1. Instance Space (X) X T BP SK x1 L - x2 N x3 H x4 x5 x6 x7 x8 x9

2. Concept Space (C)
One of the possible concepts for the concept SICK might be enumerated in the following table: X T BP SK x1 L x2 N x3 H 1 x4 x5 x6 x7 x8 x9

3. Concept Space
For any arbitrary concept C, the following table is another format for representation of each output corresponding to each instance
Each c(xi) is the output of the concept c for that particular instance
C(x3)
C(x6)
C(x9)
C(x2)
C(x5)
C(x8)
C(x1)
C(x4)
C(x7)

4. Concept Space
There two attributes and each can have 3 possible values, there are 29 unique combinations or functions. 1 1 1 1 C1 C2 C3 C4
C29

5. Training Data Set (D) D T BP SK x1 N L 1 x2 x3

6. Hypothesis Space (H)
The learner has to apply some hypothesis, that introduces a search bias to reduce the size of the concept space
This reduced concept space becomes the hypothesis space

7. Hypothesis Space
For example, the most common bias is one that uses the AND relationship between the attributes
In other words the hypothesis space uses the conjunctions (AND) of the attributes T and BP
i.e. h = <T, BP>

8. Hypothesis Space H denotes the hypothesis space
Here it is the conjunction of attributes T and BP
If written in English it would mean:
H = <t, bp>:
IF “Temperature” = t AND “Blood Pressure” = bp
Then
H = 1
Otherwise
H = 0
In other words, the function gives a 1 output for all conjunctions of T and BP, e.g., H and H, H and L, H and M, etc.

9. Hypothesis Space
h = <H, H>: <temp, bp> BP H 1 N L T

10. Hypothesis Space h = <L, L>:BP H N L 1 T
Notice that this is the C2 that we discussed earlier in the Concept Space section 1

11. Hypothesis Space H = <T, BP>
Where T and BP can take on five values
H, N, L (High, Normal, Low)
Also ? and Ø
? means that for all values of the input H = 1 (don’t care)
Ø means that there will be no value for which H will be 1

12. Hypothesis Space
For example, h1 = <?, ?>: [For any value of T and BP, the person is sick]
The person is always sick BP H 1 N L T

13. Hypothesis Space
Similarly h2 = <?, H>: [For any value of T AND for BP = High, the person is sick]
Irrespective of temperature, if BP is High, the person is sick BP H 1 N L T

14. Hypothesis Space The person is never sick
h3 = < Ø , Ø >: [For no value of T or BP, the person is sick]
The person is never sick BP H N L T

15. Hypothesis Space
Having said all this, how does this still reduce the hypothesis space to 17?
Well it’s simple, now each attribute Temp and BP can take 5 values each: L, N, H, ? and Ø
So there are 5 x 5 = 25 total number of possible hypotheses

16. Hypothesis Space
Now this is a tremendous reduction from (29 ) 512 to 25
This number can be reduced further
There are redundancies within these 25 hypotheses
Caused by Ø

17. Hypothesis Space These redundancies are caused by Ø
Whenever there is Ø in any of the inputs and we are considering conjunctions (min) the output will always be 0
If there’s this ‘Ø’ in the T or the BP or both, we’ll have the same hypothesis as the outcome is always, all zeros
For a ?: we will either get a full column of 1’s, or a full row of 1’s in the concept matrix representation.
For both ?: all 1’s

18. Hypothesis Space h = < N , Ø >: h = < Ø , Ø >: BP H N L TH N L T
h = < N , Ø >:
h = < Ø , Ø >: BP H N L T

19. Concept Learning as Search
We assume that the concept lies in the hypothesis space. So we search for a hypothesis belonging to this hypothesis space that best fits the training examples, such that the output given by the hypothesis is same as the true output of concept
Hence the search has achieved the learning of the actual concept using the given training set

20. Concept Learning as Search
In short:
Assume , search for an that best fits D, such that xi D, h(xi) = c(xi)
Where c is the concept we are trying to determine (the output of the training set)
H is the hypothesis space
D is the training set
h is the hypothesis
xi is the ith instance of Instance space

21. Ordering of Hypothesis Space
General to Specific Ordering of Hypothesis Space
Most General Hypothesis:
hg< ?, ? >
Most Specific Hypothesis:
hs< Ø , Ø >

22. Ordering of Hypothesis Space
SK = < T, BP >, T = { H, N, L } and BP = { H, N, L }
< ?, ? >
< H, ? >
< N, ? >
< L, ? >
< ?, H >
< ?, N >
< ?, L >
< H, H >
< H, N >
< H, L >
< N, H >
< N, N >
< N, L >
< L, H >
< L, N >
< L, L >
< Ø , Ø >