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Probablity Theory

A key concept in the field of pattern recognition is that of uncertainity. It arrises through:

  1. noise on measurements
  2. finite size of datasets.

Probability theory provides a consistent framework for the quantification and manipulation of uncertainity and forms one of the central foundations of Pattern Recognition.

When combined with Decision Theory, it allows us to make optimal predictions given all the informtion available to us, even though that information may be incomplete or ambiguous.

Example: Red/Blue boxes

The setup

Imagine we have two boxes red_box and blue_box. Each box contains different number of fruits oranges and apples. 

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red_box:
    - apples: 2
    - oranges: 6
blue_box:
    - apples: 3
    - oranges: 1

The Experiment

Now, we do the following steps:

  1. Select a box: Randomly pick one of the boxes (either red_box or blue_box)
  2. Pick a fruit: Then, randomly select an item of fruit for the box.
  3. Replace the fruit: Having observed the type of the picked fruit (apple or orange), now, replace it in the box from which it came.
  4. Repeat steps 1-to-3 multiple times.

The Pre-conditions / Assumptions

Now, let's suppose in doing the above experiment,

  1. we pick the red_box 40% of the times and the blue_box 60% of the times.
  2. Also, assume that that when we remove an item of fruit from a box, we are equally likely to select any of the pices of fruits in the box.

The Theory

In this experiemnt:

  1. the identity of the box to be chosen is a random variable \(B\) which can take two possible values r and b (corresponding to red or blue boxes).
  2. Similarly, the identity of fruit is also a random variable \(F\) and it can take either of the values a and o (corresponding to apple an dorange respectively).

Definition:

The Probability of an event is defined as the fraction of times, that event occurs out of the total number of trials (in the limit that the total number of trials goes to infinity). All probabilities must lie in the interval [0, 1]

Thus , the probability of selecting the red_box is \(4/10\) and that of the blue_box is \(6/10\).

\[ p(B=r) = 4/10 \]
\[ p(B=b) = 6/10 \]