HybridBayesTree - GTSAM Docs

A HybridBayesTree is the hybrid equivalent of a gtsam.GaussianBayesTree. It represents the result of multifrontal variable elimination on a HybridGaussianFactorGraph.

Like a standard Bayes tree, it’s a tree structure where each node is a ‘clique’. However, in a HybridBayesTree, each clique contains a gtsam.HybridConditional representing $P(F_k | S_k)$ , where $F_k$ are the frontal variables eliminated in that clique, and $S_k$ are the separator variables shared with the parent clique.

In hybrid Bayes trees discrete variables can only occur as frontal variables if the separator is entirely discrete, i.e., we will never have the situation that a discrete variable is conditioned on a continuous variable.

The joint probability distribution $P(X, M)$ encoded by the tree is the product of all clique conditionals:

P(X, M) = \prod_k P(F_k : S_k)

(1)

Compared to a HybridBayesNet (from sequential elimination), a HybridBayesTree is (a) tree-structured, (b) often has a shallower structure. Both are advantageous for inference tasks like marginalization and incremental updates, especially in sparse problems common in robotics (SLAM).

import gtsam
import numpy as np

from gtsam import (
    HybridGaussianFactorGraph, 
    JacobianFactor, DecisionTreeFactor, HybridGaussianFactor,
    DiscreteValues
)
from gtsam.symbol_shorthand import X, D

import graphviz

Creating a HybridBayesTree (via Elimination)¶

HybridBayesTree objects are obtained by performing multifrontal elimination on a HybridGaussianFactorGraph.

# --- Create a HybridGaussianFactorGraph ---
hgfg = HybridGaussianFactorGraph()
dk0 = (D(0), 2) # Binary discrete variable

# Prior on D0: P(D0=0)=0.6, P(D0=1)=0.4
prior_d0 = DecisionTreeFactor([dk0], "0.6 0.4")
hgfg.add(prior_d0)

# Prior on X0: P(X0) = N(0, 1)
prior_x0 = JacobianFactor(X(0), np.eye(1), np.zeros(1), gtsam.noiseModel.Isotropic.Sigma(1, 1.0))
hgfg.add(prior_x0)

# Conditional measurement on X1: P(X1 | D0)
# Mode 0: P(X1 | D0=0) = N(1, 0.25)
gf0 = JacobianFactor(X(1), np.eye(1), np.array([1.0]), gtsam.noiseModel.Isotropic.Sigma(1, 0.5))
# Mode 1: P(X1 | D0=1) = N(5, 1.0)
gf1 = JacobianFactor(X(1), np.eye(1), np.array([5.0]), gtsam.noiseModel.Isotropic.Sigma(1, 1.0))
meas_x1_d0 = HybridGaussianFactor(dk0, [gf0, gf1])
hgfg.add(meas_x1_d0)

# Factor connecting X0 and X1: P(X1 | X0) = N(X0+1, 0.1)
odom_x0_x1 = JacobianFactor(X(0), -np.eye(1), X(1), np.eye(1), np.array([1.0]), gtsam.noiseModel.Isotropic.Sigma(1, np.sqrt(0.1)))
hgfg.add(odom_x0_x1)

print("Original HybridGaussianFactorGraph:")
# hgfg.print()
display(graphviz.Source(hgfg.dot()))

# --- Elimination ---
# Use default hybrid ordering (continuous first, then discrete)
ordering = gtsam.HybridOrdering(hgfg)
print(f"\nElimination Ordering: {ordering}")

# Perform multifrontal elimination
hybrid_bayes_tree = hgfg.eliminateMultifrontal(ordering)

print("\nResulting HybridBayesTree:")
# hybrid_bayes_tree.print()
display(graphviz.Source(hybrid_bayes_tree.dot()))

Original HybridGaussianFactorGraph:


Elimination Ordering: Position 0: x0, x1, d0


Resulting HybridBayesTree:

Operations on HybridBayesTree¶

Similar to HybridBayesNet, the tree can be used for optimization, evaluation, and extracting specific conditional distributions.

hbt = hybrid_bayes_tree # Use the tree from elimination

# --- Optimization (Finding MAP state) ---
# Computes MPE for discrete, then optimizes continuous given MPE
map_solution = hbt.optimize()
print("\nMAP Solution (Optimize):")
map_solution.print()

# --- MPE (Most Probable Explanation for Discrete Variables) ---
mpe_assignment = hbt.mpe()
print("\nMPE Discrete Assignment:")
print(mpe_assignment)


MAP Solution (Optimize):
HybridValues: 
  Continuous: 2 elements
  x0: 0
  x1: 1
  Discrete: (d0, 0)
  Nonlinear
Values with 0 values:

MPE Discrete Assignment:
DiscreteValues{7205759403792793600: 0}

# --- Optimize Continuous given specific Discrete Assignment ---
dv0 = DiscreteValues()
dv0[D(0)] = 0
cont_solution_d0_eq_0 = hbt.optimize(dv0)
print("\nOptimized Continuous Solution for D0=0:")
cont_solution_d0_eq_0.print()

dv1 = DiscreteValues()
dv1[D(0)] = 1
cont_solution_d0_eq_1 = hbt.optimize(dv1)
print("\nOptimized Continuous Solution for D0=1:")
cont_solution_d0_eq_1.print()

# --- Evaluation ---
# Evaluate error (sum of errors of clique conditionals)
total_error = hbt.error(map_solution)
print(f"\nTotal Error at MAP solution: {total_error}")


Optimized Continuous Solution for D0=0:
VectorValues: 2 elements
  x0: 0
  x1: 1

Optimized Continuous Solution for D0=1:
VectorValues: 2 elements
  x0: 1.90476
  x1: 3.09524

Total Error at MAP solution: 0.023412453920796855

# --- Extract Conditional Distributions ---
# Choose a specific GaussianBayesTree for a discrete assignment
gaussian_bayes_tree_d0_eq_0 = hbt.choose(dv0)
print("\nGaussianBayesTree for D0=0:")
gaussian_bayes_tree_d0_eq_0.print()
# display(graphviz.Source(gaussian_bayes_tree_d0_eq_0.dot()))


GaussianBayesTree for D0=0:
: cliques: 3, variables: 2
- p()
  R = Empty (0x0)
  d = Empty (0x1)
  mean: 0 elements
  logNormalizationConstant: -0
  No noise model
| - p(x1)
  R = [ 2.21565 ]
  d = [ 2.21565 ]
  mean: 1 elements
  x1: 1
  logNormalizationConstant: -0.123394
  No noise model
| | - p(x0 | x1)
  R = [ 3.31662 ]
  S[x1] = [ -3.01511 ]
  d = [ -3.01511 ]
  logNormalizationConstant: 0.280009
  No noise model