Although the motivation for this research was similar to the motivation of the Bottleneck method, a different approach was taken. After the research evolved, much effort was put, in order to build a general classmate essay to deal with problems of this type. The main tool of this framework was a correspondence this web page sets with an additive function and information measures on random variables.
After the framework was constructed, we learned about the work information extraction thesis by Yeung ,  and .
Since Yeung developed the same framework and took it much further than we did, our work could not be considered original. Several different information extraction thesis were taken in order to cope with the general problem that is information extraction thesis in the first chapter.
None of these paths reached a full maturity. The result is various attempts to solve variants of the general problem in addition to the information extraction thesis framework and thess problem formalization. Encapsulating dependency IntroductionThe problem of understanding dependencies among informtaion results information extraction thesis one of the most fundamental aspects of any experimental research. When trying inormation understand information extraction thesis in nature, we frequently sense that there is a mutual influence between two different entities we observe, but it is often hard to point out what is incormation nature of this connection.
As an example we can think of a medical researcher that investigates heart faults. The researcher might notice that there is a correlation between heart attacks and the use of a certain medicine. It is much easier to detect the correlation than it is to understand why the two factors are correlated. Imagine that the researcher deduces that the reason that information extraction thesis medicine causes heart attacks is that it binds to a certain protein and deactivate it.
Information extraction thesis that deduction, the researcher informaion a big step forward, she now understands the correlation she observed before. Article source understanding can lead to practical solution for the problem and can help predicting observations in the future. The main question that we comparison essay conclusion ask, is whether it is possible to get a better understanding of a correlation, just by analyzing the statistics of various entities of the information extraction thesis. Research proposal industrial to our example, we will ask whether it is possible to get some understanding of the correlation between heart attack and the medicine, just by looking at joint frequency of various factors of the heart and the medicine.
To formalize this question mathematically, we will think of the two correlated entities as random variables. We will look information extraction thesis another random variable that encapsulates the correlation or at least some of it. The joint probability matrix contains all the statistical information about the two variables. By summing entries and normalizations one can compute the marginals and the conditional distributions. It is important to point out that the marginal probabilities P XP Y do not determine Information extraction thesis X, Y but rather set constrains and leave a certain degree of freedom.
Fixing the marginal probabilities enable to informatlon joint probabilities that ranges from independence of the variable to a high correlation among them. Information Theory supplies a measure of the quantity of this dependency.
The mutual information, denoted as I X, Y http://freey8.com/500-word-essay/subjective-literature-review.html, quantifies how much infprmation in bits each variables tells about read article other.
We can think of I X, Y as the amount of uncertainty which is reduced from the uncertainty about Y by the knowing of X and vise versa. We would like to know what each variable tells about the other, not only how much it tells. One can claim that by knowing extrachion value of X, the information we obtain is the conditional http://freey8.com/500-word-essay/children-s-homework-station.html of Yi.
Figure 3. Although this claim is correct, we would like to capture the meaning of the mutual information in a different way. We will seek for a new information extraction thesis variable Tthat encapsulates the information X and Y tell about each other. Encapsulating the dependency by information extraction thesis random variable will enable handling the mutual information in the same way we handle the information of the original correlated entities.
The demand that T will contain all the information X and Y information extraction thesis about each other, has a natural formal interpretation in terms of conditional independence. It is not hard to find such T. This is due to the fact that they contain additional information which is not relevant rxtraction the information extraction thesis between X and Y. It is easy to see we will show that shortly that in the general case, there is no T that encapsulates the mutual information.
A perfect encapsulator exists only when the joint distribution P X, Y is of a very special form. This leads us to look for an approximation: A random variable that contains as much relevant information as possible and as little irrelevant information, as possible.
Succeeding in finding a variable that encapsulates most of the dependency tehsis two random variable can be used for compression and for understanding the domain. The idea of encapsulating mutual dependency essay titles existentialism a new random variable was introduced by Tishby, Pereira and Bialek . Their idea was to extract the mutual dependency by a new variable which is a stochastic function of X that preserves information about Y inforrmation loosing information on X.
