# Continuous Bayes Definitions

In the continuous realm, the convention for the probability will be as
follows:

where x is a feature vector in d-dimensional space |R^{d} which will be referred to as *feature space*; and *ω*_{j} represent a finite set of *c* possible states (or *classes*) of existence: {*ω*_{1}, ..., *ω*_{c}}. Note the difference in the above between the *probability density function p*(*x*) whose integral

* *

and a probability *P*(*x*) represents a *probability* in the region of [0,1] in which

* .*

As was stated earlier, the Bayes rule can be thought of in the following (*simplified*) manner:

The Prior

As the name implies, the prior or *a priori* distribution is a
prior belief of how a particular system is modeled. For instance, the
prior may be modeled with a Gaussian of some estimated mean and
variance if previous evidence may suggest it be the case. Many times,
the prior may not be known so a uniform distribution is first used to
model the prior. Subsequent trials will then yield a much better
estimate.

### The Likelihood

The likelihood is simply the probability of specific class given the
random variable. This is generally known and it’s complement is
wanted - the *a posteriori *or the posterior probability.

### The Posterior

The posterior or *a posteriori* probability (or distribution)
is what results from the Bayes rule. Specifically, it states the
probability of an event occurring (or a condition being true) given
specific evidence. Hence the *a posteriori* is shown as P(*ω*|*x*) where *ω* is the particular query and *x* is the evidence given.

### The Evidence

The evidence *p*(x) is usually considered a scaling term. Bayes Theorem also states that it is equal to:

therefore, it is also possible to write:

## Independence and Conditional Independence

Briefly, it is important to discuss the terms of *independence* and *conditional independence*
since they figure prominently in the Bayesian world. Independence
essentially ensures that two (or more) random variables are not
dependent on one another. Graphically, this can be represtented as:

or mathematically by the Joint Probability: *P*(*A*,*B*,*C*) = *P*(*A*)*P*(*B*)*P*(*C*).

*Conditional independence* is more relevent to the Bayesian
world however. It states that given eveidence, one variable is
independent of another. Hence if A ╨ B|C (meaning A is independent of B
given C) means that given evidence at C, A is independent of B:

The conditional independence is witnessed in the causal relationship above