A BayesNet in GTSAM represents a directed graphical model, created by running sequential variable elimination (like Cholesky or QR factorization) on a FactorGraph or constructing from scratch.
It is essentially a collection of Conditional objects, ordered according to the elimination order. Each conditional represents , where the parents are variables that appear later in the elimination ordering.
A Bayes net represents the joint probability distribution as a product of conditional probabilities stored in the net:
The total log-probability of an assignment is the sum of the log-probabilities of its conditionals:
Like FactorGraph, BayesNet is templated on the type of conditional it stores (e.g., GaussianBayesNet, DiscreteBayesNet).
import gtsam
import numpy as np
import graphviz
# We need concrete graph types and elimination to get a BayesNet
from gtsam import GaussianFactorGraph, Ordering, GaussianBayesNet
# For the Asia example
from gtsam import DiscreteBayesNet, DiscreteConditional, DiscreteKeys, DiscreteValues, symbol
from gtsam import symbol_shorthand
X = symbol_shorthand.X
L = symbol_shorthand.LCreating a BayesNet (via Elimination)¶
BayesNets are typically obtained by eliminating a FactorGraph.
# Create a simple Gaussian Factor Graph P(x0) P(x1|x0) P(x2|x1)
graph = GaussianFactorGraph()
model = gtsam.noiseModel.Isotropic.Sigma(1, 1.0)
graph.add(X(0), -np.eye(1), np.zeros(1), model)
graph.add(X(0), -np.eye(1), X(1), np.eye(1), np.zeros(1), model)
graph.add(X(1), -np.eye(1), X(2), np.eye(1), np.zeros(1), model)
print("Original Factor Graph:")
graph.print()Original Factor Graph:
size: 3
factor 0:
A[x0] = [
-1
]
b = [ 0 ]
Noise model: unit (1)
factor 1:
A[x0] = [
-1
]
A[x1] = [
1
]
b = [ 0 ]
Noise model: unit (1)
factor 2:
A[x1] = [
-1
]
A[x2] = [
1
]
b = [ 0 ]
Noise model: unit (1)
# Eliminate sequentially using a specific ordering
ordering = Ordering([X(0), X(1), X(2)])
bayes_net = graph.eliminateSequential(ordering)
print("\nResulting BayesNet:")
bayes_net.print()
Resulting BayesNet:
size: 3
conditional 0: p(x0 | x1)
R = [ 1.41421 ]
S[x1] = [ -0.707107 ]
d = [ 0 ]
logNormalizationConstant: -0.572365
No noise model
conditional 1: p(x1 | x2)
R = [ 1.22474 ]
S[x2] = [ -0.816497 ]
d = [ 0 ]
logNormalizationConstant: -0.716206
No noise model
conditional 2: p(x2)
R = [ 0.57735 ]
d = [ 0 ]
mean: 1 elements
x2: 0
logNormalizationConstant: -1.46824
No noise model
Properties and Access¶
A BayesNet provides access to its constituent conditionals and basic properties.
print(f"BayesNet size: {bayes_net.size()}")
# Access conditional by index
conditional1 = bayes_net.at(1)
print("Conditional at index 1: ")
conditional1.print()
# Get all keys involved
bn_keys = bayes_net.keys()
print(f"Keys in BayesNet: {bn_keys}")BayesNet size: 3
Conditional at index 1:
GaussianConditional p(x1 | x2)
R = [ 1.22474 ]
S[x2] = [ -0.816497 ]
d = [ 0 ]
logNormalizationConstant: -0.716206
No noise model
Keys in BayesNet: x0x1x2
Evaluation and Solution¶
The logProbability(Values) method computes the log probability of a variable assignment given the conditional distributions in the Bayes net. For Gaussian Bayes nets, the optimize() method can be used to find the maximum likelihood estimate (MLE) solution via back-substitution.
