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\author{Jun Woo\\University of Minnesota, Minneapolis
   \And Jinhua Wang\\University of Minnesota, Minneapolis}
\Plainauthor{Jun Woo, Jinhua Wang}

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%\title{A Short Demo Article: Regression Models for Count Data in \proglang{R}}
\title{\pkg{bbl}: Boltzmann Bayes Learner for High-Dimensional Inference with
Discrete Predictors in \proglang{R}}
\Plaintitle{bbl: Boltzmann Bayes Learner for High-Dimensional Inference with 
Discrete Predictors in R}
\Shorttitle{\pkg{bbl}: Boltzmann Bayes Learner in \proglang{R}}

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\Abstract{
  Non-regression-based inferences, such as discriminant analysis, 
  can account for the effect of predictor distributions that may be 
  significant in big data modeling. We describe \pkg{bbl}, an \proglang{R}
  package for Boltzmann Bayes learning, which enables a comprehensive
  supervised learning of the association between a large number of 
  categorical predictors and multi-level response variables. Its basic underlying
  statistical model is a collection of (fully visible) Boltzmann machines
  inferred for each distinct response level. The algorithm reduces to the
  naive Bayes learner when interaction is ignored. We illustrate example use
  cases for various scenarios, ranging from modeling of a relatively small
  set of factors with heterogeneous levels to those with hundreds or more
  predictors with uniform levels such as image or genomic data. 
  We show how \pkg{bbl} explicitly quantifies the extra power provided by
  interactions via higher predictive performance of the model. In comparison
  to deep learning-based methods such as restricted Boltzmann machines,
  \pkg{bbl}-trained models can be interpreted directly via their 
  bias and interaction parameters.
}

\Keywords{Supervised learning, Boltzmann machine, naive Bayes, discriminant analysis, \proglang{R}}
\Plainkeywords{Supervised learning, Boltzmann machine, naive Bayes, discriminant analysis, R}

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\Address{
  Jun Woo\footnote{Current address: Memorial Sloan Kettering Cancer Center, New York, New York, USA} ({\it corresponding author}), Jinhua Wang\\
  Institute for Health Informatics\\
  \emph{and}\\
  Masonic Cancer Center\\
  University of Minnesota\\
  Minneapolis, Minnesota, USA\\
  E-mail: \email{wooh@mskcc.org}
}

\begin{document}
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%\section[Introduction: Count data regression in R]{Introduction: Count data regression in \proglang{R}} \label{sec:intro}
\section{Introduction}\label{sec:intro}

Many supervised learning tasks involve modeling discrete response
variables $y$ using predictors ${\bf x}$ that can occupy categorical 
factor levels \citep{hastie_etal}.
Ideally, it would be best to model the joint distribution $P({\bf x},y)$
via maximum likelihood, 
\begin{equation}
{\hat \Theta} = \argmax_\Theta \left[\ln P({\bf x},y|\Theta)\right],
\end{equation}
to find parameters $\Theta$. Regression-based methods use 
$P({\bf x},y)=P(y|{\bf x})P({\bf x})\simeq P(y|{\bf x})$. Their advantages include
the wealth of information provided for significance of fit coefficients from rigorous formal results.
An alternative is to use $P({\bf x},y)=P({\bf x}|y)P(y)$ and fit $P({\bf x}|y)$. 
Since $y$ is low-dimensional, this approach could capture extra
information not accessible from regression when there are many covarying predictors.
To make predictions for $y$ using $P({\bf x}|y)$, one uses the Bayes' formula.
Examples include linear and quadratic discriminant analyses 
\citep[pp.~106-119]{hastie_etal} for continuous ${\bf x}$. For discrete ${\bf x}$, naive Bayes 
is the simplest approach, where the covariance among ${\bf x}$ is ignored 
via
\begin{equation}
P({\bf x}|y)\simeq \prod_i P(x_i|y)
\label{eq:nbayes}
\end{equation}
with ${\bf x}=(x_1,\cdots,x_m)$.

In this paper, we focus on supervised learners taking into account the 
high-dimensional nature of $P({\bf x}|y)$ beyond the naive Bayes-level description given by Eq.~(\ref{eq:nbayes}). Namely, 
a suitable parametrization is provided by the Boltzmann machine 
\citep{ackley_etal}, which for the simple binary predictor $x_i=0,1$,
\begin{equation}
P({\bf x}|y)=\frac{1}{Z_y}\exp\left(\sum_i h_i^{(y)}x_i + \sum_{i<j} J_{ij}^{(y)}x_ix_j\right),
\label{eq:boltzmann}
\end{equation}
where $Z_y$ is the normalization constant, or partition function. Equation~(\ref{eq:boltzmann}) 
is the Gibbs distribution for Ising-type models in statistical mechanics \citep{chandler}. 
The two sets of parameters $h_i^{(y)}$ and $J_{ij}^{(y)}$ each represent single variable and 
two-point interaction effects, respectively. 
When the latter vanishes, the model leads to the naive Bayes classifier.
Although exact inference of Eq.~(\ref{eq:boltzmann}) from data is in general not possible, recent
developments led to many accurate and practically usable approximation schemes 
\citep{hyvarinen, morcos_etal,nguyen_etal, nguyen_wood-ieee,nguyen_wood}, 
making its use in supervised learning a viable alternative to regression methods. 
Two approximation methods available for use are pseudo-likelihood inference \citep{besag}
and mean field theory \citep{chandler,nguyen_etal}.

A recently described package \pkg{BoltzMM} can fit the (`fully visible') Boltzmann machine given by 
Eq.~(\ref{eq:boltzmann}) to data using pseudo-likelihood inference \citep{BoltzMM,jones_etal}. 
In contrast, classifiers based on this class of models remain largely unexplored. Supervised learners using 
statistical models of the type (\ref{eq:boltzmann}) usually take the form 
of the {\it restricted} Boltzmann machines instead \citep{hinton}, 
where (visible) predictors are augmented by hidden units and interactions 
are zero except between visible and hidden units. 
The main drawback of such layered Boltzmann machine learners, as is common in all deep learning
algorithms, is the difficulty in interpreting trained models. In contrast, with the fully visible architecture, 
$J_{ij}^{(y)}$ in Eq.~(\ref{eq:boltzmann}), if inferred with 
sufficient power while avoiding overfitting, has direct interpretation of interaction between two variables.

We refer to such learning/prediction algorithms using a generalized version
of Eq.~(\ref{eq:boltzmann}) as Boltzmann Bayes inference.
An implementation specific to genomic single-nucleotide polymorphism (SNP) data (two response groups, e.g., case and control, and uniform three-level
predictors, i.e., allele counts of $x_i= 0,1,2$) has been reported 
previously \citep{woo_etal-2016}. However, this \proglang{C++} software 
was geared specifically toward genome-wide association studies and is not
suitable for use in more general settings. 
We introduce an \proglang{R} package \pkg{bbl} (Boltzmann Bayes Learner),
which uses both \proglang{R} and \proglang{C++} for usability and 
performance, allowing the user to train and test statistical models in a
variety of different usage settings.

\section{Model and algorithm} 
For completeness and for reference to software features described in 
Section~\ref{sec:usage}, we summarize in this section key relevant formulas \citep{woo_etal-2016} used by \pkg{bbl}, 
generalized such that predictors each can have varying number of factor levels. 

\subsection{Model description}
The discrete response $y_k$ for an instance $k$ takes factor values $y$ among $G\ge 2$ groups; e.g.,
$y=\text{\code{case},\:\code{control}}$ with $G=2$; 
$k=1,\cdots, n$ denotes sample (or configuration) index. 
We also introduce weights $w_k$, each of which is integral number of times each
configuration was observed in data, such that $\sum_k w_k=n_s$ is the total 
sample size. If the data take the form of one entry per observation, 
$w_k=1$ and $n=n_s$. The use of frequency $w_k$ can lead to more efficient
learning when the number of predictors is relatively small. 
We use symbol $y$ for a particular factor value and generic response variables
interchangeably. 

