a
00:00:54
succeeded campaign B uh did not succeed
00:00:57
let's say you have two treatments to
00:01:00
Medical Treatments right and you want to
00:01:02
see the rate of uh of recovery right so
00:01:06
in with treatment treatment a a number
00:01:09
of patients recovered with treatment b a
00:01:11
different number of patients recovered
00:01:13
but you don't know that you just see the
00:01:15
patients that have recovered in
00:01:17
different instances right so then you
00:01:20
can what you would like to see is well
00:01:23
which patients were given which of these
00:01:27
recoveries due to treatment a and which
00:01:29
which are due to treatment B which sets
00:01:33
um you want to see
00:01:37
um any other case where there's two
00:01:40
treatment two conditions and conditions
00:01:44
can be success or failure this is better
00:01:47
Illustrated with or traditionally
00:01:49
Illustrated with two coins say you have
00:01:51
two coins A and B okay one of these
00:01:55
coins say a for example is more likely
00:01:57
to get to give you um to to turn heads
00:02:01
when you toss the coin the other one's
00:02:03
more likely to um to fall
00:02:08
Tails okay so they're they're imbalanced
00:02:11
coins right and the thing is if I pick a
00:02:15
coin at random and I toss it which coin
00:02:18
was it can I know by say I toss coin I
00:02:22
toss one coin several times and it gives
00:02:25
me a certain percentage of heads and a
00:02:26
certain percent a certain percentage of
00:02:28
Tails can I know whether this was coin a
00:02:32
or coin B right so so this is similar
00:02:37
again to saying you know I have two
00:02:39
marketing campaigns and uh I pick one
00:02:42
buyer right and he buys the product
00:02:46
doesn't buy the product and then on 10
00:02:48
purchases there's so many of those 10
00:02:50
purchases in which the customer buys the
00:02:52
product and a few others in which the
00:02:55
custo customer doesn't it's very similar
00:02:57
to flipping a coin in some instances it
00:03:02
uh falls on heads in some instances it
00:03:04
falls on Tails so all these examples of
00:03:07
two conditions or several conditions
00:03:10
with success and failure outcomes are
00:03:11
traditionally Illustrated with coin
00:03:14
tosses so I'm going to do the same
00:03:16
here's the problem you have two coins
00:03:19
one's more likely than the other to turn
00:03:21
up heads I pick a coin I toss it several
00:03:24
times now which coin did I pick that's
00:03:27
that's what you don't know that's what
00:03:28
you're trying to determine
00:03:32
okay so we're going to do like I said
00:03:34
we're can do this five times we're going
00:03:35
to try and do this five times and this
00:03:38
uh most of material in this tutorial was
00:03:40
taken from uh Kong do and sarapin
00:03:44
bogu um they have a tutorial online
00:03:49
now let's do this five times we'll pick
00:03:52
a coin randomly we'll toss it 10 times
00:03:54
and we'll count how many heads and how
00:03:57
many
00:03:58
Tails um where in the in the 10
00:04:02
tosses then we'll get the average number
00:04:04
of heads for each coin
00:04:07
okay and we're going to do this five
00:04:09
times and that's going to be my evidence
00:04:10
this is the customer data that I see
00:04:12
this is the patient data that I
00:04:15
see so it goes like this right so I
00:04:19
picked coin a for example I picked coin
00:04:22
B for example and I toss it 10 times and
00:04:25
it fell head Tails Tails Tails heads
00:04:27
Heads Tails heads Tails heads right so
00:04:30
five heads and five tails I picked coin
00:04:32
a and I tossed it 10 times and G me it
00:04:35
gave me nine heads and one tail I picked
00:04:38
coin a again so randomly I was picking
00:04:40
these coins now I know which coins I
00:04:43
picked right so then what I do here is I
00:04:47
count for a right the um for a I just
00:04:53
count the number of heads divide by the
00:04:57
number of heads and tails and that gives
00:04:59
me me
00:05:02
0.8 for coin B I do the same thing and
00:05:06
it gives me
00:05:08
0.45 right so that is basically how I
00:05:12
obtain the average the average number of
00:05:16
heads that coin a and coin B uh can give
00:05:20
me right so again there 24 heads and six
00:05:23
tails so I to compute the the the rate
00:05:27
