Inference I Introduction, Hardness, and Variable Elimination

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PGM 2002/03 Tirgul 4Exact Inference
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Inference in Simple Chains
X1
X2

• How do we compute P(X2)?

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Inference in Simple Chains (cont.)
X1
X2
X3

• How do we compute P(X3)?
• we already know how to compute P(X2)...

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Inference in Simple Chains (cont.)
...

• How do we compute P(Xn)?
• Compute P(X1), P(X2), P(X3),
• We compute each term by using the previous one
• Complexity
• Each step costs O(Val(Xi)Val(Xi1))
  operations
• Compare to naïve evaluation, that requires
  summing over joint values of n-1 variables

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Inference in Simple Chains (cont.)
X1
X2

• Suppose that we observe the value of X2 x2
• How do we compute P(X1x2)?
• Recall that we it suffices to compute P(X1,x2)

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Inference in Simple Chains (cont.)
X1
X2
X3

• Suppose that we observe the value of X3 x3
• How do we compute P(X1,x3)?
• How do we compute P(x3x1)?

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Inference in Simple Chains (cont.)
...
X1
X2
X3
Xn

• Suppose that we observe the value of Xn xn
• How do we compute P(X1,xn)?
• We compute P(xnxn-1), P(xnxn-2), iteratively

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Inference in Simple Chains (cont.)
...
...
X1
X2
Xk
Xn

• Suppose that we observe the value of Xn xn
• We want to find P(Xkxn )
• How do we compute P(Xk,xn )?
• We compute P(Xk ) by forward iterations
• We compute P(xn Xk ) by backward iterations

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Elimination in Chains

• We now try to understand the simple chain example
  using first-order principles
• Using definition of probability, we have

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Elimination in Chains

• By chain decomposition, we get

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Elimination in Chains

• Rearranging terms ...

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Elimination in Chains

• Now we can perform innermost summation
• This summation, is exactly the first step in the
  forward iteration we describe before

X
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Elimination in Chains

• Rearranging and then summing again, we get

X
X
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Elimination in Chains with Evidence

• Similarly, we understand the backward pass
• We write the query in explicit form

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Elimination in Chains with Evidence

• Eliminating d, we get

X
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Elimination in Chains with Evidence

• Eliminating c, we get

X
X
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Elimination in Chains with Evidence

• Finally, we eliminate b

X
X
X
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Variable Elimination

• General idea
• Write query in the form
• Iteratively
• Move all irrelevant terms outside of innermost
  sum
• Perform innermost sum, getting a new term
• Insert the new term into the product

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A More Complex Example

• Asia network

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• We want to compute P(d)
• Need to eliminate v,s,x,t,l,a,b
• Initial factors

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• We want to compute P(d)
• Need to eliminate v,s,x,t,l,a,b
• Initial factors

Eliminate v
Note fv(t) P(t) In general, result of
elimination is not necessarily a probability term
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• We want to compute P(d)
• Need to eliminate s,x,t,l,a,b
• Initial factors

Eliminate s
Summing on s results in a factor with two
arguments fs(b,l) In general, result of
elimination may be a function of several variables
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• We want to compute P(d)
• Need to eliminate x,t,l,a,b
• Initial factors

Eliminate x
Note fx(a) 1 for all values of a !!
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• We want to compute P(d)
• Need to eliminate t,l,a,b
• Initial factors

Eliminate t
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• We want to compute P(d)
• Need to eliminate l,a,b
• Initial factors

Eliminate l
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• We want to compute P(d)
• Need to eliminate b
• Initial factors

Eliminate a,b
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Variable Elimination

• We now understand variable elimination as a
  sequence of rewriting operations
• Actual computation is done in elimination step
• Exactly the same computation procedure applies to
  Markov networks
• Computation depends on order of elimination
• We will return to this issue in detail

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Dealing with evidence

• How do we deal with evidence?
• Suppose get evidence V t, S f, D t
• We want to compute P(L, V t, S f, D t)

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Dealing with Evidence

• We start by writing the factors
• Since we know that V t, we dont need to
  eliminate V
• Instead, we can replace the factors P(V) and
  P(TV) with
• These select the appropriate parts of the
  original factors given the evidence
• Note that fp(V) is a constant, and thus does not
  appear in elimination of other variables

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Dealing with Evidence

• Given evidence V t, S f, D t
• Compute P(L, V t, S f, D t )
• Initial factors, after setting evidence

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Dealing with Evidence

• Given evidence V t, S f, D t
• Compute P(L, V t, S f, D t )
• Initial factors, after setting evidence
• Eliminating x, we get