The model attempts to connect response $y$ to a set of predictors represented
by ${\bf x}$ with elements $x_i$ and the observed data for an instance $k$ denoted
by ${\bf x}^k$. We assume that predictor variables take discrete factor levels,
each with distinct effect on responses, e.g., 
$x_i=\text{\code{a},\:\code{t},\:\code{g},\:\code{c}}$ for DNA sequence data.
The overall likelihood is
\begin{equation}
L=\sum_k w_k\ln P({\bf x}^k,y_k)
=\sum_y\sum_{k \in \mathbbm{K}_y}w_k\ln P({\bf x}^k,y)\equiv \sum_y L_y,
\end{equation}
where the second summation restricts $k$ to the set $\mathbbm{K}_y$ of
all $k$ values for which $y_k=y$.
The inference is first performed for each group $y$ separately, maximizing $L_y$ given by
\begin{equation}
L_y = \sum_{k \in \mathbbm{K}_y} w_k\left[\ln P({\bf x}^k|y)+\ln P(y)\right]
=\sum_{k\in \mathbbm{K}_y} w_k\ln P({\bf x}^k|y)+n_y \ln p_y,
\label{eq:Ly}
\end{equation}
where $p_y\equiv P(y)$ is the marginal distribution of $y$ and 
$n_y=\sum_{k \in \mathbbm{K}_y} w_k$ 
is the size of group $y$.

In the parametrization we adopt for the first term in Eq.~(\ref{eq:Ly}),
the group-specific predictor distribution is written as
\begin{equation}
P({\bf x}|y)=\frac{1}{Z_y} \exp\left[ \sum_i h_i^{(y)}(x_i) + \sum_{i<j} J_{ij}^{(y)}(x_i,x_j)\right].
\label{eq:pxy}
\end{equation}
The number of parameters (d.f.) per group $y$ in $\Theta_y=\{h_i^{(y)}(x), J_{ij}^{(y)}(x,x^\prime)\}$ is 
\begin{equation}
{\rm d.f.}=\sum_i (L_i-1) + \sum_{i<j} (L_i-1)(L_j-1),
\end{equation}
where $L_i$ is the total number of levels in factor $x_i$, which contributes one less parameters to d.f. because one of the factors can be taken as reference with the rest measured against it. Internally, 
\pkg{bbl} orders factors, assigns codes $a_i=0,\cdots, L_i-1$, and set $h_i^{(y)}(a_i)=
J_{ij}^{(y)}(a_i,a_j)=0$ whenever $a_i=0$ or $a_j=0$.
We refer to $h_i^{(y)}(x)$ and $J_{ij}^{(y)}(x,x^\prime)$ as bias and interaction 
parameters, respectively.

In the special case where predictor levels are binary ($x_i=0,1$), one may use
the spin variables $s_i=2x_i-1=\pm 1$, as in the package \pkg{BoltzMM} 
\citep{jones_etal,BoltzMM}.
Its distribution \citep{jones_etal}
\begin{equation}
P({\bf s})\propto \exp\left( \frac{1}{2}{\bf s}^\top\, {\bf M}\, {\bf s} + {\bf b}^\top {\bf s}\right)
\label{eq:boltzmm}
\end{equation}
is then related to Eq.~(\ref{eq:boltzmann}) by
\begin{subequations}
\begin{eqnarray}
b_i &=& \frac{h_i}{2} + \frac{1}{4}\sum_{j\ne i} J_{ij},\\
M_{ij} &=& \frac{1}{4} J_{ij},
\end{eqnarray}
\end{subequations}
where parameter superscripts were omitted because response group is not present.

\subsection{Pseudo-likelihood inference}
One option for fitting Eq.~(\ref{eq:pxy}) to data is pseudo-likelihood maximization \citep{besag}:
\begin{equation}
L_y-n_y\ln p_y=\sum_{k\in\mathbbm{K}_y} w_k\ln P({\bf x}^k|y)
\simeq \sum_{k\in \mathbbm{K}_y} 
w_k\sum_i \ln P_i(x_i^k|y,x_{j\backslash i}^k)
\equiv \sum_i L_{iy},
\label{eq:Ly2}
\end{equation}
where the effective univariate distribution is conditional to all other predictor values:
\begin{equation}
P_i(x|y,x_{j\backslash i})=\frac{e^{{\bar h}_i^{(y)}(x|x_{j\backslash i})}}{Z_{iy}(x_{j\backslash i})},
\end{equation}
\begin{equation}
Z_{iy}(x_{j\backslash i})
=\sum_x e^{{\bar h}_i^{(y)}(x|x_{j\backslash i})}=1+\sum_{a=1}^{L_i-1} e^{{\bar h}_i^{(y)}(a|x_{j\backslash i})},
\label{eq:ziy}
\end{equation}
and 
\begin{equation}
{\bar h}_i^{(y)}(x|x_{j\backslash i})=h_i^{(y)}(x) + \sum_{j\ne i} J_{ij}^{(y)}(x,x_j).
\label{eq:barh}
\end{equation}

Including $L_2$ penalizers $(\lambda_h,\lambda)$, 
$L_{iy}$ in Eq.~(\ref{eq:Ly2}) becomes
\begin{equation}
L_{iy}=\sum_{k\in \mathbbm{K}_y} w_k
\left[{\bar h}_i^{(y)}(x_i^k|x_{j\backslash i}^k)-\ln Z_{iy}(x_{j\backslash i}^k)\right]
-\frac{\lambda_h}{2} \sum_x h_i^{(y)}(x)^2
-\frac{\lambda}{2}\sum_{j,x,x^\prime} J_{ij}^{(y)}(x,x^\prime)^2
\label{eq:Liy}
\end{equation}
with first derivatives
\begin{subequations}
\begin{eqnarray}
\frac{\partial L_{iy}/n_y}{\partial h_i^{(y)}(x)} &=& {\hat f}_i^{(y)}(x) 
- \frac{1}{n_y}\sum_{k\in \mathbbm{K}_y} w_k P_i(x|y, x_{l\backslash i}^k)
-\lambda_h h_i^{(y)}(x),
\\
\frac{\partial L_{iy}/n_y}{\partial J_{ij}^{(y)}(x,x^\prime)} &=& 
{\hat f}_{ij}^{(y)}(x,x^\prime) 
- \frac{1}{n_y}\sum_{k\in\mathbbm{K}_y} w_k\mathbbm{1}(x_j^k=x^\prime)
P_i(x|y, x_{l\backslash i}^k)
-\lambda J_{ij}^{(y)}(x,x^\prime)\nonumber\\,
\end{eqnarray}
\label{eq:partial}
\end{subequations}
\!\!where 
\begin{subequations}
\begin{eqnarray}
{\hat f}_i^{(y)}(x) \!&= &\! \frac{1}{n_y}\sum_{k\in \mathbbm{K}_y} w_k 
\mathbbm{1}(x_i^k=x),\\
{\hat f}_{ij}^{(y)}(x,x^\prime) \!&= &\! \frac{1}{n_y}\sum_{k\in
\mathbbm{K}_y} w_k \mathbbm{1}(x_i^k=x)\mathbbm{1}(x_j^k=x^\prime)
\end{eqnarray}
\end{subequations}
are the first and second moments of predictor values and $\mathbbm{1}(x)$ 
is the indicator function.
In \pkg{bbl}, Eqs.~(\ref{eq:partial}) are solved in \proglang{C++} functions 
using the quasi-Newton optimization function 
\code{gsl_multimin_fdfminimizer_vector_bfgs2} in GNU Scientific Library 
(\url{https://www.gnu.org/software/gsl}).
By default, $\lambda_h=0$ and only interaction parameters are penalized.
As can be seen from the third equality of Eq.~(\ref{eq:Ly2}), the pseudo-likelihood inference
decouples into individual predictors, and the inference for each $i$ in \pkg{bbl} is performed
sequentially. The resulting interaction parameters, however, do not satisfy the required
symmetry,
\begin{equation}
J_{ij}(x,x^\prime)=J_{ji}(x^\prime,x).
\end{equation}
After pseudo-likelihood inference, therefore, the interaction parameters are symmetrized as follows:
\begin{equation}
J_{ij}(x,x^\prime)\leftarrow \frac{1}{2}\left[ J_{ij}(x,x^\prime)+ J_{ji}(x^\prime, x)\right].
\end{equation}