of heads that coin a gives me is 24 ID
00:05:30
24 + 6 so is the number of heads divided
00:05:34
by total number of heads and tails and
00:05:36
that gives me
00:05:37
80% of the tosses are going to are going
00:05:40
to have 80% of the tosses are going to
00:05:42
fall on the head side I did the same for
00:05:45
coin B and I can compute it this is very
00:05:48
easy I know which coins I picked so I
00:05:50
can compute the average for each of
00:05:53
these for each of these uh
00:05:56
cases
00:05:57
so what if we're giving giving only the
00:06:00
result of our coin tosses so we're only
00:06:02
giv say the patient data but we don't
00:06:04
know which treatment each patient was
00:06:06
exposed to what if they give us the
00:06:09
buyer Behavior but they did the say five
00:06:13
buyers with different behaviors but they
00:06:16
didn't tell us which marketing strategy
00:06:18
they were under
00:06:20
right what if they give us again only
00:06:23
the results of our toin causes can we
00:06:26
guess the percentage of heads that each
00:06:28
coin yields and moreover can we guess
00:06:30
which coin was picked for each of these
00:06:33
10 coin
00:06:35
tosses one way to think about this is to
00:06:39
let's do this iterative process let's
00:06:41
assign a random average to both both
00:06:43
coins so let's assume you know coin a
00:06:45
has 60% heads and coin B has 55% heads
00:06:49
and that is completely random I just
00:06:51
made up those
00:06:52
numbers um and then what we're going to
00:06:55
do iter iteratively is for each of the
00:06:58
five rounds of tank of 10 coin tosses
00:07:01
we're going to check the percentage of
00:07:03
heads we're going to find the
00:07:06
probability of it coming from each coin
00:07:10
so basically we're going to if we think
00:07:12
you know that the coin a has a 60%
00:07:15
probability of of falling of falling on
00:07:18
heads right then I can see well if if a
00:07:23
if if one round of the 10 coin
00:07:26
tosses uh turns up about % heads than I
00:07:30
would think it's from coin one right it
00:07:33
would make sense so with a technique
00:07:35
similar to that we're going to try to
00:07:36
find the probability of this round of 10
00:07:39
coin tosses we're going to find the
00:07:40
probability that it came from coin a or
00:07:43
coin B given the the random averages
00:07:47
that that I
00:07:49
assigned then we will compute the
00:07:51
expected number of heads using that
00:07:54
probability as a weight and we'll
00:07:56
multiply it by the number of heads this
00:07:58
might be a little cumbersome and not
00:08:01
understandable yet but uh I will explain
00:08:04
later but this is basically we're going
00:08:05
to try
00:08:07
to to say well if I think that coin a
00:08:11
turns up 60% of the times uh
00:08:15
heads and I have this coin 10 coin
00:08:18
tosses
00:08:19
well what if if it were coin a um what
00:08:24
would be the probability that it comes
00:08:26
from coin
00:08:27
a uh and what would be the the expected
00:08:32
number of heads that I would have
00:08:33
expected from this this
00:08:35
coin uh I will explain how to compute
00:08:38
that then we'll record those numbers and
00:08:42
with those numbers I will recompute new
00:08:45
means for coin a and coin B
00:08:48
so with that then I'll go back to step
00:08:51
number two Go revisit the five rounds of
00:08:53
10 coin tosses check the percentages of
00:08:55
heads and so on and so forth this seems
00:08:57
like I'm doing the same thing in circles
00:09:00
but really it will be
00:09:02
converging to the
00:09:04
actual means uh to the actual
00:09:07
proportions of heads for coin A and B
00:09:10
you will see that this numbers this is
00:09:11
not circular this thing is actually
00:09:14
changing a little bit and the key to the
00:09:17
change is in this middle step
00:09:19
here okay Computing the expected number
00:09:22
of heads okay this is the key to the
00:09:24
change to the convergence to the actual
00:09:28
real proportion of heads for coin a and
00:09:31
coin
00:09:33
B first we need to know a little bit
00:09:36
about coin tosses okay uh because for
00:09:40
that key step that I that I mentioned
00:09:42
we're going to use what's called a
00:09:44
probability distribution and this sounds
00:09:47
uh like a hard term to grasp but it is
00:09:51