# For GaussianBayesNet, we use VectorValues
mle_solution = bayes_net.optimize()
# Calculate log probability (requires providing values for all variables)
log_prob = bayes_net.logProbability(mle_solution)
print(f"Log Probability at {mle_solution.at(X(0))[0]:.0f},{mle_solution.at(X(1))[0]:.0f},{mle_solution.at(X(2))[0]:.0f}]: {log_prob}")
print("Optimized Solution (MLE):")
mle_solution.print()Log Probability at 0,0,0]: -2.7568155996140185
Optimized Solution (MLE):
VectorValues: 3 elements
x0: 0
x1: 0
x2: 0
Visualization¶
Bayes nets can also be visualized using Graphviz.
graphviz.Source(bayes_net.dot())Loading...
Example: DiscreteBayesNet (Asia Network)¶
While the previous examples focused on GaussianBayesNet, GTSAM also supports DiscreteBayesNet for representing probability distributions over discrete variables. Here we construct the classic ‘Asia’ network example directly by adding DiscreteConditional objects.
# Define keys for the Asia network variables
A = symbol('A', 8) # Visit to Asia?
S = symbol('S', 7) # Smoker?
T = symbol('T', 6) # Tuberculosis?
L = symbol('L', 5) # Lung Cancer?
B = symbol('B', 4) # Bronchitis?
E = symbol('E', 3) # Tuberculosis or Lung Cancer?
X = symbol('X', 2) # Positive X-Ray?
D = symbol('D', 1) # Dyspnea (Shortness of breath)?
# Define cardinalities (all are binary in this case)
cardinalities = { A: 2, S: 2, T: 2, L: 2, B: 2, E: 2, X: 2, D: 2 }
# Helper to create DiscreteKeys object
def make_keys(keys_list):
dk = DiscreteKeys()
for k in keys_list:
dk.push_back((k, cardinalities[k]))
return dk# Create the DiscreteBayesNet
asia_net = DiscreteBayesNet()
# Helper function to create parent list in correct format
def make_parent_tuples(parent_keys):
return [(pk, cardinalities[pk]) for pk in parent_keys]
# P(D | E, B) - Dyspnea given Either and Bronchitis
asia_net.add(DiscreteConditional((D, cardinalities[D]), make_parent_tuples([E, B]), "9/1 2/8 3/7 1/9"))
# P(X | E) - X-Ray result given Either
asia_net.add(DiscreteConditional((X, cardinalities[X]), make_parent_tuples([E]), "95/5 2/98"))
# P(E | T, L) - Either Tub. or Lung Cancer (OR gate)
# "F T T T" means P(E=1|T=0,L=0)=0, P(E=1|T=0,L=1)=1, P(E=1|T=1,L=0)=1, P(E=1|T=1,L=1)=1
asia_net.add(DiscreteConditional((E, cardinalities[E]), make_parent_tuples([T, L]), "F T T T"))
# P(B | S) - Bronchitis given Smoker
asia_net.add(DiscreteConditional((B, cardinalities[B]), make_parent_tuples([S]), "70/30 40/60"))
# P(L | S) - Lung Cancer given Smoker
asia_net.add(DiscreteConditional((L, cardinalities[L]), make_parent_tuples([S]), "99/1 90/10"))
# P(T | A) - Tuberculosis given Asia
asia_net.add(DiscreteConditional((T, cardinalities[T]), make_parent_tuples([A]), "99/1 95/5"))
# P(S) - Prior on Smoking
asia_net.add(DiscreteConditional((S, cardinalities[S]), [], "1/1")) # or "50/50"