In \pkg{bbl}, the input data are filtered such that predictors with only one
factor level (no variation in observed data) are removed. Nevertheless, in cross-validation
of the processed data, sub-divisions into training and validation sets
may lead to instances where factor levels observed for a given predictor within $x_i$ in 
Eq.~(\ref{eq:partial}) are only a subset of those in the whole data.  
It is thus possible that optimization based on Eqs.~(\ref{eq:partial}) is 
ill-defined when any of the predictors are constant. 
In such cases, we augment the training data by an extra instance, in which constant predictors take other factor levels.

\subsection{Mean field inference}
The other option for predictor distribution inference is mean field approximation. In 
data-driven inference, the interaction parameters are approximated as \citep{nguyen_etal}
\begin{equation}
{\hat J}_{ij}^{(y)}(x,x^\prime) = -\left[\mathsf{C}^{(y)}\right]^{-1}_{ij}(x,x^\prime),
\label{eq:mf1}
\end{equation}
i.e., negative inverse of the covariance matrix,
\begin{equation}
\mathsf{C}^{(y)}_{ij}(x,x^\prime) = {\hat f}_{ij}(x,x^\prime) - {\hat f}_i(x){\hat f}_j(x^\prime).
\label{eq:Cij}
\end{equation}
Equation~(\ref{eq:mf1}) can be interpreted as treating discrete ${\bf x}$ as if it were multivariate normal:
Eq.~(\ref{eq:pxy}) would then be the counterpart of the multivariate normal p.d.f. with 
$-J_{ij}^{(y)}(x,x^\prime)$ corresponding to the precision matrix.
In real data where $n \sim {\rm d.f.}$ or less, the matrix inversion is often ill-behaved. It is regularized by 
interpolation of $\mathsf{C}^{(y)}$ 
between non-interacting (naive Bayes)
($\epsilon=0$) and fully interacting limits ($\epsilon=1$):
\begin{equation}
\mathsf{C}^{(y)}  \leftarrow \mathsf{\bar C}^{(y)}=(1-\epsilon)\frac{{\rm Tr}\:\mathsf{C}^{(y)}}
{{\rm Tr}\:\mathsf{I}}{\mathsf I} + \epsilon\mathsf{C}^{(y)},
\end{equation}
where $\mathsf{I}$ is the identity matrix of the same dimension as $\mathsf{C}^{(y)}$.
The parameter $\epsilon$ serves as a good handle for probing the relative importance of interaction 
effects.

The bias parameters are given in mean field by an analog of Eq.~(\ref{eq:barh}),
\begin{equation}
{\hat h}_i^{(y)}(x)=
{\bar h}_i^{(y)}(x)
-\sum_{j\ne i}\sum_{x^\prime}{\hat J}_{ij}^{(y)}(x,x^\prime){\hat f}_j^{(y)}(x^\prime),
\label{eq:mf}
\end{equation}
and 
\begin{equation}
{\bar h}_i^{(y)}(x)=\ln \left[{\hat f}_i^{(y)}(x)/{\hat f}_i^{(y)}(0)\right],
\label{eq:hbari}
\end{equation}
where ${\hat f}_i^{(y)}(0)$ is the frequency of (reference) factor $x_i$ for which the parameters are zero
($a_i=0$).
Equation~(\ref{eq:mf}) relates the effective bias for predictor $x_i$ (the first term on the right)
as the sum of univariate bias (left-hand side) and combined mean effects of interactions with other
variables (the second term on the right) \citep{chandler}.
The effective bias is related to frequency via Eq.~(\ref{eq:hbari}) because
\begin{equation}
{\hat f}_i^{(y)}(x)
=\frac{e^{{\bar h}_i^{(y)}(x)}}{Z_{iy}}={\hat f}_i^{(y)}(0)\,e^{{\bar h}_i^{(y)}(x)}
\label{eq:hat}
\end{equation}
where the fact that ${\bar h}_i^{(y)}(0)=0$  was used in the second equality.

As in pseudo-likelihood maximization, mean field inference also may encounter non-varying predictors
during cross-validation. To apply the same inference scheme using
Eqs.~(\ref{eq:Cij}), (\ref{eq:mf}) and (\ref{eq:hbari}) to such cases, 
the single-variable
frequency ${\hat f}_i^{(y)}(x)$ and covariance ${\hat f}_{ij}^{(y)}(x,x^\prime)$ are computed 
using data augmented by a prior count of 1 uniformly distributed among all $L_i$ factor levels for 
each predictor.

\subsection{Naive Bayes}
When interaction is ignored ($J_{ij}^{(y)}=0$), the model can be solved 
analytically. From Eqs.~(\ref{eq:mf}) and (\ref{eq:hbari}), 
\begin{equation}
{\hat h}_i^{(y)}(x)=\ln\left[{\hat f}_i^{(y)}(x)/{\hat f}_i^{(y)}(0)\right]
\label{eq:hhi}
\end{equation}
and \citep{woo_etal-2016}
\begin{equation}
L_y-n_y\ln p_y=\sum_{k\in \mathbbm{K}_y}w_k\ln P({\bf x}^k|y)
=n_y\sum_{i,x} {\hat f}_i^{(y)}(x) \ln {\hat f}_i^{(y)}(x).
\end{equation}
The likelihood ratio statistic for each predictor, where the
null hypothesis is $h_i^{(y)}(x)=h_i(x)$ with $h_i(x)$ the ``pooled''
inference parameters (same values for all response groups), is then
\begin{equation}
q_i = 2\sum_y n_y\sum_x \left[{\hat f}_i^{(y)}(x)\ln {\hat f}_i^{(y)}(x)
-{\hat f}_i(x)\ln {\hat f}_i(x)\right].
\label{eq:qi}
\end{equation}
The statistic $q_i\sim \chi^2$ with ${\rm d.f.}=(G-1)(L_i-1)$.
Another example of hypotheses that can be tested is
$h_i^{(y)}(x)=h_i^{(y)}(A)$ for $x \in X_A$, where $X_A$ is a subset $A$ of predictor
values (e.g., in \code{Titanic} model, the effects of \code{Class} are the same
for \code{2nd} and \code{3rd Class}; see Sec.~\ref{sec:usage}), for which
\begin{equation}
q_i = 2\sum_y n_y\sum_A \sum_{x\in X_A}
\left[{\hat f}_i^{(y)}(x) \ln{\hat f}_i^{(y)}(x)
-{\hat f}_i^{(y)}(A)\ln{\hat f}_i^{(y)}(A)\right]
\end{equation}
with ${\rm d.f.}=G(L_i-1-N_i)$, where $N_i$ is the number of 
predictor levels with distinct parameter values.

