pytreecat 0.1.3

A Bayesian latent tree model of multivariate multinomial data

A Bayesian latent tree model

TreeCat

Intended Use

TreeCat is an inference engine intended to power higher-level machine learning tools. TreeCat is appropriate for analyzing medium-sized tabular data with categorical and ordinal values, possibly with missing observations.

	TreeCat supports
Feature Types	categorical, ordinal
# Rows (n)	1000-100K
# Features (p)	10-1000
# Cells (n × p)	10K-10M
# Categories	2-10ish
Max Ordinal	10ish
Missing obervations?	yes
Repeated observations?	yes
Sparse data?	no, use something else
Unsupervised	yes
Semisupervised	yes
Supervised	no, use something else

Installing

First install numba (conda makes this easy). Then

$ pip install pytreecat

Quick Start

1. Format your data as a data.csv file with a header row. It’s fine to include extra columns that won’t be used.

   title	genre	decade	rating
   vertigo	thriller	1950s	5
   up	family	2000s	3
   desk set	comedy	1950s	4
   santapaws	family	2010s	1
   chinatown	mystery	1970s	4

2. Create a schema.csv with a header row name,type, where the first column is the feature name (genre, decade, rating), and the second column is the feature type (categorical or ordinal). It’s easiest to let TreeCat guess the feature types:

   $ treecat guess-schema data.csv schema.csv    # Creates schema.csv

   You can then edit the feature types in case TreeCat guessed incorrectly.

   name	type
   genre	categorical
   decade	categorical
   rating	ordinal

3. Import your csv files into treecat’s internal format. We’ll call our dataset dataset.pkz (a gzipped pickle file).

   $ treecat import-data data.csv schema.csv dataset.pkz

4. Train an ensemble model on your dataset. This typically takes ~15minutes for a 1M cell dataset.

   $ treecat train dataset.pkz ensemble.pkz

5. Load your trained model into a server

   from treecat.serving import EnsembleServer
   
   server = EnsembleServer('ensemble.pkz')

6. Run queries against the server. For example we can compute marginals

   server.sample(100, np.ones(V)).mean(axis=0)

   or compute a latent correlation matrix

   print(server.correlation())

The Server Interface

TreeCat’s server interface currently supports the two basic Bayesian operations:

• server.sample(N, counts, data=None) draws N samples from the joint posterior distribution, optionally conditioned on data.

• server.logprob(data) computes posterior log probability of data.

TreeCat’s internal data representation is multinomial, and thus supports missing and repeated measurements, and even data adding. For example to compute conditional probability of data A given data B, we can simply compute

cond = server.logprob(A + B) - server.logprob(B)

The Model

Let V be a set of vertices (one vertex per feature). Let C[v] be the dimension of the vth feature. Let N be the number of datapoints. Let K[n,v] be the number of observations of feature v in row n (e.g. 1 for a categorical variable, 0 for missing data, or k for an ordinal value with minimum 0 and maximum k).

TreeCat is the following generative model:

E ~ UniformSpanningTree(V)    # An undirected tree.
for v in V:
    Pv[v] ~ Dirichlet(size = [M], alpha = 1/2)
for (u,v) in E:
    Pe[u,v] ~ Dirichlet(size = [M,M], alpha = 1/(2*M))
    assume(Pv[u] == sum(Pe[u,v], axis = 1))
    assume(Pv[v] == sum(Pe[u,v], axis = 0))
for v in V:
    for i in 1:M:
        Q[v,i] ~ Dirichlet(size = [C[v]])
for n in 1:N:
    for v in V:
        X[n,v] ~ Categorical(Pv[v])
    for (u,v) in E:
        (X[n,u],X[n,v]) ~ Categorical(Pe[u,v])
    for v in V:
        Z[n,v] ~ Multinomial(Q[v,X[n,v]], count = K[n,v])

where we’ve avoided adding an arbitrary root to the tree, and instead presented the model as a manifold with overlapping variables and constraints.

The Inference Algorithm

This package implements fully Bayesian MCMC inference using subsample-annealed Gibbs sampling. There are two pieces of latent state that are sampled:

• Latent classes for each row for each vertex. These are sampled by single-site Gibbs sampling with a linear subsample-annealing schedule.

• The latent tree structure is sampled by randomly removing an edge and replacing it. Since removing an edge splits the graph into two connected components, the only replacement locations that are feasible are those that re-connect the graph.

The single-site Gibbs sampler uses dynamic programming to simultaneously sample the complete latent assignment vector for each row. A dynamic programming program is created each time the tree structure changes. This program is interpreted by various virtual machines for different purposes (training the model, sampling from the posterior, computing log probability of the posterior). The virtual machine for training is jit-compiled using numba.

License

Copyright (c) 2017 Fritz Obermeyer. TreeCat is licensed under the Apache 2.0 License.