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\title{A Baysian Approach to Collaborative Dish Selection}
\author{Team 10}
\date{February 23, 2012}

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\section*{Introduction}
As anyone who has ever planned a catered event can attest, attempting
to satisfy the various palates, dietary requirements and tastes of a
group of diners can be a daunting task.  This is particularly true
given the exponential number of dishes which can be created from a
small number of ingredients, as well has hard constraints such as
allergies and religious beliefs.  Many professional catering services
handle this problem by allowing guests to select from a very limited
menu. We introduce a dish recommendation system
based on Bayesian Networks modeling user preferences.
We predict the meals from a data base of recepices that most likely match the varied tastes
of the customers, using a limited set of ingredients.  This type of expert system
would be of great use to a catering service or restaurant which needs to rapdily decide on
a small number of dishes which would be acceptable for a large dinner party,
given diverse requirements and preferences.

\section*{Related Work}
Boekel and Corney propose using Bayesian Networks to model 
consumer needs in food production chains \cite{vanboekel} \cite{corney}. 
Janzen and Xiang propose an intelligent refrigerator capable of 
generating meal plans based on inventory 
and past food choices \cite{janzenxiang}.  Baysian networks have also been
applied to recommendation systems before in on-line social
networks \cite{truyen} making predictions of the form
``if you bought those items what is the probability you would like to
buy that''.  We suggest that these approaches are limited in that they
only consider the preferences of a single (or supposed 'typical') user rather than a group.

\section*{Approach}

The approached problem is to pick a single meal which best meets the requirements
and tastes of different people dining together. We learn a predictive
baysian net from a survey distributed to participants of the meal as
training data in order to capture their preferences. The dishes
in the questionaire are selected such that all ingrediants
are covered. The participants rate each dish on a scale from
one to ten and give additional information like vegetarians.
For new dishes we then predict the maximum likelihood 
rating given our model. 
In the following we will describe our approach in detail.
First we will discuss the data selection, then the
modeling of the user preference and in the
end how to train the modeled net from
gathered data and howe to predict the 
value for a new recepice.

\paragraph*{Data accuisition}
We accumulated a diverse collection of sample recipes using the open source AnyMeal application.
We converted to the freely available MealMaster format (flat file)
recipes to XML format for input into our application.
We will gathered data representing several diners' preference for
approximately 20 meals using a simple survey of the type 'rate on a
scale of 1 to 10,  10 being favorite and 1 being least favorite'.  
Furthermore we collected data for vegetarians and allergies.

%daniel is here
\paragraph*{Knowledge Engineering}


We model users' preferences using
a bayes net. In the first layer in Figure \ref{img:bayes_net}
we capture controll variables like vegan or allergies.
Those are modeled as boolean variables. 
If the user is allergic or a vegetarian, it will set the variable allergic
to 0. 

In order to model 
Our model consists of 4 layers, 
each modeling a different aspect of taste and needs.
In the first layer we capture general meal preferences, like
being vegetarian or not liking your food steamed.
The second layer models a general preference towards 
different food categories like vegetables or beef.
As one can see, the food categories are dependent 
on the general meal preference. For example 
being vegetarian will exclude beef and will 
support vegetables. The third category models 
different ingredients. Each ingredient is conditioned 
by the food category it belongs to.
In the last layer we have hard constraints like allergies
(that will exclude a particular ingredient) or 
the overall calorie content of the meal given
someone suffers from diabetes.
The overall net is shown in Figure \ref{img:bayes_net}.
Given a recipe with a list of ingredients $I = i_1,...,i_n$ 
and a Bayesian network capturing user preferences 
we can calculate the probability of users liking the dish given
the probabilities of liking each ingrediant.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{bayes}
\caption{Our Baysian net modeling user preferences}
\label{img:bayes_net}
\end{figure}

%\subsection*{implementation}
\paragraph*{Learning and Predicting}
In order to estimate the model parameters, the
system will be trained with statistics about taste 
and preferences given a set of dishes with ratings
from multiple users. From that information we can directly calculate 
the probabilities for the ingredients using Maximum Likelihood Learning \cite{murphy}.


%\subsection*{Meal Optimization}
In order to model food preferences, we implemented
a baysian net library in java. The library
uses the sum-product algorithm for
inference and maximum likelihood learning
for parameter estimation. In our implementation
we support discrete as well as continous 
probability distributions. Discrete distributions
can be modeled as tables or as trees. 
In our implementation only continous distributions with discrete parents
are supported. A continous distribution is then modeled as a mapping
of all possible combination of it' s parents to a gaussian.
Given a data set, the parameters of a discrete variable $X$ are
estimated as 
\begin{align}
P(X = x| Y_1 = y_1, ... Y_2 = y2) =\\
N(X = x| Y_1 = y_1, ... Y_2 = y2) \over N(Y_1 = y_1, ... Y_2 = y2)
\end{align}
where $N(A)$ is the number of times event $A$occurs in the data set.
We decided to implement our own Library,
so we understand what is going on and
we can debug and fix the models
and algorithms easily.

\section*{Evaluation}
The application model will be trained using a sparse subset (25-50\%) of the survey data and the optimization problem soled for the inferred constraints.
Next, we will calculate the correlation between the application's ranking of all dishes and the actual ranking as determined by the user surveys.  We suggest that a high degree of correlation indicates that the system has the potential to accurately appraise constrained group food preferences for dishes which are not part of the survey, given sufficiently detailed recipe information.

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