\subsection{Classification}
For prediction, we combine predictor distributions for all response groups via Bayes formula:
\begin{equation}
P(y|{\bf x})=\frac{P({\bf x}|y)p_y}{\sum_{y^\prime}P({\bf x}|y^\prime) p_{y^\prime}}
=\frac{1}{1+\sum_{y^\prime\ne y} P({\bf x}|y^\prime) p_{y^\prime}/
P({\bf x}|y) p_y}
=\frac{1}{1+e^{-F_y({\bf x})}},
\label{eq:py}
\end{equation}
where 
\begin{equation}
F_y({\bf x})=\ln\left[\frac{P({\bf x}|y)p_y}
{\sum_{y^\prime \ne y}P({\bf x}|y^\prime)p_{y^\prime}}\right].
\label{eq:Fy}
\end{equation}
For binary response coded as $y=0,1$, Eq.~(\ref{eq:Fy}) reduces to
\begin{eqnarray}
F_1({\bf x})&=&\ln P({\bf x}|y=1)-\ln P({\bf x}|y=0) + \ln (p_1/p_0)\nonumber\\
&=& \alpha + \sum_i \beta_i(x_i) + \sum_{i<j} \gamma_{ij}(x_i,x_j),
\label{eq:F}
\end{eqnarray}
where 
\begin{eqnarray}
\alpha &=&\ln \frac{Z_0 p_1}{Z_1p_0},\nonumber\\
\beta_i(x) &=& h_i^{(1)}(x)-h_i^{(0)}(x),\nonumber\\
\gamma_{ij}(x,x^\prime)&=& J_{ij}^{(1)}(x,x^\prime) -J_{ij}^{(0)}(x,x^\prime).
\label{eq:F2}
\end{eqnarray}
Therefore, if $J_{ij}^{(y)}(x,x^\prime)=0$ (naive Bayes), Eq.~(\ref{eq:py}) takes the form of
the logistic regression formula. However, the actual naive Bayes parameter values differ from logistic 
regression fit. No expression for $P(y|{\bf x})$ simpler than Eq.~(\ref{eq:py}) exists for data with more 
than two groups.

In pseudo-likelihood maximization inference, $Z_y$ can be approximated by
\begin{equation}
\ln Z_y = \frac{1}{n_y}
\sum_{k \in \mathbbm{K}_y}
\sum_i 
\ln \left\{\sum_x \left[e^{h_i^{(y)}(x)+\sum_{j\ne i}J_{ij}(x,x_j^k)/2}\right]\right\},
\label{eq:lnz1}
\end{equation}
or with the same expression without the factor of $1/2$ in the interaction 
term in the exponent (default). This quantity can be conveniently computed 
during the optimization process. With the mean field option, the following 
expression is used:
\begin{equation}
\ln Z_y = -\ln {\hat f}^{(y)}(0) - \frac{1}{2}\sum_{i\ne j} \sum_{x,x^\prime}J_{ij}(x,x^\prime) {\hat f}_i(x)
{\hat f}_j(x^\prime).
\label{eq:lnzy}
\end{equation}

For a test data set for which the actual group identity $y_k$ of data 
instances are known, the accuracy may be defined as
\begin{equation}
s=\frac{1}{n}\sum_k \mathbbm{1}\left[{\hat y}({\bf x}^k)=y_k\right],
\label{eq:s}
\end{equation}
where 
\begin{equation}
{\hat y}({\bf x}) = \argmax_y P(y|{\bf x}).
\label{eq:yhat}
\end{equation}
If response is binary, the accuracy defined by Eq.~(\ref{eq:s}) is sensitive to marginal distributions of 
the two groups via Eq.~(\ref{eq:F}). 
The area under curve (AUC) of receiver operating characteristic is a more robust performance measure
independent of probability cutoff. In \pkg{bbl}, the accuracy
given by Eqs.~(\ref{eq:s}) and (\ref{eq:yhat}) is used in general with the option 
to use AUC for binary response using \proglang{R} 
package \pkg{pROC} \citep{robin_etal,proc}.

\section{Software Usage and Tests}
\label{sec:usage}
\subsection[Logistic regression]{Logistic regression}
\label{sec:lr}
To motivate the use of \pkg{bbl} and highlight differences, we first 
consider the use of logistic regression using \code{glm}. 
We use the base \proglang{R} data \code{Titanic} as an example:
%
<<data>>=
titanic <- as.data.frame(Titanic)
titanic
@
%
Although more detailed versions of the same data set are available
[see, e.g., \pkg{titanic} \citep{titanic} or 
\pkg{stablelearner} \citep{stablelearner,philipp_etal}], 
the simpler version above
only including factor variables suffices for our purposes because
\pkg{bbl} requires discrete factors as predictors.
Input data can either be of the form above with unique combinations of predictors
in each row along with frequency (input to \code{weights} argument of \code{glm})
or raw data (one observation per row) we generate using the utility function \code{freq2raw}:
%
<<raw>>=
library('bbl')
titanic_raw <- freq2raw(data = titanic, freq = Freq)
head(titanic_raw)
summary(titanic_raw)
@
%
We train a logistic regression model using \code{glm}:
%
<<lr>>=
gfit0 <- glm(Survived ~ Class + Sex + Age, family = binomial(), 
  data = titanic, weights = Freq)
coef(summary(gfit0))
@
%
The fit above included linear terms only. It indicates that survival was strongly
associated with class status, sex (female heavily favored), and age.
The model below includes all interactions:
%
<<glm2>>=
gfit1 <- glm(Survived ~ (Class + Sex + Age)^2, family = binomial(), 
  data = titanic, weights = Freq)
coef(summary(gfit1))
@
%
A comparison of the linear coefficients and significance levels in the two models
suggest that interaction plays important roles; in particular, marginal effects on 
the linear level remained significant only for the \code{Female} status.

To illustrate training and prediction, we divide the sample into 
train and test sets:
%
<<div>>=
set.seed(159)
nsample <- NROW(titanic_raw)
flag <- rep(TRUE, nsample)
flag[sample(nsample, nsample/2)] <- FALSE
dtrain <- titanic_raw[flag,]
dtest <- titanic_raw[!flag,]
@
%

We train a \code{glm} model with interactions and make prediction on the test data:
<<lr>>=
gfit2 <- glm(Survived ~ Class * Sex + Sex * Age, family = binomial(), 
  data = dtrain)
prl <- predict(gfit2, newdata = dtest)
yhat <- ifelse(prl > 0, 'Yes', 'No')
mean(yhat == dtest$Survived)
gauc <- pROC::roc(response = dtest$Survived, predictor = prl, 
  direction = '<')$auc
gauc
@
%
In the above, the interaction \code{Class:Age} was omitted because it was rank-deficient
(no \code{Crew} among children) and prediction from a rank-deficient fit is ill-defined.

For comparison with \pkg{bbl}, which by default includes regularization, we
also consider penalized logistic regression fit using \pkg{glmnet} 
\citep{friedman_etal,glmnet}
<<glmnet>>=
if(!require('glmnet'))
  install.packages('glmnet')
library('glmnet')
xdat <- data.matrix(dtrain[, 1:3])
y <- dtrain[, 4]
gnet <- cv.glmnet(x = xdat, y = y, family = 'binomial', alpha = 1,
  nfolds = 5, type.measure = 'auc')
plot(gnet)
@
%

\setkeys{Gin}{width=0.6\textwidth}
\begin{figure}[t!]
\centering
<<glmnet, echo=FALSE, fig=TRUE, height=4, width=4>>=
plot(gnet)
@
%
\caption{\small \label{fig:glmnet}
Cross-validation run of \code{glmnet} on \code{Titanic} data.}
\end{figure}
Note that the above fit used the non-interacting model of three predictors
and $L_1$ penalization ($\alpha=1$). The input matrix contains
integer-coded terms in the linear model (columns):
<<glmnet2,>>=
head(xdat)
@
%
Figure~\ref{fig:glmnet} indicates that the effect of regularization is minimal
for this model.

