Imprecise Probability and Network Quality of Service

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Imprecise Probability and Network Quality of
Service

• Martin Tunnicliffe

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Two Kinds of Probability

• Alietory Probability The probability of chance.
• Example When throwing an unweighted die, the
  probability of obtaining a 6 is 16.
• Epistemic Probability The probability of belief.
• Example The defendant on trial is probably
  guilty.

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Probability and Betting Odds

• A fair bet is a gamble which, if repeated a
  large number of times, returns the same amount of
  money in winnings as the amount of money staked.
• Example, if there is a 110 chance of winning a
  game, then the odds for a fair gamble would be
  101.
• Problems arise when we do not know exactly what
  the chance of winning is. Under such
  circumstances, how can we know what constitutes a
  fair gamble?
• Behavioural interpretation of probability (Bruno
  de Finetti, 1906-1985) Probability in such
  cases refers to what people will consider or
  believe a fair bet to be.
• Belief stems from experience, i.e. inductive
  learning.

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Inductive Learning

• Induction is the opposite of deduction, which
  infers the specific from the general.
• Example All dogs have four legs. Patch is a
  dog. Therefore Patch has four legs.
• Induction is the opposite It infers the general
  from the specific.
• Example Patch is a dog. Patch has four legs.
  Therefore all dogs have four legs.

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Inductive Learning
The last statement has little empirical support.
However, consider a larger body of evidence
The statement all dogs have four legs now has
significant plausibility or epistemic
probability. However, it remains uncertain Even
with a hundred dogs, there is no categorical
proof that the hundred-and-first Dalmatian will
not have five legs!
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Approaches to Inductive Learning.

• Frequentist statistics disallows the concept of
  epistemic probability (We cannot talk about the
  probability of a five-legged Dalmatian). Thus
  it offers very little framework for inductive
  learning.
• The Objective Bayesian approach allows epistemic
  probability, which it represents as a single
  probability distribution. (This is the Bayesian
  Dogma of Precision).
• The Imprecise Probability approach uses two
  distributions representing upper probability
  and lower probability.

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Marble Problem
Example (shamelessly ripped off from P. Walley,
J. R. Stat. Soc. B, 58(1), pp.3-57,
1996) Marbles are drawn blindly from a bag of
coloured marbles. The event ? constitutes the
drawing of a red marble. The composition of the
bag is unknown. For all we know, it could contain
no red marbles. Alternatively every marble in the
bag may be red. Nevertheless, we are asked to
compute the probability associated with a fair
gamble on ?, both a priori (before any marble is
drawn) and after n marbles are drawn, j of which
are red. (Marbles are replaced before the next
draw.)
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Binomial Distribution
If ? is the true (unknown) chance of drawing a
red marble. The probability of drawing j reds in
n draws is
This is proportional to the Likelihood of ?
given that j red marbles have been drawn
Walley actually considers a more complex
multinomial situation, where three or more
outcomes are possible. However, I am only going
to consider two possibilities ? red marble and
? any other coloured marble.
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Bayes Theorem
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Beta Model
The hyper-parameter s is the prior strength,
the influence this prior belief has upon the
posterior probability.
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Beta Model Prior Distributions
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Beta Model Posterior Distributions
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Posterior Expectation
Under the behavioural interpretation, this is
viewed as the posterior probability P(?j,n) of a
red.
Thus in the light of the new information, a fair
gamble now requires odds of 41 on red, and 43
against red.
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Dirichlet Distribution
Walleys paper uses the generalised Dirichlet
distribution.
The beta distribution is the special case of the
Dirichlet for which the number of possible
outcomes is 2. (Sample set has cardinality 2.)
This leads to the Imprecise Dirichlet Model or
IDM. The simpler Beta-function model may be
called the Imprecise Beta Model (IBM).
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Objective Bayesian Approach
We need an initial value for t, to represent our
belief that ? will occur when we have no data
available ( j n 0). This is called a
non-informative prior.
Under Bayes Postulate (in the absence of any
information, all possibilities are equally
likely) t 0.5
However, a value for s is still needed.
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Non-Informative Priors
Bayesians favour setting s to the cardinality of
the sample space (in this case 2) to give a
uniform prior.
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Problems with Bayesian Approach
Problem Bayesian formula assigns finite
probabilities to events which have never been
known to happen, and might (for all we know) be
physically impossible.
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Problems with Bayesian Approach
Strict application of Bayes Postulate yields
prior (and hence posterior) probabilities which
depend on the choice of sample space (which
should be arbitrary).
The experiment is identical in all three cases
Only its representation is altered. Thus the
Representation Invariance Principle (RIP) is
violated.
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A Quote from Walley
The problem is not that Bayesians have yet to
discover the truly noninformative priors, but
rather that no precise probability distribution
can adequately represent ignorance. (Statistical
Reasoning with Imprecise Probabilities, 1991)
What does Walley mean by precise probability?
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The Dogma of Precision

• The Bayesian approach rests upon de Finettis
  Dogma of Precision.
• Walley (1991) ..for each event of interest,
  there is some betting rate which you regard as
  fair, in the sense that you are willing to accept
  either side of a bet on the event at that rate.
• Example If there is a 14 chance of an event ?,
  I am equally prepared to bet 41 on ? and 43
  against ?.

