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Tensor Belief Propagation

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Tensor Belief Propagation (TBP) is an experimental algorithm for approximate inference in discrete graphical models [1]. It takes a factor graph in .uai or .fg format and outputs approximate marginals for each variable.

[1] Wrigley, Andrew, Wee Sun Lee, and Nan Ye. "Tensor Belief Propagation." International Conference on Machine Learning. 2017.


  • Linux or OSX
  • Python 3.6+


Install libDAI prerequisites:

# Linux
$ sudo apt-get install g++ make doxygen graphviz libboost-dev libboost-graph-dev libboost-program-options-dev libboost-test-dev libgmp-dev cimg-dev

$ brew install boost gmp doxygen graphviz

Install tbp with the Python package manager pip:

$ pip install tbp
Successfully installed tbp-X.X.X

This will take a while as libDAI must be compiled.


TBP takes a factor graph in either .fg or .uai format as input, and outputs the approximate marginal distribution of each variable in .MAR format. This involves two steps — first, all potential functions in the graph must be decomposed into sums of rank-1 tensors yielding a decomposed factor graph (.dfg). Then, the message passing procedure must be run on the decomposed graph to give approximate marginals.

Command line

After installation, the command line utility tbp is available to do either or both of these steps. For usage instructions, run tbp --help.


Decompose the factor graph ising_8x8.fg and find marginals:

$ tbp tests/ising_8x8.fg
64 2 0.594961 0.405039 2 ... 0.608573 0.391427

Decompose input potentials into 3 rank-1 components and save the resulting decomposed graph (but don't find marginals):

$ tbp tests/ising_8x8.fg -r 3 -o tests/ising_8x8.dfg --verbosity 2
Reading graph tests/ising_8x8.fg (libDAI format)...
Decomposing input graph (r=3 terms per factor)...
Successfully saved decomposed graph to tests/ising_8x8.dfg.

Decompose the factor graph Promedus_11.uai after applying some evidence, find marginals using TBP with sample size of 1000, and save the output to out.MAR:

$ tbp tests/uai/MAR_prob/Promedus_11.uai -e tests/uai/MAR_prob/Promedus_11.uai.evid -k 1000 -o out.MAR --verbosity 2
Reading graph tests/uai/MAR_prob/Promedus_11.uai (UAI format)...
Applying evidence file tests/uai/MAR_prob/Promedus_11.uai.evid...
Decomposing input graph (r=4 terms per factor)...
Running TBP with sample size K=1000...
Successfully saved marginals to out.MAR.

Python library

The tbp package can also be used directly from Python, for example:

import tbp

# Load a factor graph in .uai format
g = tbp.load_uai_graph('tests/uai/MAR_prob/linkage_11.uai')

# Apply evidence (fixed variable assignments)

# Decompose each factor into a weighted sum of 4 rank-1 tensors
dg = g.decompose(r=4)

# Run TBP to find marginals with sample size of 10000
mar = dg.tbp_marg(K=10000)


Installing into a virtual environment

If pip install has issues with dependencies or version conflicts, you can install the necessary packages into a virtual environment (a project-specific folder rather than globally on your system):

$ sudo pip3 install virtualenv  # pip or pip3, depending on your system
$ virtualenv -p python3 venv    # create venv folder to store packages
$ source venv/bin/activate      # activate virtual environment
$ pip install tbp               # install tbp into venv folder

Now when you invoke tbp, the local versions will be used.

Building from GitHub clone

To use the tbp Python package from source without installation via pip install, libDAI must first be compiled:

$ git clone
$ cd tbp/libdai
$ cp Makefile.<platform> Makefile.conf  # Choose <platform> according to your platform
$ make
libDAI built successfully!

This produces a utility libdai/utils/dfgmarg which is symlinked from tbp/dfgmarg and used during inference. See libDAI README for full installation instructions.

Using MATLAB for the decomposition

The decomposition of potential functions uses the non-negative CP decomposition algorithm in the Tensorly tensor library. As an alternative to TensorLy, the MATLAB Tensor Toolbox can be used (this was what we used in [1]). To use this instead of Tensorly:

You can now replace method='tensorly' with method='matlab' when calling decomposition functions in

File formats

.dfg (decomposed factor graph)

We created the .dfg file format based on libDAI's .fg file format to represent decomposed factor graphs. A decomposed factor graph is a factor graph with all factors represented as sums of rank-1 tensors rather than multidimensional tables.

The first line of a .dfg file contains the number of factors in the graph, followed by a blank line. Then, factors are described in turn by blocks separated by a single blank line. Each factor block is structured as follows:

 1. n_terms
 2. <weights>
 3. n_variables
 4. <variable indices>
 5. <variable cardinalities>
 6. n_nonzero_1
 7. 1 0.5
 8. 3 0.1
 9. 4 0.1
10. ...
11. n_nonzero_2
12. 1 0.5
13. 3 0.1
14. 4 0.3
15. ...

In the header section of the factor block (lines 1-5), n_terms is the number of terms in the decomposition and <weights>, <variable indices> and <variable cardinalities> are self-explanatory space-separated lists of length n_terms, n_variables and n_variables respectively.

The remainder of the factor block (line 6 onwards) describes a series of n_variables 2D matrices that together describe the n_terms rank-1 tensors. Each matrix corresponds to a single variable and has shape (cardinality, n_terms), where cardinality is the cardinality of the variable and n_terms is the number of rank-1 terms in the decomposition (constant for all variables). Each matrix begins with the number of nonzero values in the matrix, followed by a series of index value pairs describing the nonzero entries of the matrix in column-major order. See libDAI's documentation for examples of how to reshape these lists back into matrices.

The ith rank-1 tensor is constructed by taking the outer product of the ith columns of all matrices. The complete factor is then reconstructed by adding up these rank-1 tensors and weighting according to <weights>.

Other file formats

Other file formats used in this project are:

To do

  • ICML experiments - finish cleaning code used for experiments (see for partial code)
  • Rewrite code that loads .uai files to handle all problems (currently breaks on some)
  • Deal with Z <= 0 warning from C++ code
  • Clean up C++ code and compiler warnings
  • Add more tests


Bug reports, suggestions and comments are welcome. Please email or use the issue tracker.


See LICENSE.txt (MIT).



Tensor Belief Propagation - algorithm for approximate inference in discrete graphical models







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