\subsection[Boltzmann Bayes learning]{Boltzmann Bayes learning}
\label{sec:BB}
The logistic regression shown in Section~\ref{sec:lr} allowed for
inference and significance testing of linear and interaction coefficients 
in association with the response variable.
However, the regression fit did not provide any further information regarding 
the source of association: in the examples in Section~\ref{sec:lr},
the survival of Titanic passengers was seen to be associated with
being \code{Female} and not being \code{Crew} members.
The corresponding linear regression coefficients, which have the same functional form as 
in Eq.~(\ref{eq:F}) [$\beta_i(x)$ in Eq.~(\ref{eq:F2}) if interactions are neglected],
are measures of the {\it difference} in coefficients $h_i^{(y)}$ between the two response groups [Eq.~(\ref{eq:F2})].
The two terms, $h_i^{(1)}$ and $h_i^{(0)}$, whose difference yielded the coefficient 
$\beta_i(x)$ remained unknown. How were the sub-groups distributed among survivor and 
non-survivor groups? Were there very few \code{Female} \code{3rd}-class passengers
among the survivor group compared to non-survivor, or were they found in both 
groups but more so among non-survivors? 

The \pkg{bbl} inference estimates the
individual distributions of predictors in response groups separately 
and subsequently combines them to make predictions. 
For binary response, this inference provides estimates of the two coefficients 
[$h_i^{(1)}$, $h_i^{(0)}$ for linear effects and $J_{ij}^{(1)}$, $J_{ij}^{(0)}$ 
for interactions] in Eq.~(\ref{eq:F}) whose 
difference corresponds to the logistic regression coefficients.
More generally, the availability of the direct estimates of predictor distributions
in each response group given by Eq.~(\ref{eq:pxy}) facilitates model interpretations in a way not possible for regression-based models, 
as we show below in this section and Section~\ref{sec:image}.

With this comparison in mind, we use the same Titanic data below to illustrate
the Boltzmann Bayes inference. As in \pkg{glm}, \pkg{bbl} uses formula input to 
train an \code{S3} object of class \code{bbl}: 
%
<<class>>=
bfit0 <- bbl(Survived ~ Class + Sex + Age, data = titanic, weights = Freq,
  prior.count = 0)
@
%
which by default triggers a pair of pseudo-likelihood inferences, solving
the maximum pseudo-likelihood equations (\ref{eq:partial}) first
under the alternative hypothesis (individual groups have 
distinct distributions) and then the null hypothesis (all samples 
have the same distribution).

The argument \code{prior.count} can be used to add prior counts to frequencies of 
occurrence of each predictor level. One may observe that when interaction is neglected,
the naive Bayes model involves categorical distributions for each predictor.
In this special case, therefore, the prior count can be regarded as the hyperparameter
of the conjugate Dirichlet prior, making the overall treatment of the model a fully
Bayesian extension.

The \code{print} method on \code{bbl} shows the structure of model
and (subsets) of inferred parameters:
%
<<print>>=
bfit0
@
%
where \code{dh} represents parameters $\Delta h_i^{(y)}
=h_i^{(y)}-h_i$; i.e., 
individual group parameters offset by the pooled values.
Internally, the parameters $h_i^{(y)}$ and $J_{ij}^{(y)}$ are stored
as lists with argument order $(y,i)$ and $(y,i,j)$, respectively. 
The inner-most elements of the lists are vectors and matrices of dimension 
$L_i-1=\text{\code{c(3,1,1)}}$ and $(L_i-1,L_j-1)$, respectively.
The \code{summary} method on \code{bbl} object prints out parameters
and their significance test outcomes under the naive Bayes approximation
(no interactions) as a rough overview of model under consideration:
%
<<summary>>=
summary(bfit0)
@
%
The test results are those from likelihood ratio test applied to the
naive Bayes result, Eq.~(\ref{eq:qi}), with the null hypothesis
$h_i^{(y)}(a)=h_i(a)$. The tables of bias parameters shown above 
include those for two survival status groups. Their signs and magnitudes,
along with the computed significance levels, clearly indicate the
associations of lower \code{Class} status and being \code{Male} with non-survivors.
There are few children among both survivors and non-survivors; hence highly negative
bias parameters in all groups, although less so in survivor group,
as expected. We note that the \code{summary} method displays naive Bayes results, 
for which simple analytic expressions for test results are available, even for models 
containing interactions.

One may compare the naive Bayes parameter $\beta_i(x)$
with the logistic regression coefficients:
%
<<nbvslr>>=
cb0 <- coef(bfit0)
beta <- list(Class = cb0$h$Yes$Class - cb0$h$No$Class, 
  Sex = cb0$h$Yes$Sex - cb0$h$No$Sex, Age = cb0$h$Yes$Age - cb0$h$No$Age)
unlist(beta)
coef(summary(gfit0))[,'Estimate']
@
%
and observe that they are largely consistent (with different signs depending on which
factor level was used as reference) but not identical.

We now fit an interacting model using \code{bbl}:
%
<<survival>>=
bfit <- bbl(Survived ~ Class * Sex + Sex * Age, data = titanic, 
  weights = Freq)
bfit
@
%

\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t!]
\centering
<<plot, echo=FALSE, fig=TRUE, height=5.0, width=5>>=
oldpar <- par(mar = c(6, 4.5, 3, 4), tck = -0.05, cex.axis = 0.8)
plot(bfit)
par(oldpar)
@
\caption{\small \label{fig:plot}
Plot of \code{bbl} object displays bias (top) and interaction parameters (bottom).
All parameters are offset by their pooled (singe-group) values.}
\end{figure}
%
<<plot2, echo=TRUE, eval=FALSE>>=
plot(bfit)
@
%
The parameters printed include those for interactions.
The \code{plot} method shows a barplot of bias parameters and a heatmap of
interaction parameters (Fig.~\ref{fig:plot}).
Note that \code{Male} members were predominant (bias parameters; top), while
\code{Male} \code{3rd}-class passengers were under-represented 
(interactions; bottom left), among non-survivors. In addition, 
\code{Male}-\code{Child} class had enhanced survival (bottom right).

We now fit the training data and make prediction on test data:
%
<<predict.bbl>>=
bfit2 <- bbl(Survived ~ Class * Sex + Sex * Age, data = dtrain)
pr <- predict(bfit2, newdata = dtest, type = 'prob')
head(pr)
auc <- pROC::roc(response = dtest$Survived, predictor = pr[, 2], 
  direction = '<')$auc
auc
@
%
Here, Eq.~(\ref{eq:py}) was used with ${\bf x}$ from the supplied \code{newdata}.
The \code{predict} method returns a data frame containing predicted group
probabilities and the most likely group for each row.

One can do cross-validation applied to \code{dtrain} data, dividing it into 
\code{nfold = 5} train/validation subsets of 4:1 proportion, and aggregating
predictions for validation sets using the trained model:

%
<<cvsim, echo=TRUE>>=
cv <- crossVal(Survived ~ .^2, data = dtrain, method = 'pseudo', 
  lambda = 10^seq(-5, -2, 0.2), verbose = 0)
cv
plot(cv, mar=c(4, 4, 3, 3), tck = -0.04, bty = 'n')
@
%

\setkeys{Gin}{width=0.6\textwidth}
\begin{figure}[t!]
\centering

<<plotcv, echo=FALSE, fig=TRUE, height=4, width=4>>=
plot(cv, mar = c(4, 4, 3, 3), tck = -0.04, bty = 'n')
@
%
\caption{\small \label{fig:plotcv}
Cross-validation run of \code{Titanic} data in \pkg{bbl}.}
\end{figure}

Here, the model included all interaction terms and returned an object with a \code{data.frame} of AUCs for multiple \code{lambda} values
as well as 95\% confidence intervals and optimal values 
with maximum AUC. We use this information to make prediction as follows:
%
<<pr2>>=
model <- bbl(Survived ~ .^2, data = dtrain, lambda = cv$regstar)
pr2 <- predict(model, newdata = dtest)
bscore <- mean(dtest$Survived == pr2$yhat)
bscore
bauc <- pROC::roc(response = dtest$Survived, predictor = pr2[,2], 
  direction = '<')$auc
bauc
@
%
Alternatively, \code{predict(cv, ...)} will apply the optimal model within
cross-validation to test data. The difference compared to the re-training 
step above is that the optimal model stored in \code{cv} was trained
on 4/5 of the sample, while \code{model} above used the whole training set.