not and I want to explain that very
00:09:53
quickly here
00:09:55
so let's see how do coin tosses behave
00:09:59
coin toss behave like this if I have one
00:10:02
coin let's focus on this on this example
00:10:05
over here okay if I have one
00:10:08
coin and I toss it
00:10:11
right one time is going to fall there's
00:10:14
the probability that it falls on Tails
00:10:17
which is zero heads these numbers at the
00:10:19
bottom indicates the number of heads
00:10:21
zero heads there's one case in which it
00:10:24
will do that and the number of heads
00:10:27
well I can toss it and it can fall on
00:10:29
heads right so there's also one one pro
00:10:33
possibility that one possibility that it
00:10:35
has one one head right that it falls on
00:10:39
heads this is basically as you see these
00:10:41
are two squares is 50% that it has no
00:10:44
heads 50% that it falls on heads if I
00:10:48
have one
00:10:50
toss now if I flip two coins right I can
00:10:54
get for example the first coin heads the
00:10:57
second coin heads or the first coin
00:10:59
heads the second coin Tails or the first
00:11:02
coin tails the second coils head or both
00:11:06
falling on Tails right so if we count if
00:11:10
those are my possibilities and if we
00:11:12
count how many how many coin of these
00:11:15
coin tosses that I can possibly do how
00:11:17
many coin tosses can fall can have
00:11:20
zero uh heads in them only one the case
00:11:24
in which I toss the two coins and they
00:11:25
both fail on
00:11:27
Tails how many cases of my coin tosses
00:11:31
can I have where there's two heads well
00:11:34
also one the case in which I toss the
00:11:36
first coin and its heads and the second
00:11:37
coin and its heads right so then I I saw
00:11:41
two heads in my two coin in my flipping
00:11:43
of two
00:11:44
coins now what's the case in which you I
00:11:47
can see one head in this flip well
00:11:49
there's two cases when head when the
00:11:51
first coin Falls in heads and the second
00:11:53
one on Tails or when the first one falls
00:11:56
on tails and the second one falls on
00:11:58
heads there's two cases in which I can
00:12:01
see uh um in which I can
00:12:04
see head one head right so to recap now
00:12:10
if we look at these probabilities
00:12:11
there's 25% chance that I will get no
00:12:15
heads 25% chance that I will get one
00:12:18
head and 50% chance that I will see one
00:12:23
head um two heads did I I mean the here
00:12:27
25% chance that I see two heads now if I
00:12:30
flip three
00:12:32
coins the combinations are a few more
00:12:35
right it can have the first one fall
00:12:37
heads the second one tails the third one
00:12:40
tails the first one fall heads the
00:12:43
second heads the third one tails and so
00:12:45
on and so forth all combinations right
00:12:47
if I analyze these combinations I can
00:12:50
see that there's one chance in W in
00:12:52
which I get no heads which is where the
00:12:55
three coins fall tails and there's one
00:12:58
chance in which I see three heads in
00:13:01
which all three coins fall heads and
00:13:04
there's equal number of chances that I
00:13:06
see one head or two heads and so on and
00:13:10
so forth if I do do this for uh five
00:13:13
coins this is what it's looking like
00:13:15
right this is what my counts are looking
00:13:17
like and you can see that this starts
00:13:20
forming a curve right like a a
00:13:24
bell-shaped curve
00:13:26
here
00:13:27
right
00:13:31
in the the the the N the the larger the
00:13:35
number of coins that I toss and
00:13:38
interestingly if I toss five coins for
00:13:41
example in this
00:13:43
five and I think that the coins are
00:13:46
balanced so there's 50% chance of
00:13:48
getting heads and tails right 50% chance
00:13:51
then if I TOS five coins well 50% of
00:13:54
five is you know between two and three
00:13:56
and the vast majorities of of heads the
00:13:59
vast majorities of Trials I will see two
00:14:02
or three heads this is what this graph
00:14:04
is
00:14:05
indicating this is how the coin toss
00:14:08
behaves
00:14:11
now this is called a binomial
00:14:14
distribution okay many cases with with
00:14:17
uh uh success or failure with a certain
00:14:20
probability of success will look like
00:14:23
this
00:14:25
okay so for example if I cost 15 coins
