Conditional & Joint Probability
A brief digression back to joint probability:
i.e. both events O and H occur
Pr{H∩O} = Pr{H|O}Pr{O}
Again, we can express joint probability in terms of their
separate conditional and unconditional probabilities
This key result turns out to be exceedingly useful!
Conditional Probability
Converting expressions of joint probability
The intersection operator makes no assertion regarding order:
Pr{H∩O} =Pr{H|O}Pr{O}
=Pr{O|H}Pr{H}
We can therefore express everything only in terms of
reciprocal conditional and unconditional probabilities:
Pr{O|H}Pr{H} =Pr{H|O}Pr{O}
This is usually expressed in a slightly rearranged form…
Conditional Probability
Bayes’ theorem expresses the essence of inference
Pr{H|O} = Pr{O|H} · Pr{H}
Pr{O}
We can think of this as allowing us to compute the probability of some hidden event H
given that some observable event O has occurred, provided we know the probability of
the observed event O assuming that hidden event H has occurred
Bayes’ theorem is a recipe for problems involving conditional probability
Conditional Probability
Normalizing the probabilities
Pr{H|O} = Pr{O|H} · Pr{H}
𝐻
Pr{O|H} · Pr{H}
For convenience, we often replace the probability of the observed event O with the
sum over all possible values of H of the joint probabilities of O and H. Whew! But
consider that if we now calculated Pr{H|O} for every H, the sum of these would be
one, which is just as a probability should behave…
This summing of the expression in numerator “normalizes” the probabilities
Conditional Probability
Bayes’ theorem as a recipe for inference
The likelihood model
The prior
We’ve seen already that the probability of an
observation given a hidden parameter is really a
likelihood. Choosing a likelihood model is akin to
proposing some process, H, by which the
observation, O, might have come about
Our best guess about H before any observation is
made. Often we will make neutral assumptions,
resulting in an uninformative prior. But priors can
also come from the posterior results from some
earlier experiment. How best to choose priors is
an essential element of Bayesian analysis
Pr{H|O} = Pr{O|H} · Pr{H}
The posterior
Think of this perhaps as the
evidence for some specific model
H given the set of observations O.
We are making an inference about
H on the basis of O
Pr{O}
The observable probability
Generally, care must be taken to ensure that our observables
have no uncertainty, otherwise they are really hidden!!!
Bayes’ theorem is so important that each part of this recipe has a special name
The Backward Algorithm
Many paths give rise to the same sequence X
We would often like to know the total
probability of some sequence:
P(x) =
𝑃(𝑥, 𝜋)
𝜋
But wait! Didn’t we already solve this problem using the
forward algorithm!?
Well, yes, but we’re going to solve it again by iterating
backwards through the sequence instead of forwards
Sometimes, the trip isn’t about the destination. Stick with me!
The Backward Algorithm
Defining the backward variable
“The backward variable for state k at position i”
bk(i) = P(xi+1 … xL | pi = k)
“the probability of the sequence from the end to the symbol
at position i, given that the path at position i is k”
Since we are effectively stepping “backwards” through the event
sequence, this is formulated as a statement of conditional probability
rather than in terms of joint probability as are forward variables
The Backward Algorithm
What if we had in our possession all of the backward
variables for 1, the first position of the sequence?
P(X) =
𝒆𝒍(𝒙𝟏)𝒃𝒍(𝟏) 𝒂𝑺 → 𝒍
𝒍
To get the overall probability of the sequence, we would need
only sum the backward variables for each state (after
correcting for the probability of the initial transition from Start)
We’ll obtain these “initial position” backward variables in a manner
directly analogous to the method in the forward algorithm…
The Backward Algorithm
A recursive definition for backward variables
bk(i) = P(xi+1 … xL | pi = k)
As always with a dynamic programming algorithm, we
recursively define the variables.. But this time in terms of
their own values at later positions in the sequence…
el(xi) · 𝒃𝒍(𝒊) · 𝒂𝒌 → 𝒍
bk(i-1) =
𝒍
The termination condition is satisfied and the basis case
provided by virtue of the fact that sequences are finite and in this
case we must eventually (albeit implicitly) come to the End state
The Backward Algorithm
If you understand forward, you already understand backward!
Initialization:
bk(L) = 1 for all states k
Recursion (i = L-1,…,1):
bk(i) =
el(xi + 1) · bl(i +1) · 𝒂𝒌 → 𝒍
𝒍
Termination:
P(X) =
𝒆𝒍(𝒙𝟏)𝒃𝒍(𝟏) 𝒂𝒔𝒕𝒂𝒓𝒕 → 𝒍
𝒍
We usually don’t need the termination step, except to check that the result is the same
as from forward… So why bother with the backward algorithm in the first place?