A major advantage of the \code{bbl} fit compared to regression is
the availability of predictor distributions in each response group,
$P({\bf x}|y)$, given by Eq.~(\ref{eq:pxy}). In addition to using
the model to make predictions of response groups, one can also examine
the predictor distributions and identify configurations
dominant in each response group. Since the total number of configurations
${\bf x}$ grows exponentially with the number of predictors, Markov
chain Monte Carlo (MCMC) sampling is necessary for exploration of 
these distributions except for very low dimensions. 
The function \code{mcSample} performs
Gibbs sampling of the predictor distributions using \code{bbl} parameters
and outputs the most likely configuration in each response group:
%
<<mcmc>>=
map <- mcSample(bfit, nstep = 1000, progress.bar = FALSE)
map
@
%
The return value is a list containing the predictor configurations with the
highest probability in each response group (columns in \code{map$xmax} above)
and the corresponding ``energy'' values, which are exponents of 
Eq.~(\ref{eq:pxy}). 

\subsection{Simulated data}
We next use simulated data to show the effect of penalizers on \code{bbl} 
inference as well as its usefulness under varying sample sizes.
%
<<sim1>>=
predictors <- list()
m <- 5
L <- 3
for(i in 1:m) predictors[[i]] <- seq(0, L-1)
par <- randompar(predictors)
names(par)
@
%
The utility function \code{randompar} generates random parameters for 
predictors. We have set the total number of predictors as $m=5$, each taking
values $0,1,2$ ($L_i=L=3$). 
%
<<sample>>=
xi <- sample_xi(nsample = 10000, predictors = predictors, h = par$h, 
  J = par$J, code_out = TRUE)
head(xi)
@
%
The function \code{sample_xi} will list all possible predictor states and 
sample configurations based on the distribution (\ref{eq:pxy}). 
The total number of states here is $L^m=3^5$, which is amenable for 
exhaustive enumeration. However, this is possible only for small $m$ and $L$.
If either are even moderately larger, \code{sample_xi} will hang.

Because there is only one response group, we call the main engine 
\code{mlestimate} of \pkg{bbl} inference directly instead of \code{bbl}:
%
<<mle>>=
fit <- mlestimate(xi = xi, method = 'pseudo', lambda = 0)
@
%

\begin{figure}[t!]
\centering
%
<<par, echo=FALSE, fig=TRUE, height=4.0, width=4.5>>=
oldpar <- par(mar = c(4, 4, 1, 2), lwd = 0.5, cex.axis = 0.8, 
  cex.lab = 1.0, mgp = c(2.2 ,0.9, 0), tck = -0.03)
range <- range(par$h, par$J, fit$h, fit$J)
plot(x = unlist(par$h), y = unlist(fit$h), bg = 'cornflowerblue', 
  xlim = range, ylim = range, pch = 21, cex = 0.8, xlab = 'True', 
  ylab = 'Inferred', lwd = 0.7, xaxt = 'n', yaxt = 'n', bty = 'n')
axis(side = 1, at = seq(-1.5, 1.5, 0.5), lwd = 0.5, las = 1)
axis(side = 2, at = seq(-1.5, 1.5, 0.5), lwd = 0.5, las = 1)
segments(x0 = -1, x1 = 1, y0 = -1, y1 = 1, lty = 2, lwd = 0.7)
points(x = unlist(par$J), y = unlist(fit$J), pch = 24, bg = 'orange', 
  cex = 0.8, lwd = 0.7)
legend(x = 0.5, y = -0.5, legend = expression(italic(h), italic(J)), 
  cex = 0.8, pch = c(21, 24), pt.bg = c('cornflowerblue', 'orange'))
par(oldpar)
@
%
\caption{\small \label{fig:par}
Comparison of true parameters and those inferred from pseudo-likelihood 
Boltzmann Bayes inference. See the text for conditions.}
\end{figure}

In contrast to \code{bbl} function, which fits a model of 
multiple response groups and predictors in factors,
\code{mlestimate} is for a single group and requires input matrix 
\code{xi} whose elements are integral codes of factors: $a_i=0,\cdots,L_i-1$.
Figure~\ref{fig:par} compares the true and inferred parameters. Here, the
sample size was large enough that no regularization was necessary.

We next simulate a full binary response data set with four-level predictors:
%
<<atgc>>=
nt <- c('a', 'c', 'g', 't')
set.seed(135)
for(i in 1:m) predictors[[i]] <- nt
names(predictors) <- paste0('v', 1:m)
par <- list()
par[[1]] <- randompar(predictors)
par[[2]] <- randompar(predictors, h0 = 0.1, J0 = 0.1)
dat <- randomsamp(predictors, response = c('ctrl', 'case'), par = par,
  nsample = 1000)
@
%
The function \code{randomsamp} generates random samples of 
\code{predictor}-\code{response} pairs using the supplied \code{par}.
We perform a cross-validation using mean field inference,
%
<<cr-mf>>=
cv <- crossVal(y ~ .^2, data = dat, method = 'mf', eps = seq(0, 1, 0.1),
  verbose=0)
cv
@
%
Here, \code{bbl} is called inside \code{crossVal} as before but with 
\code{method = `mf'}, which triggers mean field inference with 
Eqs.~(\ref{eq:mf1}) and (\ref{eq:mf}).

As shown in Fig.~\ref{fig:mfcv}{\bf a}, prediction AUC is optimized near 
$\epsilon=0.7$. The difference between AUC at $\epsilon=0$ (naive Bayes limit) 
and the maximum is a measure of the overall effect of interaction.  We select
three values of $\epsilon$ and examine the fit:
%
<<mf-par>>=
fit <- list()
eps <- c(0.2, 0.7, 1.0)
for(i in seq_along(eps))
  fit[[i]] <- bbl(y ~ .^2, data = dat, method = 'mf', eps = eps[i], 
    verbose = 0)
@
%

\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t!]
\centering
<<cv, echo=FALSE, fig=TRUE, height=5.5, width=6>>=
oldpar <- par(mfrow = c(2, 2), mar = c(4, 4, 2, 2), lwd = 0.5, 
  cex.axis = 0.8, cex.lab = 0.9, mgp = c(2.2, 0.8, 0), tck = -0.03, 
  las = 1)
estar <- cv$regstar
plot(cv, xlab = expression(epsilon), ylab = 'AUC', lwd = 0.7, cex = 0.7,
  bty = 'n', log = '')
segments(x0 = estar, x1 = estar, y0 = 0, y1 = cv$maxscore, lty = 2, 
  lwd = 0.5, col = 'red')
title(adj = 0, cex.main = 1.2, font = 2, main = 'a')

for(i in 1:3){
  plot(x = c(unlist(par[[2]]$h), unlist(par[[1]]$h)), 
    y = unlist(coef(fit[[i]])$h), bg = 'cornflowerblue', 
    xlim = c(-1.5, 1.5), ylim = c(-1.5, 1.5), pch = 21, cex = 0.7, 
    xlab = 'True', ylab = 'Inferred', lwd = 0.7, xaxt = 'n', yaxt = 'n',
    bty = 'n')
  axis(side = 1, at = seq(-1.5, 1.5, 0.5), lwd = 0.5, las = 1)
  axis(side = 2, at = seq(-1.5, 1.5, 0.5), lwd = 0.5, las = 1)
  segments(x0 = -2, x1 =2 , y0 = -2, y1 = 2, lty = 2, lwd = 0.7)
  points(x = c(unlist(par[[2]]$J), unlist(par[[1]]$J)), 
    y = unlist(coef(fit[[i]])$J), pch = 24, bg = 'orange', cex = 0.7, 
    lwd = 0.7)
  if(i==1) legend(x = 0.5, y = -0.5, legend = expression(italic(h), 
    italic(J)), cex = 0.8, pch = c(21, 24), 
    pt.bg = c('cornflowerblue', 'orange'))
  title(adj = 0,main = letters[i + 1], cex.main = 1.1, font = 2)
  mtext(side = 3, line = 1.0, cex = 0.8, bquote(epsilon == .(eps[i])),
    adj = 0.5)
}
par(oldpar)
@
\caption
{\small 
Regularized mean field inference using simulated data.
({\bf a}) Cross-validation AUC with respect to regularization parameter $\epsilon$.
({\bf b}-{\bf d}) Comparison of true and inferred parameters under three $\epsilon$ values. 
Best fit is achieved when AUC is maximum.}
\label{fig:mfcv}
\end{figure}

