Exercise 10.6 - Fishing nets

Answers

For question (a), we are to compute the posterior:

p ( X 2 | X 4 = thin , X 1 ∼ π ) ,

where:

π = ( 0.5 , 0 , 0 , 0.5 ) T .

This shall be done straightforwardly:

p ( X 2 = salmon | X 4 = thin , X 1 ∼ π ) = ∑ x 1 , x 3 p ( X 2 = salmon , X 1 = x 1 , X 3 = x 3 | X 4 = thin ) = p ( X 1 = winter , X 2 = salmon , X 3 = light , X 4 = thin ) p ( X 4 = thin ) + p ( X 1 = winter , X 2 = salmon , X 3 = medium , X 4 = thin ) p ( X 4 = thin ) + p ( X 1 = winter , X 2 = salmon , X 3 = dark , X 4 = thin ) p ( X 4 = thin ) + p ( X 1 = autumn , X 2 = salmon , X 3 = light , X 4 = thin ) p ( X 4 = thin ) + p ( X 1 = autumn , X 2 = salmon , X 3 = medium , X 4 = thin ) p ( X 4 = thin ) + p ( X 1 = autumn , X 2 = salmon , X 3 = dark , X 4 = thin ) p ( X 4 = thin ) = 0.5 ∗ 0.6 ∗ 1.7 p ( X 4 = thin ) .

While that for sea bass is:

p ( X 2 = salmon | X 4 = thin , X 1 ∼ π ) = 0.5 ∗ 0.05 ∗ 0.3 p ( X 4 = thin ) ,

hence the fish is likely to be salmon.

For question (b),

p ( X 1 = winter | X 3 = medium , X 4 = thin ) = ∑ x 2 p ( X 1 = winter , X 2 = x 2 | X 3 = medium , X 4 = thin ) = 0.25 ∗ 0.9 ∗ 0.33 ∗ 0.6 + 0.25 ∗ 0.1 ∗ 0.1 ∗ 0.05 p ( X 3 = medium , X 4 = thin ) = 0.04468 p ( X 3 = medium , X 4 = thin ) .

Analogously, we have:

p ( X 1 = spring | X 3 = medium , X 4 = thin ) = 0.01573 p ( X 3 = medium , X 4 = thin ) ,

p ( X 1 = summer | X 3 = medium , X 4 = thin ) = 0.02055 p ( X 3 = medium , X 4 = thin ) ,

p ( X 1 = autumn | X 3 = medium , X 4 = thin ) = 0.03985 p ( X 3 = medium , X 4 = thin ) .

Therefore the season is likely to be winter. Intuitively, a fish that is thin and medium lightness is likely to be a salmon. And the season in which a salmon is caught is most likely to be winter.

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2021-03-24 13:42
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