3 consider the concept learning algorithm findg which outputs
a maximally general consistent hypothesis eg some maximally
general member of the version space a give a distribution for
ph and pd1h under which findg is guaranteed to output a

Question

3 .

Consider the concept learning algorithm FindG, which outputs

a maximally general consistent hypothesis

(

e

.

g

.,

some maximally

general member of the version space

) . (

a

)

Give a distribution for

P

(

h

)

and P

(

D

1

h

)

under which FindG is guaranteed to output a MAP

hypothesis.

(6)

Give a distribution for P

(

h

)

and P

(

D

1

h

)

under which

FindG is not guaranteed to output a MAP

.

hypothesis.

(

c

)

Give a

distribution for P

(

h

)

and P

(

D

1

h

)

under which FindG is guaranteed to

output a ML hypothesis but not a MAP hyp

. 3 .

Consider the concept learning algorithm FindG, which outputs

a maximally general

consistent hypothesis

(

e

.

g

.,

some maximally general member of the

version space

) .

(

a

)

Give a distribution for P

(

h

)

and P

(

D

1

h

)

under which FindG is

guaranteed to

output a MAP hypothesis.

(6)

Give a distribution for P

(

h

)

and P

(

D

1

h

)

under which FindG is

not guaranteed

to output a MAP

.

hypothesis.

(

c

)

Give a distribution for P

(

h

)

and P

(

D

1

h

)

under which FindG is

guaranteed to

output a ML hypothesis but not a MAP hyp

Craig Gilmore · Answer

This question relates to understanding the behavior of the FindG algorithm in the context of Maximum A Posteriori (MAP) and Maximum Likelihood (ML) estimation in concept learning. Let's address each part of the question in detail:

(a) Distribution under which FindG is guaranteed to output a MAP hypothesis

For FindG to output a MAP hypothesis, FindG needs to find the hypothesis that maximizes the posterior probability $($ . The MAP hypothesis is found by maximizing $P(h \mid D) = \frac{P(D \mid h) \cdot P(h)}{P(D)}$ .

To simplify, it's equivalent to maximizing $P(D \mid h) imes P(h)$ , since $)$ is constant for all hypotheses $h$ .

For FindG to be guaranteed to output a MAP hypothesis:

$P(h)$ (the prior probability of $h$ ) should be uniform over the version space, meaning $P(h)$ is constant for all hypotheses within the version space.
$($ should favor the most general hypothesis within the version space since FindG outputs the maximally general consistent hypothesis.

Thus, if we assume:

$P(h) = ext{constant}$ for all $h$ in the version space.
$P(D \mid h)$ favors the most general $h$ within the version space.

Under these conditions, FindG is guaranteed to output a MAP hypothesis, as the hypothesis with the highest posterior probability will also be the most general consistent hypothesis.

(b) Distribution under which FindG is not guaranteed to output a MAP hypothesis

FindG is not guaranteed to output a MAP hypothesis if the conditions stated above are not met. Specifically, this can happen if:

$P(h)$ is not uniform. For example, when some hypotheses are deemed more probable a priori than others, FindG might not maximize the posterior probability.
$P(D \mid h)$ does not consistently favor the most general hypothesis. For instance, if there is a specific hypothesis that fits the data well but is less general, FindG would not select it even though it has a higher posterior probability.

Consider:

$P(h)$ is non-uniform, such that some hypotheses have higher prior probabilities than others.
$P(D \mid h)$ does not particularly favor generality but rather correctness.

Under these conditions, FindG would not necessarily output the hypothesis with the highest posterior probability (the MAP hypothesis).

(c) Distribution under which FindG is guaranteed to output a ML hypothesis but not a MAP hypothesis

Maximum Likelihood (ML) estimation finds the hypothesis that maximizes the likelihood $P(D \mid h)$ without considering the prior probability $P(h)$ . For FindG to output an ML hypothesis but not a MAP hypothesis:

$P(h)$ should vary across hypotheses.
$P(D \mid h)$ favors the most general hypothesis consistent with the data.

Consider:

$P(h)$ is non-uniform.
$P(D \mid h)$ still favors more general hypotheses as in the case of ML estimation.

In this scenario, FindG will find the most general hypothesis that is still consistent with data (hence maximizing $P(D \mid h)$ ), but this hypothesis will not be guaranteed to maximize $P(h \mid D)$ because $P(h)$ introduces a non-uniform prior.

Summary

(a) Uniform $P(h)$ and $P(D \mid h)$ favoring general hypotheses ensure FindG outputs a MAP hypothesis.
(b) Non-uniform $P(h)$ or $P(D \mid h)$ not favoring general hypotheses mean FindG might not output a MAP hypothesis.
(c) Non-uniform $P(h)$ and $P(D \mid h)$ favoring general hypotheses ensure FindG outputs an ML hypothesis but not necessarily a MAP hypothesis.

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