Figure~\ref{fig:mfcv}{\bf b}-{\bf d} compares the three inferred parameter sets (\code{coef(fit[[i]])$h, coef(fit[[i]])$J}) 
with the true values (\code{par[[iy]]\$h, par[[iy]]\$J}). As $\epsilon$ increases from 0 to 1, 
interaction parameter $J$ grows from zero to large, usually overfit levels. We verify that
the bias and variance strike the best balance under $\epsilon=0.7$ 
(Fig.~\ref{fig:mfcv}{\bf c}),
as suggested by cross-validation AUC in Fig.~\ref{fig:mfcv}{\bf a}.

\subsection{Genetic code}
We consider a different learning task example with a much larger space of
response groups, namely those of amino acids; $K=21$, which
include 20 amino acids 
plus stop signal (`*'), encoded by DNA sequences ($x_i=\sf{a, c, g, t}$).
In DNA sequences, three nucleotides combine to encode specific amino acids. 
We will train a model attempting to re-discover the mapping from nucleotide 
sequences to amino acids.
%
<<ntaa>>=
set.seed(351)
n <- 2000
dat <- data.frame(b1 = sample(nt, size = n, replace = TRUE),
  b2 = sample(nt, size = n, replace = TRUE),
  b3 = sample(nt, size = n, replace = TRUE))
head(dat)
@
%
In the above, we generated random instances of triplet codons for training.
We use the package \pkg{Biostrings} \citep{pages_etal}
to translate it into amino acids:
%
<<biostrings>>=
if(!require('Biostrings')){
  if(!require('BiocManager'))
    install.packages('BiocManager')
  BiocManager::install('Biostrings')
}
aa <- Biostrings::DNAString(paste(t(dat), collapse = ''))
aa
aa <- strsplit(as.character(Biostrings::translate(aa)), split = '')[[1]]
xdat <- cbind(data.frame(aa = aa), dat)
head(xdat)
@
%

We now cross-validate using \code{bbl}:
%
<<aacv>>=
cv <- crossVal(aa ~ .^2, data = xdat, lambda = 10^seq(-3, 1, 0.5), 
  verbose = 0)
cv
@
%
Note that with the multinomial response group, the accuracy defined by 
Eq.~(\ref{eq:s}) is used. The class \code{cv.bbl} extends \code{bbl} and
stores the model with the optimal $\lambda$.
In contrast to Section~\ref{sec:BB}, we do not refit the model under this $\lambda$
because accuracy is maximum.
Testing can use all possible codon sequences ($4^3=64$ total):
%
<<codon>>=
panel <- expand.grid(b1 = nt, b2 = nt, b3 = nt)
head(panel)
dim(panel)
p <- predict(cv, panel)
ap <- Biostrings::DNAString(paste(t(panel), collapse = ''))
ap <- strsplit(as.character(Biostrings::translate(ap)), split = '')[[1]]
accuracy <- mean(ap == p$yhat)
accuracy
@
%
The trained model has perfect accuracy of 1 and will not 
make mistakes in any translation of DNA sequences. 

\subsection{Image data}
\label{sec:image}

\setkeys{Gin}{width=0.5\textwidth}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.8]{mnist_mf}
\caption{\small Cross-validation of Boltzmann Bayes inference on MNIST data using mean field option.}
\label{fig:mnist}
\end{figure}

We next consider learning examples with data sets containing 
predictors numbering $\sim 100$ or more. The  
MNIST data set (\url{http://yann.lecun.com/exdb/mnist/}),
widely used for benchmarking classification algorithms \citep{lecun_etal},
contains image data of grayscale levels ($x_i=[0,255]$) derived from 
hand-written digits ($y_k=0,\cdots,9$) for $m=28\times 28=784$ pixels.
We use down-sampled training ($n=1,\!000$) and test ($n=500$) data sets, 
where grayscale has been transformed into binary predictors ($x_i=0,1$):
%
<<mnist, eval=TRUE>>=
dat0 <- read.csv(system.file('extdata/mnist_train.csv', package = 'bbl'))
dat <- removeConst(dat0)
dat[1:5, 1:10]
@
%
%
<<mnist_cval, eval=FALSE, echo=TRUE>>=
cv <- crossVal(y ~ .^2, data = dat, method = 'mf', eps = 0.05)
@
%
Note that before calling \code{crossVal}, we removed predictors without factor
variations (pixels that are always empty) using the utility function 
\code{removeConst}. By default, error will occur inside \code{crossVal} 
otherwise. 

\begin{table}[t!]
\centering
\begin{tabular}{llll}
\hline
Algorithm           & Method           & Error rate (\%) & Reference/package   \\ \hline
Linear classifier   & 1-layer NN       & $12.0$        & \cite{lecun_etal}\\
K-nearest neighbors & Euclidean (L2)   & $5.0$         & \cite{lecun_etal}\\
2-layer NN          & 300 hidden units & $4.7$         & \cite{lecun_etal}\\
RBM                 & 2-layer          & $0.95$        & \cite{salakhutdinov_hinton}\\
Naive Bayes         & Mean field ($\epsilon=0$) & $15.7$ &  \pkg{bbl}                 \\
BB                  & Mean field ($\epsilon=0.05$)      & $8.5$         &    \pkg{bbl}               \\
\hline
\end{tabular}
\caption{\small \label{tb:mnist} Performance comparison of BB inference and other models on MNIST data set.
The \pkg{bbl} inferences used the full MNIST training and test data sets (see text).
BB, Boltzmann Bayes; NN, neural network; RBM, restricted Boltzmann machine.}
\end{table}


The above run will take a few minutes and yield a prediction score of $0.89$. 
By feeding a vector of 
$\epsilon$ values, one can obtain the profile shown in Fig.~\ref{fig:mnist}. The jump in performance under $\epsilon^*\sim 0.05$
over $\epsilon \rightarrow 0$ (naive Bayes) limit gives a measure of interaction
effects. The relatively small value of $\epsilon^*$ at the optimal condition,
compared to e.g., Fig.~\ref{fig:mfcv}{\bf a}, reflects the sparseness of 
image data. 

We now retrain the model without cross-validation under $\epsilon^*$ and 
classify test set images:
%
<<mnist3, eval=FALSE>>=
mnist <- bbl(y ~ .^2, data = dat, method = 'mf', eps = 0.05)
dtest <- read.csv(system.file('extdata/mnist_test.csv', package = 'bbl'))
dtest <- dtest[, colnames(dtest) %in% colnames(dat)]
pr <- predict(mnist, newdata = dtest[, -1], progress.bar = TRUE)
accuracy <- mean(pr$yhat == dtest$y)
@
%
%
<<mnist3_2, eval=TRUE, echo=FALSE>>=
accuracy <- 0.916
@
%
%
<<mnist3_3, eval=TRUE, echo=TRUE>>=
accuracy
@
%

Since \code{mnist} dropped a subset of original predictors, the test data must
be filtered accordingly. Note the increase in accuracy compared to 
cross-validation value because of the use of full training data. 

