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rapport/main.pdf
rapport/main.pdf
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rapport/proofs.typ
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rapport/proofs.typ
···
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$ forall s in S, Q'(s, 2) = 1/2 Q(s, 2) $ <dkl-zero-a2>
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$ forall s in S, forall a in A - {1, 2}, Q'(s, a) = Q(s, a) $ <dkl-zero-else>
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#let kl = (a, b) => $#a log #a / #b$
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#let crossout = (content, why) => $underbracket(cancel(#content), "d'après " #why)$
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#let why = (content, reference) => $underbracket(#content, "d'après " #ref(reference))$
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#let crossout = (content, reference) => $why(cancel(#content), reference)$
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On a
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$
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D_"KL" ( Q || Q' )
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&= sum_((s, a) in S times A) kl(Q(s, a), Q'(s, a)) \
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&= sum_((s, a) in S times A) Q(s, a) log Q(s, a) / (Q'(s, a)) \
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&"On découpe la somme selon les valeurs de " A ":" \
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&= sum_(s in S)
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crossout(
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sum_(a in A - {1, 2}) kl(Q(s, a), Q(s, a)) ,
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#[@dkl-zero-else]
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)
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+ kl(Q(s, 1), 2Q(s, 1)) + kl(Q(s, 2), 1/2 Q(s, 2)) \
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sum_(a in A - {1, 2}) [ Q(s, a) log Q(s, a) / (Q'(s, a)) ]
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+ Q(s, 1) log Q(s, 1) / (Q'(s, 1))
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+ Q(s, 2) log Q(s, 2) / (Q'(s, 2)) \
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&= sum_(s in S)
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crossout(sum_(a in A - {1, 2}) Q(s, a) log Q(s, a) / (Q(s, a)), #<dkl-zero-else>)
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+ Q(s, 1) log Q(s, 1) / why( 2 Q(s, 1), #<dkl-zero-a1>)
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+ Q(s, 2) log Q(s, 2) / why( 1/2 Q(s, 2), #<dkl-zero-a2>) \
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&= sum_(s in S)
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Q(s, 1) lr([ log Q(s, 1) - log Q(s, 1) - log 2 ], size: #200%) +
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Q(s, 2) [ log Q(s, 2) - log Q(s, 2) - log 1/2 ] \
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&= sum_(s in S) - Q(s, 1) log 2 + Q(s, 2) log 2 wide "d'après" #[@dkl-zero-a1 et @dkl-zero-a2] \
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&= sum_(s in S) log 2 thin (cancel(Q(s, 2) - Q(s, 1))) wide wide thin "d'après" #[@dkl-zero-qeq] \
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&= sum_(s in S) - Q(s, 1) log 2 + Q(s, 2) log 2 \
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&= sum_(s in S) log 2 thin crossout((Q(s, 2) - Q(s, 1)), #<dkl-zero-qeq>) \
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&= sum_(s in S) 0 = 0
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$