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Dealing with Evidence

• Given evidence V t, S f, D t
• Compute P(L, V t, S f, D t )
• Initial factors, after setting evidence
• Eliminating x, we get
• Eliminating t, we get

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Dealing with Evidence

• Given evidence V t, S f, D t
• Compute P(L, V t, S f, D t )
• Initial factors, after setting evidence
• Eliminating x, we get
• Eliminating t, we get
• Eliminating a, we get

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Dealing with Evidence

• Given evidence V t, S f, D t
• Compute P(L, V t, S f, D t )
• Initial factors, after setting evidence
• Eliminating x, we get
• Eliminating t, we get
• Eliminating a, we get
• Eliminating b, we get

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Complexity of variable elimination

• Suppose in one elimination step we compute
• This requires
• 
  multiplications
• For each value for x, y1, , yk, we do m
  multiplications
• additions
• For each value of y1, , yk , we do Val(X)
  additions
• Complexity is exponential in number of variables
  in the intermediate factor!

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Numeric example Green Network
Rain
Hiker
Gazlan
Car
Waste
Pollution
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Example

• Suppose we wish to calculate P(c0p1,w0), using
  the algorithm shown in class.
• Answer
• First, Well calculate P(C,p1,w0).
• We have the initial factors
• P(R)fR (R) r1 0.2
• r0 0.8

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Initial Factors

• P(GR) fG(G,R) r0 g0 0.1
• r0 g1 0.9
• r1 g0 0.3
• r1 g1 0.7
• P(HR) fH(H,R) h0 r0 0.2
• h0 r1 0.7
• h1 r0 0.8
• h1 r1 0.3

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Initial Factors (cont.)

• P(w0G,H) fW(w0,G,H) w0 h0 g0 0.9
• w0 h0 g1 0.7
• w0 h1 g0 0.8
• w0 h1 g1 0.15
• P(CH) fC(C,H) h0 c0 0.95
• h0 c1 0.05
• h1 c0 0.4
• h1 c1 0.6

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Initial Factors (cont.)

• P(p1C) FP(p1,C) p1 c0 0.2
• 
  p1 c1 0.9
• We will now choose an elimination order
• P(F,p1,w0)

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Calculation

• The steps are gwghr fWXfG w0 g0 h0 r0
  0.9X0.1 0.09
• 
  w0 g0 h0 r1 0.9X0.3 0.27
• w0 g0
  h1 r0 0.8X0.1 0.08
• 
  w0 g0 h1 r1 0.8X0.3 0.24
• 
  w0 g1 h0 r0 0.7X0.9 0.63
• 
  w0 g1 h0 r1 0.7X0.7 0.49
• 
  w0 g1 h1 r0 0.15X0.9 0.135
• 
  w0 g1 h1 r1 0.15X0.7 0.105

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Calculation (cont.)

• mhwr w0 h0 r0 0.090.63
  0.72
• w0 h0 r1 0.27 0.49 0.76
• w0 h1 r0 0.080.135 0.215
• w0 h1 r1 0.240.105 0.345
• gwhr fH x fR x mhwr w0 h0 r0 0.8x0.2x0.72
  0.1152
• w0 h0 r1
  0.2x0.7x0.76 0.1064
• w0 h1 r0
  0.8x0.8x0.215 0.1376
• w0 h1 r1
  0.2x0.3x0.345 0.0207

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Calculation (cont.)

• mhw w0 h0 0.11520.1064
  0.2216
• w0 h1 0.13760.0207
  0.1583
• gwhc fC x mhw w0 h0 c0 0.2216x0.95
  0.21052
• w0 h0 c1 0.2216x0.05
  0.01108
• w0 h1 c0 0.1583x0.4
  0.06332
• w0 h1 c1 0.1583x0.6
  0.09498
• mwc w0 c0 0.210520.06332
  0.27384
• w0 c1 0.011080.09498
  0.10606

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Calculation (cont.)

• gwcp fP x mwc w0 c0 p1 0.2x0.27384
  0.054768
• w0 c1 p1 0.8x0.10606
  0.095454
• Now, we have

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The Computational Cost

• Computing gwghr fW X fG requires 8
  multiplications.
• Computing mhwr requires 4
  additions.
• Computing gwhr fH x fR x mhwr requires 8
  multiplications.
• Computing mhw requires 2 additions.
• Computing gwhc fC x mhw requires 4
  multiplications.
• Computing mwc requires 2 additions.
• Computing gwcp fP x mwc requires 2
  multiplications.
• For a total of 8842 22 multiplications
• and 422 8 additions.

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Elimination order does matter

• We can choose another elimination order, say
  R,G,H
• For a total of 16442 26 multiplications,
• and 422 8 additions.