00:14:30
and the probability of heads is 0.5 so
00:14:32
basically it's a fair coin right I will
00:14:35
see 15 what's half of 15 is about around
00:14:38
seven and eight right so I see the bulk
00:14:44
of of the
00:14:46
heads between seven and eight heads
00:14:49
there are rare cases few cases in which
00:14:51
I see you know 12 13 or 14 heads or 15
00:14:56
they're rare cases in which I see
00:14:59
say between four 3 2 one or no heads at
00:15:03
all right so these are these are
00:15:07
the this is how the distribution behaves
00:15:11
now if the coin is not
00:15:14
fair excuse me for example the
00:15:17
probability of heads is only 20% so it
00:15:19
always falls on Tails this curve is
00:15:24
skewed okay it's not the Bell shape
00:15:27
curve that you saw earlier but it's CED
00:15:30
and it is likely that I will see in this
00:15:32
15 coin tosses somewhere in the vicinity
00:15:35
of two three or four heads only for the
00:15:38
most part it's going to be very rare
00:15:41
that I see seven or eight heads okay
00:15:45
because the probability of heads is
00:15:48
lower this is how coin tosses behave now
00:15:52
with this in
00:15:54
mind um with this in
00:15:57
mind we will see that there are formulas
00:16:01
to compute say for example well what's
00:16:03
the probability of heads if if the
00:16:06
probability of heads is 0.2 and um I
00:16:10
don't
00:16:11
know I have 15 uh coin tosses or 100
00:16:15
coin tosses well what's the probability
00:16:18
that I see for
00:16:19
example seven heads there are formulas
00:16:23
to compute that and we're going to use
00:16:25
them so let's go back to our example
00:16:29
so the five rounds of 10 coin tosses so
00:16:32
I
00:16:33
just uh did 10 coin I picked randomly a
00:16:36
coin without knowing this time without
00:16:38
knowing which coin it was and I tossed
00:16:41
it 10 times right and for example the
00:16:44
first time I picked a coin and and I
00:16:46
tossed it it
00:16:48
yielded this number of heads and and
00:16:50
Tails right the second time it yielded
00:16:53
the the rest and so on and so forth
00:16:57
so if that's
00:17:00
the let's say that's the case now and I
00:17:04
made up absolutely made up that coin a
00:17:09
will have 60% chance of heads and coin B
00:17:11
will have uh 55 55% chance of uh of
00:17:17
taals
00:17:20
so this is wrong this should be a little
00:17:24
uh a 55 not a
00:17:26
five now
00:17:30
these are the tosses right so for each
00:17:32
coin these are my 10 tosses for each
00:17:34
coin that I picked coin A and B I don't
00:17:36
know which one of these was coin a was
00:17:38
was coin B but I just saw the data
00:17:41
basically this is saying I have my
00:17:44
patient data I have my buyer data but I
00:17:47
don't know which marketing campaign I
00:17:48
don't know which treatment okay and I
00:17:51
and and and also I don't know what's the
00:17:53
mean ratio of success for the treatment
00:17:56
for the marketing campaign for the coin
00:17:57
toss so I just made up these
00:18:00
two now let's take the first round this
00:18:03
one the round number one there's five
00:18:05
heads and five tails so the proportion
00:18:08
is 5 over 10 heads and 5 over 10 Tails
00:18:12
now we'll compute the likelihood that it
00:18:14
was from coin a and coin B using the
00:18:16
binomial
00:18:18
distribution okay so to compute the
00:18:21
likelihood let me go back
00:18:23
to to
00:18:26
um to the this graph for example
00:18:31
okay in this
00:18:34
graph if I find for example let's say my
00:18:38
coin has um 50% chance if I find five
00:18:42
heads so
00:18:45
here five heads well what is the
00:18:49
probability well the question is what is
00:18:50
the probability that these five heads
00:18:54
came from a coin that behaves like
00:18:57
this as supposed to for example a coin
00:18:59
that behaves like
00:19:01
this right where most of the most of the
00:19:06
of the of the times it will produce five
00:19:09
heads right so what's the probability
00:19:11
that I found five heads in a coin in a
00:19:13
Fair coin for example in 10 tosses of a
00:19:16
fair coin that is what I need to ask
00:19:18
myself and then with the probability of
00:19:21
0.6 what's the probability that I found
00:19:23
five heads in a coin that behaves like a
00:19:26
coin that has a probability of 0.