We performed similar cross-validation and test analyses of the full MNIST data 
(training $n=60,\!000$ and test $n=10,\!000$) and obtained the accuracy of $0.915$ 
(classification error rate $8.5\%$), which compares favorably 
with other large-scale neural network algorithms (Table~\ref{tb:mnist}).

As with Titanic data, we leverage the unique advantage of \code{bbl} fit
of providing predictor distributions and estimate dominant
configurations of each response group (Fig.~\ref{fig:mnist_map}):
%
<<mnist_map1, eval=FALSE, echo=TRUE>>=
mnist_map <- mcSample(mnist, nstep = 20, progress.bar = TRUE)
oldpar <- par(mfrow = c(2, 5), mar = c(1, 1, 1, 1))
xvar <- colnames(dat0[, -1])
xmap <- apply(mnist_map$xmax, 1:2, as.numeric)
xf <- matrix(0, nrow = length(xvar), ncol = 10)
rownames(xf) <- xvar
for(i in 1:10) xf[rownames(xmap), i] <- xmap[, i]
for(i in 1:10){
  mat <- matrix(t(xf[, i]), nrow = 28, ncol = 28)
  image(x = 1:28, y = 1:28, z = mat[, 28:1], col = c('white', 'black'), 
    xaxt = 'n', yaxt = 'n', xlab = '', ylab = '')
}
par(oldpar)
@
%
\setkeys{Gin}{width=0.6\textwidth}
\begin{figure}[t!]
\centering
\includegraphics[scale=1]{mnist_map}
\caption
{\small Maximum probability configurations of digits ($0,\cdots,9$) 
estimated from \code{bbl} fit coefficients using Gibbs sampling.}
\label{fig:mnist_map}
\end{figure}

It is interesting to note that the model for handwritten digit ``1'' is a combination of
two versions, one slated forward and the other backward.
The images shown in Fig.~\ref{fig:mnist_map} illustrate examples of model interpretation
made possible by the Bayesian formulation used by \pkg{bbl}, a significant advantage
compared to regression-based methods and 
other deep learning models whose interpretations are challenging \citep{montavon_etal}.


\subsection{Transcription factor binding site data}
\label{sec:tfbs}
One of machine learning tasks of considerable interest in biomedical
applications is the detection of transcription factor binding sites within
genomic sequences \citep{wasserman_sandelin}. Transcription factors are
proteins that bind to specific DNA sequence segments and regulate
gene expression programs. Public databases, such as 
JASPAR \citep{jaspar2018}, host known transcription factors and their 
binding sequence motifs. Supervised learners allow users to leverage these
data sets and search for binding motifs among candidate sequences.

\setkeys{Gin}{width=0.8\textwidth}
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{tfbf}
\caption{\small Cross-validation of transcription factor binding motif 
model using \pkg{bbl} with control sequences generated by 3 nucleotide
mutations. Data set is from \cite{jaspar2018} (sample ID MA0014.3; see text).
({\bf a}) Pseudo-likelihood and ({\bf b}) mean field inferences.}
\label{fig:jaspar}
\end{figure}

Here, we illustrate such an inference using an example set 
(MA0014.3) of binding motif sequences from JASPAR
(\url{http://jaspar.genereg.net}):
%
<<jaspar>>=
seq <- readFasta(system.file('extdata/MA0014.3.fasta', package = 'bbl'))
head(seq)
dim(seq)
@
%

The data set consists of common nucleotide segments from $n=948$ raw sequences
used for motif discovery. We simulate a training set by generating non-binding sequences with random mutation of 3 nucleotides:
%
<<jaspar2>>=
set.seed(561)
nsample <- NROW(seq)
m <- NCOL(seq)
nt <- c('A', 'C', 'G', 'T')
ctrl <- as.matrix(seq)
for(k in seq_len(nsample))
  ctrl[k, sample(m, 3)] <- sample(nt, 3, replace = TRUE)
colnames(ctrl) <- 1:m
data <- rbind(data.frame(y = rep('Binding', nsample), seq), 
  data.frame(y = rep('Non-binding', nsample), ctrl))
data <- data[sample(NROW(data)), ]
@
%
We assess the performance of pseudo-likelihood and mean field inferences below using cross-validation:
%
<<jaspar3>>=
ps <- crossVal(y ~ .^2, data = data, method = 'pseudo', 
  lambda = 10^seq(-2, -1, 0.2), verbose = 0)
ps
mf <- crossVal(y ~ .^2, data = data, method = 'mf', 
  eps = seq(0.1, 0.4, 0.1), verbose = 0)
mf
@
%
In both cases, there is an optimal, intermediate range of regularization with
maximum AUC (Fig.~\ref{fig:jaspar}). 
The level of performance attainable with non-interacting models, 
such as position frequency matrix \citep{wasserman_sandelin}, 
corresponds to the $\epsilon=0$ limit in Fig.~\ref{fig:jaspar}{\bf b}.
The AUC range obtained above is representative of the sensitivity and 
specificity levels one would get when scanning a genomic segment using 
a trained model for detection of a binding site to within resolution of
$\sim 3$~base pairs.

\section{Summary} \label{sec:summary}
We introduced a user-friendly \proglang{R} package \pkg{bbl}, 
implementing general Boltzmann Bayes classifiers applicable to 
heterogeneous, multifactorial predictor data associated with a discrete 
multi-class response variable. The currently available \proglang{R} 
package \pkg{BoltzMM} is limited to fitting data into a single fully 
visible Boltzmann distribution without reference to response variables, 
and assumes binary predictors. The package \pkg{bbl} employs a more general
statistical distribution accommodating heterogeneous, factor-valued 
predictors via Eq.~(\ref{eq:pxy}), embedding it in a Bayesian classifier to
build supervised learning and prediction models. The basic implementation
architecture of \pkg{bbl} follows those of standard base \proglang{R}
packages such as \pkg{glm}.

Compared to more widely applied restricted Boltzmann machine algorithms
\citep{hinton}, the Boltzmann Bayes model explicitly infers interaction
parameters for all pairs of predictors, making it possible to interpret
trained models directly, as illustrated in Figs.~\ref{fig:plot} and 
\ref{fig:mnist_map}, the latter using MCMC sampling of predictor 
distributions. The \pkg{bbl} inference is especially suited to
data types where a moderate number of unordered features (such as nucleotide
sequences) combine to determine class identity, as in transcription factor
binding motifs (Section~\ref{sec:tfbs}). 
Among the two options for inference methods, mean field
(\code{method = `mf'}) is faster but can become memory intensive for models 
with a large number of predictors. Pseudo-likelihood maximization 
(\code{method = `pseudo'}) is slower but usually provides better performance.

%% -- Optional special unnumbered sections -------------------------------------

\section*{Computational details}

The current version of \pkg{bbl} is available at the
Comprehensive \proglang{R} Archive Network (CRAN) at
\url{https://CRAN.R-project.org/package=bbl}.
Installation of \pkg{bbl} requires the GNU Scientific library 
\url{https://www.gnu.org/software/gsl} installed. The results in this paper 
were obtained using \proglang{R}~\Sexpr{paste(R.Version()[6:7], collapse = ".")}.
\proglang{R} itself and all packages used are available from the CRAN
at \url{https://CRAN.R-project.org} and Bioconductor at
\url{https://bioconductor.org}.

\bibliography{refs}

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%% - With proper section titles and _not_ just "Appendix".

\newpage

%\begin{appendix}
%
%\section{More technical details} \label{app:technical}
%
%\end{appendix}

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\end{document}
