CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND OF STUDY

In some applications of Bernoulli trials, the underlying success probability could change from one trial to another, that is, likelihood based techniques should be modified. This is common in situations where there are unmodeled influences that affect all the components of the binomial sum which count the number of successes in a fixed number n of Bernoulli trials. This happens, for example, in biological or agriculture applications, when we have batch effects (see for example, Kleinman (2008); Williams (2005); Crowder (2008); Haseman and Kupper (2009) or Morgan (2002)). A litter effect is related to the tendency that members of a group respond in a more similar way to some treatment than members of other groups. These random effects should be included in the modelling of the data set besides the usual covariates to account for this overdispersion. The beta-binomial model was introduced by Pearson (2005) and more formally described by Skellam (2008). It is a popular method for explicitly account for the over dispersion. We can find several applications of this model in various areas, such as Chatfield and Goodhardt (2006), who described the buying behaviour of the consumer, and Gange et al. (2006), who studied the effect of policy changes on appropriateness of hospital admissions. In addition, Aeschbacher et al. (2007) showed that a beta-binomial distribution provided a better fit than the usual distributions in biological experiments using mices when the data used were based on a large number of counts of dead.

1.2 STATEMENT OF THE PROBLEM

Our motivation is that a mixture based on documents can select background data more like the topic training data, and therefore generalize that data in a more realistic way compared to a mixture based on coarse clusters. One problem that can result when targeting to individual stories is the assignment of a large proportion of the mixture probability to a small number of stories, yielding a mixture distribution which is it sparse. To measure the sparseness of the mixture (recall it is a probability distribution built from a large number of components, and so may not actually contain zeros), we assign it a total count B according to

B = exp ()

Where ei is the total count of background story I (to understand this formular consider the case in which all background stories have the same total count c. The expression for B then reduces to

B = C exp (-)

1.3 AIM AND OBJECTIVES OF THE STUDY

The main aim of the research work is to examine Beta-Binomial mixture model and its application. Other specific objectives of the study are:

1. to determine the effect of beta-mixture model on trafficking
2. to investigate on the factors affecting the application of beta-mixture models
3. to determine the effect of speed on the application Beta-Binomial Mixture model
4. to determine the whether the application of beta-mixture model is effective

1.4 RESEARCH QUESTIONS

The study came up with research questions so as to ascertain the above stated objectives of the study. The research questions for the study are:

1. What is the effect of beta-mixture model on trafficking?
2. What are the factors affecting the application of beta-mixture models?
3. What is the effect of speed on the application Beta-Binomial Mixture model?
4. Is the application of beta-mixture model effective?

1.5 SIGNIFICANCE OF THE STUDY

The study on Beta-Binomial mixture model and its application will be of immense benefit to the entire mathematics/statistics and other departments in the sense that the study will derive the bet-binomial mixture model and educate the respondents on how to apply it effectively. The study will serve as a repository of information to other researchers that desire to carry out similar research on the above topic as the findings of the study will contribute to the body of the existing literature on Beta-Binomial mixture model and its application

1.6 SCOPE OF STUDY

The study on Beta-Binomial mixture model and its application will cover on beta-mixture model on trafficking, the effect of speed on the application Beta-Binomial Mixture model and other factors affecting beta-binomial mixture model application

CHAPTER TWO: REVIEW OF RELATED LITERATURE

2.0 INTRODUCTION

This chapter gives an insight into various studies conducted by outstanding researchers, as well as explained terminologies with regards to Beta-Binomial mixture model and its application. The chapter also gives a resume of the history and present status of the problem delineated by a concise review of previous studies into closely related problems.

2.1 BINOMIAL MIXTURES

The Binomial distribution has two parameters n and p, either, or both of which may be randomized, to give a binomial mixture. This project discusses cases in which the parameter p has a continuous mixing distribution with probability density g (p) so that

f(x) =  px (1-p)n-x g(p) dp

Where f(x) is a Binomial mixture distribution

2.2 REVIEWS ON BETA-BIONOMIAL MIXTURE MODEL

Many authors have investigated interval estimation with the binomial distribution. The standard textbook method is based on the asymptotic properties of the MLEs. This interval, however, performs poorly when the proportions are close to zero or one and also when the samples are small. Hence, other approaches are warranted. Beal (2007) constructed and evaluated several asymptotically-based confidence intervals for the difference between two binomial parameters and compared them to the standard textbook method. A reparameterization defined by a = p1 + p2 and b = p1  p2 is used in each case. Beal finds that intervals presented by Mee (2004) and Miettinen & Nurminen (2005) (based on the above reparameterization) are good overall choices in that they provide a significant improvement over the standard textbook method. Beal also notes however, that these two intervals must be constructed numerically, and in general are hard to compute. Beal (2007) proposes a simpler approach by defining a Jeffreys-Perks interval. The Jeffreys-Perks interval uses a Bayesian approach to estimate a = p1 + p2. The Jeffreys-Perks interval proves much easier to compute and provides considerable improvement over the standard textbook method.