00:19:30
six and for that there's um there's a
00:19:35
little formula here which is the
00:19:36
binomial distribution probability
00:19:42
okay and the idea here is that this
00:19:44
formula will give us if my coin has
00:19:47
probability Theta on N trails and in
00:19:51
this case 10 trails with K successes
00:19:54
that's the number of heads right five
00:19:57
this is the probability
00:19:59
that these K successes came from a coin
00:20:06
with with this probability okay and the
00:20:10
question is well what's the probability
00:20:11
that it came from coin a and what's the
00:20:13
probability that it came from coin B
00:20:15
right and the one that has the higher
00:20:17
probability so far is the most likely
00:20:18
coin to have been picked that's the
00:20:21
that's the
00:20:22
idea
00:20:26
so now we move down so we have let's
00:20:29
let's compute these
00:20:30
things so because oh by the way a little
00:20:33
math here because this thing this term
00:20:36
here is constant I'm going to ignore it
00:20:38
okay because it it won't affect the
00:20:40
computation it won't affect will affect
00:20:42
the computation it won't affect which
00:20:43
one's bigger which one's lower and it
00:20:45
won't affect the the step that I want to
00:20:47
do
00:20:49
next so we have recapping my mean
00:20:52
invented mean of 0.6 and my invented
00:20:54
mean of 0.5 for the other
00:20:56
coin I was plus 0.5 not 0.55
00:21:01
so uh we take the first round with uh 5
00:21:04
over 10 heads and 5 over 10 Tails we
00:21:06
will see we'll compute the likelihood of
00:21:09
coming from coin a using this formula
00:21:13
okay and H now stands for
00:21:16
heads and that gives us
00:21:20
0.0079 okay or eight and the likelihood
00:21:23
of coin B is
00:21:27
0.0000 976 so
00:21:30
0.1 mostly right now right now we can
00:21:34
say that it's more likely to have come
00:21:36
from coin B which is correct right uh
00:21:39
than from coin a but the thing is that
00:21:42
we now because we only have two coins we
00:21:44
need to convert this into probabilities
00:21:47
these likelihoods we need to convert
00:21:49
them into probabilities so what we do is
00:21:52
we
00:21:54
add these two numbers right and and then
00:21:59
we divide this guy by the sum and then
00:22:02
we divide this guy by the sum and we get
00:22:06
0.45 for a and 0.55 for B this procedure
00:22:11
is called
00:22:12
normalization okay again take the sum of
00:22:16
these two that's going to give you a
00:22:17
number and then divide this first number
00:22:21
by that sum and this second number by
00:22:23
that sum if you divide the first number
00:22:24
by that sum you're going to get 0.45 if
00:22:27
you divide the second number by the sum
00:22:28
you're going to get
00:22:29
0.55 okay and this is the probability
00:22:33
that the first 10 coins came from
00:22:37
a and the first 10 coins came from B
00:22:40
given this means that I that I gave
00:22:43
right and it makes sense if it had five
00:22:45
taals and B has 50% of success well it
00:22:49
looks like it hit the mark for B right
00:22:52
and not quite for a this is what these
00:22:54
probabilities are
00:22:56
saying so we're going to do this
00:22:58
this
00:23:00
um for all coin tosses okay we're going
00:23:03
to do this for all of our trials for all
00:23:05
of our five
00:23:07
trials
00:23:09
now let's recap the probability that the
00:23:12
first round came from coin a was 0.45
00:23:16
the probability they came from coin B
00:23:17
was 0.55 now let's estimate the likely
00:23:21
number of heads and tails from those
00:23:23
different coins so from coin a the
00:23:27
likely number of heads or the estimated
00:23:29
number of heads is 0.45 which is you
00:23:32
know the the likelihood that it came
00:23:34
from coin a times the actual number of
00:23:37
heads right that gives you 2.2 heads so
00:23:40
if this had came from coin a then then I
00:23:45
I should have seen 2.2
00:23:47
heads now and Tails is 2.2 Tails because
00:23:52
it's the same basically it's the same
00:23:54
computation now if it came from coin B
00:23:58