The algorithm to find such variable is called "The information bottleneck method". Although the motivation of this research is similar, we will information extraction thesis to handle it in a different way. We information extraction thesis a general optimization problem that represents the seek for a good extractor -T.
The information bottleneck problem is a special case of that problem. We will try to deal with other information extraction thesis cases as well as with the general problem. Our hope is that this research http://freey8.com/500-word-essay/critical-thinking-worldviews.html lead to new algorithms for extraction of mutual information and for a better understanding of the concept of dependency between random variables.
The first condition represents the fact that T contains all the mutual information and the second condition represents the information extraction thesis that it doesn't visit web page any additional information.
Suppose that T satisfies those two conditions. The go here is the data processing inequality see Using the same arguments on Y we obtain that T is information extraction thesis a deterministic function of Y.
If we group the values of X according to the values of g and group the values of Y according to the values of h, the joint distribution P X, Y must have the same structure as in Figure 3. In the general case, the best we can do is to look for an approximation of a perfect extractor. The optimization problemAs we have seen, we must compromise on the original demands on T.
Instead of insisting that T will contain all the mutual information, we wish to capture most of it. Similarly, instead of insisting that T will contain no irrelevant information, thesis absorption spectroscopy will try to minimize it. In the next chapter we will examine a genral framework to deal with mutual relations of random variables.
This general tool will give some additional insights and will enable to refine and to generalize this optimization problem. Chapter 4A framework for dealing with mutual relation of random variableSince our optimization problem deals with mutual relations among three random variables, we wish to gain more understanding about the domain of variables interrelations.
We want to know what are the possible information measures that a set of variable can have, what are the implications from knowing a certain fact on the variables, and so forth. For example, the fact that two variables are independent conditioning on a third variable, implies that the entropy of the third variable is greater or equal than the mutual information of the two variables. We will try to construct a general framework in which a fact information extraction thesis this, will have a clear and intuitive essay guidelines philosophy. The main tool we are going to use is an interesting correspondence between mutual relations among information extraction thesis variables and mutual relations among sets.
The main idea could be illustrated by an example. Let X 1 and Xetraction 2 be two arbitrary random variables.
This illustration appears in  but information extraction thesis any graduation speech goldsmiths formalization. At this point it is not clear that such sets indeed exist, but we will prove that shortly. The following corresponding equations are easily verified by examining lower part of Figure 4. The set-structure of the information quantities, gives insight and tools to deal with inference and optimization problems that consists of more info random variables.
In the simple example of two variables, this set-structure does not give additional insights to the well known basic equalities. After we prove the general correspondence, we will give several examples of how this correspondence could be beneficial in the case of three variables.
I-measureTheorem 1 information-set-correspondence for any set of random variable X 1. Any of the above equation will hold also for sets of variables. The correspondence extrachion will be unions of the analogical sets. Before we give this proof, we will try to illustrate the general idea information extraction thesis a more intuitive manner. The proof is based on the fact that information quantities can be decomposed into information extraction thesis or subtraction of joint entropies in the same manner that a value of an additive function on set-expression can be decomposed into sum or subtraction of union expressions.
The sets could be any sets that have a full algebra i. In order to show why this is indeed true, we define an additive function by setting its values on the atoms. Figure 4. It is clear that an additive function can take any value on an atom and that the values on the atoms fully characterize the additive function.
The information extraction thesis step is to this web page the function extractino on union expression in terms of values on atom and vise versa. We show that there exist a regular linear transformation from the atom values into the union values. This part of the proof contains the technical information extraction thesis that are somewhat hard to follow. The existence of the regular linear transformation implies that the union expressions can take any value and family misunderstanding essay they, like the values on atoms, fully characterize the additive function.
The similar decomposition and the similar values on the union expression are sufficient to show that any two correspondence expressions has irrigation engineering thesis same value. Given an arbitrary set of random information extraction thesis X 1.
To obtain the rest of the equations, we can make similar choices of non singleton sets.