# Add conditional probability tables (CPTs) using C++ sugar syntax
# P(A) - Prior on Asia
asia_net.add(DiscreteConditional((A, cardinalities[A]), [], "99/1"))
print("Asia Bayes Net:")
asia_net.print()Asia Bayes Net:
DiscreteBayesNet
size: 8
conditional 0: P( D1 | E3 B4 ):
Choice(E3)
0 Choice(D1)
0 0 Choice(B4)
0 0 0 Leaf 0.9
0 0 1 Leaf 0.2
0 1 Choice(B4)
0 1 0 Leaf 0.1
0 1 1 Leaf 0.8
1 Choice(D1)
1 0 Choice(B4)
1 0 0 Leaf 0.3
1 0 1 Leaf 0.1
1 1 Choice(B4)
1 1 0 Leaf 0.7
1 1 1 Leaf 0.9
conditional 1: P( X2 | E3 ):
Choice(X2)
0 Choice(E3)
0 0 Leaf 0.95
0 1 Leaf 0.02
1 Choice(E3)
1 0 Leaf 0.05
1 1 Leaf 0.98
conditional 2: P( E3 | T6 L5 ):
Choice(T6)
0 Choice(L5)
0 0 Choice(E3)
0 0 0 Leaf 1
0 0 1 Leaf 0
0 1 Choice(E3)
0 1 0 Leaf 0
0 1 1 Leaf 1
1 Choice(L5)
1 0 Choice(E3)
1 0 0 Leaf 0
1 0 1 Leaf 1
1 1 Choice(E3)
1 1 0 Leaf 0
1 1 1 Leaf 1
conditional 3: P( B4 | S7 ):
Choice(S7)
0 Choice(B4)
0 0 Leaf 0.7
0 1 Leaf 0.3
1 Choice(B4)
1 0 Leaf 0.4
1 1 Leaf 0.6
conditional 4: P( L5 | S7 ):
Choice(S7)
0 Choice(L5)
0 0 Leaf 0.99
0 1 Leaf 0.01
1 Choice(L5)
1 0 Leaf 0.9
1 1 Leaf 0.1
conditional 5: P( T6 | A8 ):
Choice(T6)
0 Choice(A8)
0 0 Leaf 0.99
0 1 Leaf 0.95
1 Choice(A8)
1 0 Leaf 0.01
1 1 Leaf 0.05
conditional 6: P( S7 ):
Leaf 0.5
conditional 7: P( A8 ):
Choice(A8)
0 Leaf 0.99
1 Leaf 0.01
# Visualize the network structure
dot_string = asia_net.dot()
display(graphviz.Source(dot_string))
# Evaluate the log probability of a specific assignment
# Example: Calculate P(A=0, S=0, T=0, L=0, B=0, E=0, X=0, D=0)
values = DiscreteValues()
for key, card in cardinalities.items():
values[key] = 0 # Assign 0 to all variables to start
log_prob_zeros = asia_net.logProbability(values)
print(f"Log Probability of all zeros: {log_prob_zeros}")
# Sample from the Bayes Net
sample = asia_net.sample()
print("Sampled Values (basic print):")
print(sample)
# --- Pretty Print ---
print("Sampled Values (pretty print):")
# Create a reverse mapping from integer key to string like 'A8'
# We defined A=symbol('A',8), S=symbol('S',7), etc. above
symbol_map = { A: 'A8', S: 'S7', T: 'T6', L: 'L5', B: 'B4', E: 'E3', X: 'X2', D: 'D1' }
# Iterate through the sampled values and print nicely
# Sort items by the symbol string for consistent order (optional)
for key, value in sorted(sample.items(), key=lambda item: symbol_map.get(item[0], str(item[0]))):
symbol_str = symbol_map.get(key, f"UnknownKey({key})") # Get 'A8' from key A
print(f" {symbol_str}: {value}")Loading...
Log Probability of all zeros: -1.2366269421045588
Sampled Values (basic print):
DiscreteValues{4683743612465315848: 0, 4755801206503243780: 1, 4899916394579099649: 1, 4971973988617027587: 0, 5476377146882523141: 0, 5980780305148018695: 1, 6052837899185946630: 0, 6341068275337658370: 0}
Sampled Values (pretty print):
A8: 0
B4: 1
D1: 1
E3: 0
L5: 0
S7: 1
T6: 0
X2: 0