In likelihood ratio formula, what does the exponent mean?

Question

In section 3.2.1 of Kevin Murphy's Machine Learning textbook, the likelihood of the dataset is denoted by the following:

$p(D|h) = [{1 \over size(h)}]^N = [{1 \over |h|}]^N$

What does the N term represent and why?

Also, in the event that it is the probability of choosing N terms from the concept extension h (which is a subset of our range of numbers from which we are allowed to pick from), then why is it 1/size(h) and not size(h)/size([entire range of numbers])? It seems the latter would make more sense as far as probability goes.

Thanks in advance

Answer 1

The author means by $p(D|h) = [{1 \over size(h)}]^N = [{1 \over |h|}]^N$ - the probability of independently chosen $N$ neccessary items with replacements from $h$ - which is our sample space, not $N$, $N$ - is a number of trying in our experiment.

Let $D= \{3\}$ is our example chosen from $h=\{3,6,9,12\}$. then $p(D|h)=\tfrac{1}{4}$ - the prob. to chose $\{3\}$ from $|h|=4$ elements. By independency condition after $N$ such trials with replacements: $p(D|h\bigcap D|h\bigcap D|h\bigcap...\bigcap D|h)=p(D|h)\bigcap...p(D|h)=[\tfrac {1}{4}]^N$.