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Exact Lifted Inference with Distinct Soft Evidence on Every Object Hung Hai Bui, Tuyen N. Huynh, Rodrigo de Salvo Braz Artificial Intelligence Center SRI International Menlo Park, CA, USA
July 26, 2012
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Outline
1 Outline
2 Distinct Soft Evidence is Problematic
3 LIDE (Lifted Inference with Distinct Evidence)
4 Experiments
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Lifted Inference and the Problematic Soft Evidence • The main idea of lifted inference is to exploit symmetry of the
probabilistic models. This leads to algorithms that can be very efficient on high-tree width, but symmetric models • Soft evidence at the level of every object destroys the model’s
symmetry • Everyone has different weight, cholesterol level, etc
Symmetric Symmetry destroyed
• Aim: lifted inference with distinct soft evidence on every object 3/18
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Distinct Soft Evidence on a Unary Predicate • The simplest form of distinct soft evidence: on every
grounding of a single unary predicate • Consider an MLN M consists of • An MLN M0 with a unary predicate q. • A set of soft evidence of the form wi : q(i) for every object i.
Evidence
M0 1.4
:
¬Smokes(x)
w1
:
Cancer (P1 )
2.3
:
¬Cancer (x)
w2
:
Cancer (P2 )
4.6
:
¬Friends(x, y )
1.5
:
Smokes(x) ⇒ Cancer (x)
w1000
:
Cancer (P1000 )
1.1
:
Smokes(x) ∧ Friends(x, y )
...
⇒ Smokes(y )
(tree-width = 1000) 4/18
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LIDE (Lifted Inference with Distinct Evidence)
• Most lifted inference methods applied to M would completely
shatter the model, thus reverting to ground inference. • LIDE’s approach 1 2
Perform lifted inference on M0 only Use special operations to absorb the soft evidences • Instead of exploiting symmetry of the model, we exploit symmetry of the partition function
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Symmetric Function
Definition A n-variable function F (t1 , . . . , tn ) is symmetric if for all permutation π, permuting the variables of F by π does not change the output value, that is, F (t1 , . . . , tn ) = F (tπ(1) . . . , tπ(n) ).
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• F depends only on the histogram of its arguments. • If ti ∈ {0, 1}, the set {ck }, k = 0, . . . , n, where ck = F (t) for
any t such that ktk1 = k is termed the counting representation of the symmetric function F . • An exchangable distribution is a symmetric function, so it
admits a counting representation.
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Exchangeability of Groundings of a Unary Predicate Theorem Let D∗ = {d1 , . . . , dn } be the set of individuals that do not appear as constants in the MLN M0 and q be a unary predicate in M0 . Let P0 (.) = Pr(q(d1 ) . . . q(dn ) | M0 ). Then, the random vector (q(d1 ) . . . q(dn )) is exchangeable under P0 .
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• Proof is in the paper. • This seems trivial: d1 , . . . dn do not appear in M0 so they are
“indistinguishable”. But beware, “indistinguishable” does not necessarily imply exchangeable: groundings of an n-ary predicate in general are NOT exchangeable when n > 1.
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LIDE as a Wrapper
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Step 1: apply any applicable lifted inference technique on M0 to compute the counting representation {ck } of P0 (). • One natural method is counting elimination.
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Step 2: Absorb the soft evidence • Equivalent to compute the posterior of a set of exchangable
binary random variables n
P(q1 , . . . , qn ) =
Y 1 P0 (q1 . . . qn ) φi (qi ) Z i=1
where qi = q(di )
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Posterior of Exchangeable Binary RVs
n
Pr (q1 , . . . , qn ) =
Y 1 P0 (q1 . . . qn ) φi (qi ) Z i=1
We discuss three related problems, to compute • The MAP configuration q under the marginal Pr(q) (a.k.a the
marginal-map problem) • The partition function Z • The marginal Pr(qi ) for each individual di
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MAP Inference Let αi =
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φi (1) φi (0) ,
Φ=
Q
φi (0). Then n
P(q) =
Y q Φ P0 (q1 . . . qn ) αi i Z i=1
max P(q) = q
n Y Φ max ck max αiqi Z k q:kqk1 =k i=1
• Observation: the 2nd maximization simply picks k largest
elements of α. • By sorting the vector α, the MAP problem can be solved in
O(n log(n)) given {ck } as input.
