Maximum likelihood estimation

1
Maximum likelihood estimation

• Example X1,,Xn i.i.d. random variables with
  probability pX(x?) P(Xx) where ? is a
  parameter
• likelihood function L(?x) where x(x1,,xn) is
  set of observations
• maximum likelihood estimate
• maximizer of L(?x)

2

• typically easier to work with log-likelihood
  function, C(?x) log L(?x)

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Properties of estimators

• estimator is unbiased if
• is asymptotically unbiased if
• as n?8

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Properties of MLE

• asymptotically unbiased, i.e.,
• asymptotically optimal, i.e., has
  minimum variance as n?8
• invariance principle, i.e., if is MLE
  for ? then is MLE for any function
  t(?)

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Network Tomography

• Goal obtain detailed picture of a
  network/internet from end-to-end views

• infer topology /connectivity

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Network Tomography

• Goal obtain detailed picture of a
  network/internet from end-to-end views

• infer link-level
• loss
• delay
• available bandwidth
• . . .

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Brain Tomography
unknown object
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Network Tomography
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Why end-to-end

• no participation by network needed
• measurement probes regular packets
• no administrative access needed
• inference across multiple domains
• no cooperation required
• monitor service level agreements
• reconfigurable applications
• video, audio, reliable multicast

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Naive Approach I

• D0 D1 M1
• D0 D2 M2

2 equations, 3 unknowns
?
M1
M2
Di not identifiable
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Naive Approach II

• bidirectional tree

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Naive Approach II

• bidirectional tree

D2 D1
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Naive Approach II

• bidirectional tree

D1D2
D2 D1
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Naive Approach II
D0 D1 D0 D2

• bidirectional tree

D0
D2
D1
D1D2
D2 D1
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Naive Approach II

• bidirectional tree

• 6 equations, 6 unknowns

• not linearly independent! (not
  identifiable)

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Naive Approach III

• Round trip link delays

RAB R0 R1
RAC R0 R2

• RBC R1 R2
• Linear independence! (identifiable)
• true for general trees
• can infer some link delays within general graph

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Bottom Line

• similar approach for losses
• yields round trip and one way metrics for subset
  of links
• approximations for other links

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MINC (Multicast Inference of Network
Characteristics)
source

• multicast probes
• copies made as needed within network

• receivers observe correlated performance
• exploit correlation to get link behavior
• loss rates
• delays

receivers
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MINC (Multicast Inference of Network
Characteristics)

• multicast probes
• copies made as needed within network

• receivers observe correlated performance
• exploit correlation to get link behavior
• loss rates
• delays

?
?
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MINC (Multicast Inference of Network
Characteristics)

• multicast probes
• copies made as needed within network

• receivers observe correlated performance
• exploit correlation to get link behavior
• loss rates
• delays

?
?
?
?
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MINC (Multicast Inference of Network
Characteristics)

• multicast probes
• copies made as needed within network

• receivers observe correlated performance
• exploit correlation to get link behavior
• loss rates
• delays

?
?
? ?
? ?
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MINC (Multicast Inference of Network
Characteristics)

• multicast probes
• copies made as needed within network

• receivers observe correlated performance
• exploit correlation to get link behavior
• loss rates
• delays

estimates of a1, a2, a3
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Multicast-based Loss Estimator

• tree model
• known logical mcast topology
• tree T (V,L) (nodes, links)
• source multicasts probes from root node
• set R ? V of receiver nodes at leaves
• loss model
• probe traverses link k with probability ak
• loss independent between links, probes
• data
• multicast n probes from source
• data YY(j,i), j ? R, i1,2,,n
• Y(j,i) 1 if probe i reaches receiver j, 0
  otherwise
• goal
• estimate set of link probabilities a ak k
  ?V from data Y

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Loss Estimation on Simple Binary Tree

• each probe has one of 4 potential outcomes at
  leaves
• (Y(2),Y(3)) ? (1,1), (1,0), (0,1), (0,0)
• calculate outcomes theoretical probabilities
• in terms of link probabilities a1, a2, a3
• measure outcome frequencies
• equate
• solve for a1, a2, a3, yielding estimates
• key steps
• identification of set of externally measurable
  outcomes
• knowing probabilities of outcomes ?? knowing
  internal link probabilities

Source
0
a1
1
a2
a3
2
3
Receivers
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General Loss Estimator Properties

