What does it mean that an hypotesis is consistent?

Question

I am studying concept learning, and I am focusing on the concept of consistency for an hypotesis.

Consider an Hypotesis $h$, I have understood that it is consistent with a training set $D$ iff $h(x)=c(x)$ where $c(x)$ is the concept and this has to be verified for every sample $x$ in $D$.

For example consider the following training set:

and the following hypotesis:

$h_1=<?,?,?,Strong,?,?>$

I have that this is not consistent with D because for the example $3$ in $D$ we have $h(x)!=c(x)$.

I don't understand why this hypotesis is not consistent.

Infact, consider the following hypotesis:

$h=<Sunny,Warm,?,Strong,?,?>$

this is consistent with $D$ because for each example in $D$ we have $h(x)=c(x)$.

But why the first hypotesis $h_1$ is not consistent while the second,$h$, is consistent?

Can somebody please explain this to me?

Answer 1

I'm not especially familiar with this but from the example provided we can deduce that:

• An hypothesis is a partial assignment of values to the features. That is, by "applying the hypothesis" we obtain a subset of instances for which the features satisfy the hypothesis.
• An hypothesis is consistent with the data if the target variable (called "concept" apparently, here EnjoySport in the example) has the same value for any instance in the subset obtained by applying it.

First case: $h_1=<?,?,?,Strong,?,?>$. All 4 instances in the data satisfy $h_1$, so the subset satisfying $h_1$ is the whole data. However the concept EnjoySport can have two values for this subset, so $h_1$ is not consistent.

Second case: $h_2=<Sunny,Warm,?,Strong,?,?>$. This hypothesis is more precise than $h_1$: the subset of instances which satisfy $h_2$ is $\{1,2,4\}$. The concept EnjoySport always have value Yes for every instance in this subset, so $h_2$ is consistent with the data.

Intuitively, the idea is that an hypothesis is consistent with the data if knowing the values specified by the hypothesis gives a 100% certainty about the value of the target variable.