then I multiply this proportion by five
00:24:00
heads which is what I saw and that gives
00:24:02
me
00:24:04
2.8 and 2.8 Tails I will tally these
00:24:08
numbers and I will do this for all five
00:24:11
runs and I will end up with something
00:24:13
like
00:24:14
this these are my
00:24:17
cases then for the first case I got
00:24:21
0.45 times the likelihood that times my
00:24:24
belief that it's from coin a and 0.55
00:24:28
times my belief that it was from coin B
00:24:30
I got from coin a 2.2 heads and 2.2
00:24:34
tails and I got from coin B 2.8 heads
00:24:37
and 2.8 Tails okay I did this for all
00:24:42
five tosses and I got these numbers here
00:24:44
then I add I add these numbers the
00:24:46
numbers of heads I add the number of
00:24:48
tails for each coin and I get for coin 8
00:24:52
21.3 heads and 8.6 tails and here here I
00:24:58
get 11.7 heads and 8.4 tails and what I
00:25:02
will do is with these
00:25:04
numbers I will compute the new Theta the
00:25:08
new means for these coins so for example
00:25:11
for the first
00:25:12
coin and I'm going to use this formula
00:25:14
the number of heads divided by the
00:25:16
number of heads plus Tails so for
00:25:19
example for the first number there for
00:25:22
the first coin for coin a I will add uh
00:25:26
21 3 plus 8.6 and that's going to go in
00:25:31
the lower bottom of the fraction right
00:25:33
so and that um
00:25:37
21.3 and
00:25:39
8.6 that is basically
00:25:42
29.9 and the number of heads here which
00:25:45
is 21.3 I'm going to divide the number
00:25:48
of
00:25:49
heads by
00:25:51
[Music]
00:25:52
um by the number of tail by the by the
00:25:55
sum of heads and tails I will do the
00:25:57
same for coin B and this will give me
00:25:59
the new thetas for the
00:26:04
coins see 21.3 divided by the sum 21.3 +
00:26:08
8.6 so this these
00:26:11
numbers here come from these numbers
00:26:15
here
00:26:17
okay I just want to point out but I'm
00:26:20
covering them okay and that gives you 71
00:26:23
if you do the same for coin B you get
00:26:25
the new Theta of 0.58
00:26:28
now with these new
00:26:30
thetas this new thetas this will go will
00:26:35
be put in place of
00:26:37
this right and this number will go in
00:26:41
place of this
00:26:43
one you see how the thetas now are
00:26:45
changing
00:26:46
slightly and then I'll go over the whole
00:26:49
thing again I will again compute the
00:26:52
probability using the binomial formula
00:26:54
the probability that it came from these
00:26:57
that that uh each of these each of these
00:26:59
runs came from coin a and coin B I will
00:27:02
compute the weights in absolute
00:27:05
probability terms I will register the
00:27:07
number of heads the expected number of
00:27:08
heads and the expected number of tails
00:27:10
for coin A and B and I will compute a
00:27:12
new mean and so on and so forth I will
00:27:14
do this cycle many many times until the
00:27:17
the computation of means don't change at
00:27:19
some point they settle okay or they
00:27:22
start changing you know in the fifth
00:27:24
decimal which you don't really care too
00:27:26
much right well uh long story short
00:27:32
these coins
00:27:34
settle
00:27:35
um these coins will settle with these
00:27:39
means 0 52 for coin B and 80 for coin a
00:27:44
so coin a is going to be pretty
00:27:47
um pretty uh um pretty skewed okay so
00:27:54
once you have the means right once you
00:27:57
have the means
00:27:58
then
00:27:59
again Computing the the Computing doing
00:28:03
this step right the the estimation step
00:28:05
okay you will see whether this first run
00:28:08
is more likely to have come from coin A
00:28:10
or B and so on and so forth okay and you
00:28:13
can determine using expectation
00:28:16
maximization you can determine whether
00:28:19
the data that you see came from coin a
00:28:22
or coin B and what is the mean uh
00:28:28
probability of um of success for the
00:28:32
first condition and the second condition
00:28:34
or the first coin and the second coin
00:28:36
and that is basically estimation
00:28:38
maximization