• Can be done, details see
• R. Cáceres, N.G. Duffield, J. Horowitz, D.
  Towsley, Multicast-Based Inference of
  Network-Internal Loss Characteristics,'' IEEE
  Transactions on Information Theory, 1999

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Statistical Properties of Loss Estimator

• model is identifiable
• distinct parameters a k ? distinct
  distributions of losses seen at leaves
• Maximum Likelihood Estimator
• strongly consistent (converges to true value)
• asymptotically normal
• (MLE ?efficient minimum asymptotic variance)

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Impact of Model Violation

• mechanisms for dependence between packets losses
  in real networks
• e.g. synchronization between flows from TCP
  dynamics
• expect to manifest in background TCP packets more
  than probe packets
• temporal dependence
• ergodicity of loss process implies estimator
  consistency
• convergence of estimates slower with dependent
  losses
• spatial dependence
• introduces bias in continuous manner small
  correlation result in small bias
• can correct for with a priori knowledge of
  typical correlation
• second order effect
• depends on gradient of correlation rather than
  absolute value

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MINC Simulation Results

• accurate for wide range of loss rates
• insensitive to
• packet discard rule
• interprobe distribution beyond mean

inferred loss
probe loss
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MINC Experimental Results

• background traffic loss and inferred losses
  fairly close
• over range of loss rates, best when over 1

inferred loss
background loss
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Validating MINC on a real network

• end hosts on the MBone
• chose one as source, rest as receivers
• sent sequenced packets from source to receivers
• two types of simultaneous measurement
• end-to-end loss measurements at each receiver
• internal loss measurements at multicast routers
• ran inference algorithm on end-to-end loss traces
• compared inferred to measured loss rates
• inference closely matched direct measurement

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MINC Mbone Results

• experiments with 2- 8 receivers
• 40 byte probes 100 msec apart
• topology determined using mtrace

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Topology Inference
Probe source

• problem
• given
• multicast probe source
• receiver traces (loss, delay, )
• identify (logical) topology
• motivation
• topology may not be supplied in advance
• grouping receivers for multicast flow control

?
Receivers
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General Approach to Topology Inference

• given model class
• tree with independent loss or delay
• find classification function of nodes k which is
• increasing along path from root
• can be estimated from measurements at R(k)
  leaves descended from k
• examples
• 1-Ak Probprobe lost on path from root 0 to k
• mean of delay Yk from root to node k
• variance of delay Yk from root to node k
• build tree by recursively grouping nodes
  r1,r2,,rm
• to maximize classification function on putative
  parent

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BLTP Algorithm

• 1. construct binary tree based on losses
• estimate shared loss L 1-Ak seen from receiver
  pairs
• aggregate pair with largest L
• repeat till one node left

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Example

• 1. construct binary tree
• estimate shared loss L seen from receiver pairs
• aggregate pair with largest L
• repeat till one node left

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Example

• 1. construct binary tree
• estimate shared loss L seen from receiver pairs
• aggregate pair with largest L
• repeat till one node left

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Example

• 1. construct binary tree
• estimate shared loss L seen from receiver pairs
• aggregate pair with largest L
• repeat till one node left

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Example

• 1. construct binary tree
• estimate shared loss L seen from receiver pairs
• aggregate pair with largest L
• repeat till one node left

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Example

• 1. construct binary tree
• estimate shared loss L seen from receiver pairs
• aggregate pair with largest L
• repeat till one node left

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Example

• 1. construct binary tree
• estimate shared loss L seen from receiver pairs
• aggregate pair with largest L
• repeat till one node left

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Example

• 1. construct binary tree
• estimate shared loss L seen from receiver pairs
• aggregate pair with largest L
• repeat till one node left

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BLTP Algorithm

• 1. construct binary tree
• 2. prune links with 1-aklte

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Theoretical Result

• 1. construct binary tree
• 2. prune links with 1-aklte
• if e lt min 1-ak, topology identified with prob ?
  1 as n ? ?

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Results

• Simulation of Internet-like topology
• (min ak .12)
• BLTP is
• simple, efficient
• nearly as accurate as Bayesian methods
• can combine with delay measurements

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Issues and Challenges

• relationship between logical and physical
  topology
• relation to unicast
• tree layout/composition
• combining with network-aided measurements
• scalability