WEBVTT
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In this sequence I want to
remind you a few concepts in the
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theory of probability and then
in the next one we finally derive
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the equations of the Bayes filter.
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So the concept that I
want to remind you are 3: the
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Markov assumption, the theorem
of a total probability and the
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Bayes theorem. So let us start
with the Markov assumption, the
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Markov assumption
regard a stochastic process.
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What is a stochastic process?
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A stochastic process is a
random quantity which also depends
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on time so in this
case, this would be a random
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quantity which takes a value at
the timestep one, the timestep
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two, so on and so
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forth. The Markov assumption
tells that the statistical properties
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of these random quantities
do not depend on the past,
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so the process has not memory.
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So let us consider for
instance the case of a proprioceptive
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measurement, we have that the
measurement at the timestep i
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is a function of this state
plus this random quantity Wi and
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what we assume is that
the Markov assumption
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holds so this means that
the statistical property of W
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are independent of
the past value of this
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quantity at a previous step and
so this means that statistical
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properties of Z only depends on the
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current values over the
current random quantity W.
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If you want, we assumed
already these assumptions when we say
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that we could build the
statistical distribution P(Z by knowing S).
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Why? Because we can
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build the statistical property
of Z by just knowing S and without
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knowing all the values of
the measurements that occurred
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before the timestep i.
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Regarding the case of
proprioceptive measurements we have Ui
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equal to the measurement Um
at timestep i plus the random
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quantity Vi and again the Markov
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assumption tell us that
the statistical property of V
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at a timestep i are
independent of the previous value of this
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quantity and so the same holds for U.
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And what we also assume with
the Markov assumption is that the
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statistical property over
both U and Z are independent of
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the values that occur
before the other variables.
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That means that the value of
Z, the statistical poverty of Zi
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are independent on the values
that occured before the timestep
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i for the proprioceptive
measurements and it's the same for
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the statistical property of Ui.
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And so for the case of the
proprioceptive measurements, actually
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we've already done these
assumption when we say that
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we could build the
distribution P(Si+1 by knowing Si and
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Um i+1). We don't need to include here
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all the previous measurements.
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So now, let us keep to the
theorem of total probability.
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This theorem says that the
probability of having a given event
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A can be written as the sum of all
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the possible events B of
the joint probability of
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having A and B. This is the
theorem of total probability.
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The Bayes theorem regards the
conditional probabilities and
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say that the probability
of having P(A by knowing B)
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is equal to the joint probability
of having A and B divided by P(B).
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We can also write the contrary
so we can invert B and A then
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we have that: P of having
B by knowing A is equal to
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P of having A and B divided by P(A).
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If we use this set on
the quantity here in
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order to eliminate P of having A
and B, we obtain that the probability
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of having A by knowing B is
equal to the probability of having
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B by knowing A times
P(A) divided by P(B).
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We can further condition
this quantity with another event
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and we have that P of
having A by knowing B
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and another event C is equal to P
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of having B and these
other events C times P of
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having A by knowing C divided by P
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of having B by knowing C.
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Finally if we use this equation
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to eliminate P(A,B) here in
the theorem of total probability,
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we obtain that this is the
sum on all the event B, of
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P of having A by knowing B times P(B).
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And if we also here condition all
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these two guys by
another event C, we obtain
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that P of having A by
knowing C is equal to the sum
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on B of P having A and B and
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C times P of having B by knowing C.
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So as we will see
these 2 equations are
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exactly the equations that we
will use in the next sequence
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to derive the
equations of the Bayes filter.