Fast Exact Bayes Net Structure Learning

1
Fast Exact Bayes NetStructure Learning
relatively-speakingly-

• Daniel Eaton
• Tuesday Oct 31, 2006

2
What structure learning is

• (SL)

Data
A
B
optimize
Space of models
) .1
p(
A
B
integrate
p(
) .2
A
B
) .7
p(
A
B
eg. dirichlet-multinomial DAGs
3
What SL is for this talk

• Have complete exchangeable data
• M cases, N variables
• Continuous or discrete
• Model space DAGs with CPD for each node (ex.
  multinomial or linear gaussian)
• plus a prior on parameters p(?G), and DAGs p(G)

4
What SL is for this talk

• To determine posterior over DAGs given data
• For each DAG G
• compute marginal likelihood p(dataG) by
  integrating out parameters ?
• easy for conjugate prior models
• ie. dirichlet-multinomial
• p(Gdata) a p(dataG)p(G)
• Unfortunately ...

5
Why SL is hard

• Number of DAGs is superexponential in of nodes
• 7 node DAGs...
• Even with a tight graph representation (1 b/edge)
  takes 6.3 GB to store
• Posterior prob takes 8.2 GB

6
Possible resolutions

• RW MCMC on DAG space
• "Posterior landscape" too big and bumpy bad
  mixing, doesn't work for Ngt10
• Never represent DAGs Condition on node orderings
  (Buntine 1991)
• Compute marginal probability of "graph features"
• Friedman Koller 2003 -gt MCMC
• Koivisto Sood 2004 -gt Exact!

7
Graph features

• Just an indicator function on DAGs (f(G)) which
  is 1 iff a particular structure (ie. an edge)
  exists in the graph, 0 otherwise
• Assume N3, want to know marginal probability
  that edge from A -gt B exists
• Of course, naively, this sum is difficult for ngt5
  ...

A
B
A
B
A
B
fab(
) 1
fab(
) 0
fab(
) 1
C
C
C
8
Conditioning on orders (order-trick)

• Order on variables, just a permutation of index
  set
• ie. 3 node graph, (1,2,3), (2,1,3), (3,2,1), ...
• Only N! of these!
• For a fixed order
• Intuition Can consider each node independently
  of the others (acyclicity ensured by ordering)

9
Order-trick feature probability

• Ultimately we want to determine
• For now, assume order known so we go for
• know how to do denominator
• numerator depends on feature f
• for f a single edge (A-gtB), it's easy

10
How to use order-trick in practice

• Except in very special (perhaps bio) cases we
  don't know the node order a priori
• Must sum over N! orders
• Ouch, still super-exponential!
• 1. Sample orders with MCMC (Koller/Friedman)
• Sampler mixes much better than in DAG-space
• Order-space smaller and less bumpy
• 2. Do it exactly with dynamic programming!
• Koivisto Sood (2004)

11
Koivisto

• Recognize that although there are N! orders to
  sum over, there is much redundant computation
• Will allow us to compute the marginal probability
  of a particular feature in O(N2N N2N-1C(M)) time
• All edge marg probs in O(N32N) time (naive)
• Or, O(N2N) time with a recent (2006) extension
  that I won't cover today

12
Koivisto

• Consider
• just need a way to evaluate
• since
• can be computed by setting f 1

13

• To simplify derivation, assume uniform
• Please accept that (proof later, if requested)
• so that
• Introduce
• so that

key each term is modular
14

• Brute force

()
O(NN!)
(1)
(2)
(3)
ordered sets
(1,2)
(1,3)
(2,1)
(2,3)
(3,1)
(3,2)
(1,2,3)
(1,3,2)
(2,1,3)
(2,3,1)
(3,1,2)
(3,2,1)


a1()a2(1)a3(1,2)
a2()a1(2)a3(1,2)


E
DP recurrence ...
a3(1,2) x ( a1()a2(1) a2()a1(2) )
15

• DP

O(N2N)
2
3
1
unordered sets
1,3
2,3
1,2
1,2,3
16
alpha
naive O(N3N)

• We assumed alpha was easy to compute
• for each i, there are 2N values ai()
• "Möbius transform"
• Exists clever algorithm to compute each ai in
  O(2N) time (so O(N2N) overall)
• Intuition (2 node)

precompute
3 node
111
11
10
11
10
11
101
010
110
10
100
00
01
00
01
010
00
01
001
000
17
Koivisto summary

• For each nodes, for each possible parent set,
  compute ML(i,Gi), fi, p(Gi) (each Nx2N arrays)
• Compute Betas
• ßf ML . f . p(G)
• ß1 ML . p(G) (trivial feature f1, used for
  normalization)
• Compute Alphas
• For each node, compute af,i() and a1,i() (1x2N
  vectors)
• Compute gf(V) using af and g1(V) using a1

O(N2N-1C(m))
O(N2N)
O(N2N)
O(N2N)
O(N2N N2N-1C(m))
18
Timing
30s
30m
day
19
Order-trick limitations

• Graph structure prior must be modular
• Breaks markov equivalence
• Cannot have arbitrary priors (ie. uniform
  impossible)
• Cannot query arbitrary features (f must be
  modular)
• ie. Directed path between nodes A B
• Resolution use MCMC with proposal based on
  sampling from marginal edge probabilities
• An independence sampler
• Works well for N5

20
Improvement by MCMC
"cancer" network
21
Improvement by MCMC
22
Follow-up

• To read Exact Bayesian structure learning from
  uncertain interventions
• Kevin Daniel
• Submitted to AIStats06
• To run
• Code in CVS Aline/koivisto
• Probably best to wait till Dec to use

Happy Halloween