The Backward Algorithm
The probability of a sequence summed across all possible state paths
0.1
A: 0.30
0.5
_
P(x) =
0.1 * 0.3 * 0.05825
S
0.9
0.5
A: 0.20
C: 0.25
C: 0.35
G: 0.15
G: 0.25
T: 0.30
T: 0.20
State “+”
0.6
State “-”
A
C
0.5 * 0.25 * 0.2
0.5 * 0.15 * 1
0.5 * 0.35 * 0.19
0.5 * 0.25 * 1
S = 0.05825
0.4
G
1
S = 0.20.2
End
0.9 * 0.2 * 0.0566
S = 0.0119355
0.6 * 0.25 * 0.2
0.6 * 0.15 * 1
0.0119355
0.4 * 0.35 * 0.19
0.4 * 0.25 * 1
S = 0.0566
S = 0.19
1
The backward algorithm takes its name from its backwards iteration through the sequence
The Decoding Problem
The most probable state path
The Viterbi algorithm will calculate the most probable state
path, thereby allowing us to “decode” the state path in cases
where the true state path is hidden
Sequence
_646223114453161463432454523162625244256
1323352435544245413424666621512464652653
6662616666426446665162436615266416215651
True state path
SFFFFFFFFFFFFFLLLLFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLFFFFFFFFF
Viterbi path
a.k.a MPSP
SFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLL
LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL
p*= argmax P(x, p)| θ)
p
The Viterbi algorithm generally does a good job of recovering the state path…
The Decoding Problem
Limitations of the most probable state path
…but the most probable state path might not always be the
best choice for further inference on the sequence
• There may be other paths, sometimes several, that result in
probabilities nearly as good as the MPSP
• The MPSP tells us about the probability of the entire path, but
it doesn’t actually tell us what the most probable state might
be for some particular observation xi
• More specifically, we might want to know:
P(pi = k|x)
“the probability that observation xi resulted from being in state k, given the observed sequence x”
This is the posterior probability of state k at position i when sequence x is known
Calculating the Posterior Probabilities
The approach is a little bit indirect….
We can sometimes ask a slightly different or related
question, and see if that gets us closer to the result we seek
Maybe we can first say something about the probability of
producing the entire observed sequence with observation xi
resulting from having been in state k…..
P(x,pi = k) =
P(x1…xi, pi = k)·P(xi+1…xL|x1…xi, pi = k)
=
P(x1…xi, pi = k)·P(xi+1…xL|pi = k)
Does anything here look like something we may have seen before??
Calculating the Posterior Probabilities
Limitations of the most probable state path
P(x,pi = k)
=
P(x1…xi, pi = k)·P(xi+1…xL|pi = k)
=
fk(i)·bk(i)
These terms are exactly our now familiar forward and backward variables!
Calculating the Posterior Probabilities
Putting it all together using Baye’s theorem
fpki(i)·b
(i)
P(pi = k|x) =P(x|P(x,
=pk)·P(
=
k)
i k pi = k)
Remember, we can get P(x) directly from
either the forward or backward algorithm!
P(x)
We now have all of the necessary ingredients required to apply
Baye’s theorem… We can now therefore find the posterior
probability of being in state k for each position i in the sequence!
We need only run both the forward and the backward algorithms to
generate the values we need.
Python: we probably want to store our forward and backward values as instance variables rather than method variables
OK, but what can we really do with these posterior probabilities?
Posterior Decoding
Making use of the posterior probabilities
Two primary applications of the posterior state path probabilities
• We can define an alternative state path to the most probable state path:
˰
pi = argmax P(pi = k | x)
k
This alternative to Viterbi decoding is most useful when we are most
interested in the what the state might be at some particular point or
points. It’s possible that the overall path suggested by this might not
be particularly likely
In some scenarios, the overall path might not even be a permitted path through the model!
Posterior Decoding
Plotting the posterior probabilities
Key: true path / Viterbi path / posterior path
_44263143366636346665525516166566666224264513226235262443416
SFFFFFFFFLLLLLLLLLLLFFLLLLLFFLLLLLLLLFFFFFFFFFFFFFFFFFFFLLLL
SFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFF
FFFFFFFFFLLLLLLLLLLLLFFFFLLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFF
myHMM.generate(60,143456789)
Since we know individual probabilities at each position we can easily plot the posterior probabilities
Posterior Decoding
Plotting the posterior probabilities with matplotlib
Assuming that we have list variables self._x containing the range of
the sequence and self._y containing the posterior probabilities…
from pylab import *
# this line at the beginning of the file
. . .
class HMM(object):
. . . # all your other stuff
def show_posterior(self):
if self._x and self._y:
plot (self._x, self._y)
show()
return
Note: you may need to convert the log_float probabilities back to normal floats
I found it more convenient to just define the __float__() method in log_float