Lesson5.6. Hypothesis - Satisfies vs Consistent

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Example

Here's a set of example fruits:

1. Red, Big (Apple)

2. Green, Small (Not Apple)

3. Red, Small (Apple)

4. Green, Big (Not Apple)

 Imagine we are trying to learn a rule to predict whether a fruit is an apple or not, based on its characteristics. Our training examples (D) consist of different fruits with two features: color (red or green) and size (big or small), along with their classification (apple or not apple).

Satisfies

An example ( x ) is said to satisfy a hypothesis ( h ) if ( h(x) = 1 ). This is a simpler condition and does not consider whether ( x ) is a positive or negative example. Essentially, it’s like saying the hypothesis ( h ) gives a positive signal (1) for the example ( x ), without worrying about whether this is correct or not.

Let's apply this to each of our examples:

1. Example: Red, Big (Apple)

   - Hypothesis: "If a fruit is red, it is an apple."

   - Does it satisfy? Yes, because the hypothesis predicts this example as an apple (( h(x) = 1 )), which is a positive indication.

2. Example: Green, Small (Not Apple)

   - Hypothesis: "If a fruit is red, it is an apple."

   - Does it satisfy? No, because the hypothesis does not make a positive prediction for this example (( h(x) = 0 )). The fruit is green, so it does not satisfy the hypothesis of being red and thus an apple.

3. Example: Red, Small (Apple)

   - Hypothesis: "If a fruit is red, it is an apple."

   - Does it satisfy? Yes, similar to the first example, this example is red, and the hypothesis predicts it as an apple (( h(x) = 1 )), fulfilling the condition for satisfaction.

4. Example: Green, Big (Not Apple)

   - Hypothesis: "If a fruit is red, it is an apple."

   - Does it satisfy? No, as with the second example, this example is green and does not meet the hypothesis's criteria for being red and thus an apple.

In summary, in the context of the hypothesis "If a fruit is red, it is an apple," only the red fruit examples (1st and 3rd) satisfy the hypothesis because they meet the condition set by the hypothesis (being red) and thus receive a positive prediction (( h(x) = 1 )). The green fruit examples (2nd and 4th) do not satisfy the hypothesis since they do not meet the criteria (they are not red) and thus do not receive a positive prediction from the hypothesis. Remember, "satisfies" is solely about whether the hypothesis gives a positive prediction for the example, not whether the prediction is correct.

Hypothesis Consistency: 

A hypothesis is considered consistent with the training examples if it accurately classifies each example. In more straightforward terms, if your hypothesis is a rule for predicting an outcome, it is consistent if it always gives the correct prediction for the examples you have.

   - Definition: 

For a hypothesis ( h ) and a set of training examples ( D ), the hypothesis ( h ) is consistent if for every example ( (x, c(x)) ) in ( D ), the hypothesis' prediction ( h(x) ) is equal to the actual outcome ( c(x) ). Here, ( x ) is an input from your examples, and ( c(x) ) is the true outcome or classification for that input.




Consistency is a stricter and more accurate measure. An example is consistent with a hypothesis if the prediction made by the hypothesis ( h(x) ) matches the actual classification or outcome ( c(x) ) of that example. This means the hypothesis doesn’t just give a positive signal, but it gives the correct signal (whether positive or negative) as per the actual classification of the example.

Now, let's apply the definitions:

1. Hypothesis (h): 

A rule that attempts to predict if a fruit is an apple based on its color and size. An example hypothesis could be "If a fruit is red, it is an apple."

2. Consistent Hypothesis: 

A hypothesis is consistent if it correctly classifies each example in our training data. For our example hypothesis "If a fruit is red, it is an apple," it is consistent because:

   - It correctly classifies the 1st example (Red, Big) as an apple.

   - It doesn't misclassify the 2nd example (Green, Small) as an apple.

   - It correctly classifies the 3rd example (Red, Small) as an apple.

   - It doesn't misclassify the 4th example (Green, Big) as an apple.

   Here, ( h(x) = c(x) ) for each example ( x ) (where ( h(x) ) is the hypothesis's prediction and ( c(x) ) is the actual classification).

 


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