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Partition Function Z
Z (α1 , . . . , αn ) = Φ
X
P0 (q1 , . . . , qn )
q1 ...qn
n Y
αiqi
i=1
• Observation: Z is a polynomial in α. More importantly Z is a
symmetric polynomial. • According to the fundamental theorem of symmetric
polynomials, it can be expressed in terms of a small number of building units called elementary symmetric polynomials.
Z (α) = Φ
n X
ck ek (α)
k=0
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Elementary Symmetric Polynomials • ek (α) is the k-th order elementary symmetric polynomial in α,
the sum of all products of distinct k elements of α X ek (α) = αi1 . . . αik 1≤i1 <...
n
• Sum of k terms, so naive evaluation is a bad idea. P • Newton’s Identity: Let pk (α1 . . . αn ) = ni=1 αik be the k-th
power sum. Then
ek (α) =
1 k
Pk
i=1 (−1)
i−1 e k−i (α)pi (α)
• This yields a recursive method to compute all ek (α) in O(n2 ). • Thus, Z can be computed in O(n2 ) given {ck }. AAAI 2012
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Marginal on Each Individual
• As usual, the marginals Pr(qi ) can be computed in a way
similar to the computation of the normalization term Z , as the following theorem shows. (i)
• Let α(i) be the vector such that αi
(i)
= 0 and αj = αj for
every j 6= i. Then
Pn ck ek (α(i) ) Z (α(i) ) = Pk=0 Pr(qi = 0) = n Z (α) k=0 ck ek (α)
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Experimental Setup • Friends and Smokes domain. • Task: compute the marginal probability of having cancer of
each person given the cancer test readings of whole population. • Individual soft evidence uniformly sample from [0,2]. Thus,
lifted BP reduces to ground BP • Two versions of the “Friends & and Smokes” MLN: • Original Friends-and-Smokes: encourage Smokes(x) and
Smokes(y ) to be the same if Friends(x, y ) is unknown. • Attractive potential between Smokes(x) and Smokes(y ) • Friends-and-Smokes-Neg:
−1.1 : Smokes(x) ∧ Friends(x, y ) ⇒ Smokes(y ). • Repulsive potential between Smokes(x) and Smokes(y ) • Difficult test case for BP
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Running Time on “Friends & Smokes” 800 700 600 500 400
LIDE
300
BP
200 100 0
10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Running time (seconds)
900
Number of persons
*Note • Use a slightly modified C-FOVE for lifted inference without evidence • C-FOVE time dominates evidence absorbing time • Junction-tree ran out of memory for N = 30.
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Running Time on “Friends & Smokes-Neg” 9 Running time (seconds)
8 7 6 5
LIDE
4 BP with damping (damping = 0.1)
3 2 1 0 10 20 30 40 50 60 70 80 90 100 Number of persons
*Note: BP did not converge when N≥ 200 AAAI 2012
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Prob(Cancer)
Evidence Strength vs Probability 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
0.5 1 1.5 Evidence strength w
2
*Note:
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• This is a scatter plot, not a function. • Distribution of Pr(Cancer) spreads, so quantization will loose
accuracy. 17/18
Conclusion and Future Direction • We propose a new strategy for handling distinct soft evidence • Perform lifted inference (e.g. C-FOVE) without the distinct
soft evidence • Absorb the soft evidence by exploiting the symmetry of the
partition polynomial Z • Future direction • Soft evidence on multiple (L) unary predicates. • Polynomial in domain size N, but super-exponential in L • Need to depart from exact inference and derive efficient approximation. • Soft evidence on one binary predicate • Intractable in general • Are there efficient approximations that can be derived from this approach?
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