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The Imprecise Probability Approach
The Imprecise Probability approach solves the
problem by removing the dogma of precision, and
thus the requirement for a noninformative prior.
It does this by eliminating the need for a single
probability associated with ?, and replaces it
with an upper probability and a lower
probability.
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Upper and Lower Probabilities
Walley Before any marbles are drawn I do not
have any information at all about the chance of
drawing a red marble, so I do not see why I
should bet on or against red at any odds. This
is not a very exciting answer, but I believe that
it is the correct one.
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Upper and Lower Probabilities
Lower Probability The degree to which we are
confident that the next marble will definitely be
red. Upper Probability The degree to which we
are worried that the next marble might be red.
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Posterior Upper and Lower Probabilities
However, the arrival of new information (j
observed reds in n trials) allow these two
probabilities to be modified. The prior upper and
lower probabilities (1 and 0) can be substituted
for t in the Bayesian formula for posterior mean
probabvility. Thus we obtain the posterior lower
and upper probabilities
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Properties of Upper and Lower Probabilities
The amount of imprecision is the difference
between the upper and lower probabilities, i.e.
This does not depend on the number of successes
(occurrences of ?). As n??, the imprecision tends
to zero and the lower and upper probabilities
converge towards j/n, the observed success
ratio. As s ? ?, the prior dominates The
imprecision becomes 1, and the lower and upper
probabilities return to 0 and 1 respectively. As
s ? 0, the new data dominates the prior and and
the lower and upper probabilities again converge
to j/n (Haldanes model).
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Interpretation of Upper and Lower Probabilities
How do we interpret these upper and lower
probabilities? Which do we take as the
probability of red? It depends on whether you
are betting for or against red. If you are
betting for red then you take the lower
probability, since this represents the most
cautious expectation of the probability of
red. However, if you are betting against red, you
take the upper probability, since this is
associated with the lower probability of not-red.
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Interpretation of Upper and Lower Probabilities
(For consistency, we continue to assume that s
2.)
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Analogy with Possibility Theory
Thus upper probability is analogous to
possibility and lower probability to necessity.
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Choosing a Value of s
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Choosing a Value of s
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Confidence Intervals for ?
You might be tempted to think that the upper and
lower probabilities represent some kind of
confidence interval for the true value of
?. This is not the case. Upper and lower
probabilities are the mean values of belief
functions for ?, relevant to people with
different agendas (betting for and against ?).
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Confidence Intervals for ?
Suppose we want to determine a credible
interval (? -(?),?(?)) such that we are
at least ? ?100 per cent sure that ? -(?) lt ? lt
?(?)
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Confidence Intervals for ?
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Calculating the Confidence Interval
Integrating the two probability distributions, we
find that we can compute the confidence intervals
by solving the equations
I indicates the Incomplete Beta Function. No
analytic solution exists, but numerical iteration
using the partition method is quite
straightforward.
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Frequentist Confidence Limits
Binomial distribution for ? ?-(?)
Binomial distribution for ? ?(?)
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Frequentist Confidence Limits
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Comparison Frequentist vs. Imprecise Probability
When s 1 (Perks), Imprecise Probability agrees
exactly with Frequentism on the upper and lower
confidence limits.
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Applications in Networking

• Network Management and Control often requires
  decisions to be made based upon limited
  information.
• This could be viewed as gambling on imprecise
  probabilities.
• Monitoring Network Quality-of-Service.
• Congestion Window Control in Wired-cum-Wireless
  Networks.

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Quality of Service (QoS)
Different types of applications have different
QoS requirements. FTP and HTTP can tolerate
delay, but not errors/losses (transmitted and
received messages must be exactly
identical). Real time services (Voice/Video) can
tolerate some data losses, but are sensitive to
variations in delay.
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QoS Metrics
Loss Percentage of transmitted packets which
never reach their intended destination (either
due to noise corruption or overflow at a queuing
buffer.)
Throughput The throughput is the rate at which
data can be usefully carried.
Latency A posh word for delay the time a
packet takes to travel between end-points.
Jitter Loosely defined as the amount by which
latency varies during a transmission. (Its
precise definition is problematic.) Most
important in real-time applications.
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Quality of Service (QoS) Monitoring
Failure Probability ?
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Simulation Data
Heavily Loaded Network (Average utilisation 97)
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Simulation Data
Lightly Loaded Network (Average utilisation 46)
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Jitter Definition 1
Ref http//www.slac.stanford.edu/comp/net/wan-mon
/dresp-jitter.jpg
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Jitter Definition Two
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Jitter Profiles
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Wired-Cum-Wireless Networks
Wired Network Congestion Only
Wireless Network Congestion Plus Random Noise
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WTCP Identifying the Cause of Packet Loss using
Interarrival Time
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WTCP Identifying the Cause of Packet Loss using
Interarrival Time
Assume we already know the mean M and standard
deviation s of the interarrival time when the
network is uncongested.
If M - Ks lt ?i,j ltM Ks (where K is a constant),
then the losses are assumed to be random. The
sending rate is not altered.
Otherwise, we infer that queue-sizes are varying
An indication that congestion is occurring. The
sending rate is reduced to alleviate the problem.
Much work still to be done on this optimising
mechanism to maximise throughput.