a. How to start: b. How to continue: c. How to stop: b EAP 2. EAP EAP (expected a posteriori) (posteriori distribution) (θ) MAP (maximum a posteriori)

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1 LET (pp ) EAP EAP (, 2013) (, 2014) PROX (, 2015) (computer-adaptive testing) EAP (expected a posteriori) Keywords: EAP 1. (, 1996, p. 273; Thissen & Mislevy, 2000, p. 101) 25

2 a. How to start: b. How to continue: c. How to stop: b EAP 2. EAP EAP (expected a posteriori) (posteriori distribution) (θ) MAP (maximum a posteriori) (Bayesian modal) (, 2011, p. 83) MAP (, 2011, p. 82) MAP (maximum likelihood estimation method) MAP (2010, p. 190) (2002, p. 35) MAP (2009, p. 56) (2011, p. 83) EAP (, 2011, p. 85;, 2009, p. 56) 2.1 (2012, p. 39) U A B 1 26

3 U A B B A P(B A) = P(A B) P(A) (1) (1) P(B A) A A B 1 A B A B (1) P(A B) 1 (1) P(B A) P(A) ( : ) P(A B) ( : 52 3 ) ( ) P(B A) = P(A B) : P(A) : = =

4 1 2 1 B A A B 1 U A B A B P(A B) = P(A B) P(B) (2) = = 3 12 = (2) (2) P(A B) P(B) ( : ) P(A B) ( : 52 3 ) ( ) (1) (1) : P(B A) = P(A B) P(A) P(A) P(A B) = P(B A)P(A) (3) 28

5 (2) (2) : P(A B) = P(A B) P(B) P(B) P(A B) = P(A B)P(B) (4) (3) (4) P(A B) (3) : P(A B) = P(B A)P(A) (4) : P(A B) = P(A B)P(B) P(B A)P(A) = P(A B)P(B) (5) (5) P(B) P(A B) (6) P(A B) = P(B A)P(A) P(B) (6) 2.2 (2010, p ) (2012, p ) U H D ( 3) 29

6 D D H U H 3. H D 1 3 A H B D (6) : P(A B) = P(H D) = P(B A)P(A) P(B) P(D H )P(H) P(D) (7) D H i (i = 1, 2, 3,, N) ( 4) H i D D H1 D H2 D H3 D H D HN U H1 H2 H3 H HN 4. H D 2 30

7 4 D H i P(D) = P(D H 1 ) + P(D H 2 ) + P(D H 3 ) + + P(D H N ) (8) (3) P(A B) = P(B A)P(A) (8) P(D H i ) P(A B) P(D) 8 : P(D) = P(D H 1 ) + P(D H 2 ) + P(D H 3 ) + + P(D H N ) P(D H i ) = P(D H i )(H i ) P(D) =P(D H 1 )P(H 1 ) + P(D H 2 )P(H 2 ) + P(D H 3 )P(H 3 ) + + P(D H N )P(H N ) (9) (9) (7) : P(H D) = P(D H)P(H) P(D) P(H i D) = P(D H i )P(H i ) P(D H 1 )P(H 1 ) + P(D H 2 )P(H 2 ) + P(D H 3 )P(H 3 ) + + P(D H N )P(H N ) P(H i D) = P(D H i)p(h i ) N P(D H i )P(H i ) i=1 (10) (10) 31

8 (2002, pp ) P(H i D) = P(D H i )P(H i ) P(D H i )P(H i ) dx (11) (11) P(H i D) D H i P(D H i ) H i D U D H i H i P(H i ) D H i (11) P(D) = (11) P(D H i )P(H i ) P(H i D) P(D H i )P(H i ) (i = 1, 2, 3,..., N) (12) (12) U = H D = 32

9 2.3 EAP EAP EAP (expected a posteriori) i (, 2011, p. 84) (expected value) x 1, x 2,...x n, 2007, p. 234, p. 239;, 2015, p. 97 EAP 2 E(X) Khan Academy (2009) 6 2, 2, 3, 5, 5, 6 U ( ) 6 = 3.8 (13) (13) % (2)2 + 1(3) + 2(5) + 1(6) 6 = 1 6 ( ) = = = 33% % % % 6 (14) = =

10 (14) X P(X = x) (x = 1, 2, 3,..., n) E(X) E(X) = n x i p i i=1 (15) E(X) E(X) = x f (x) dx (16) (11) (11) : P(H i D) = P(D H i )P(H i ) P(D H i )P(H i ) dx (11) P(D H i ) likelihood L P(H i D) = L(D H i )P(H i ) L(D H i )P(H i ) dx (17) E(X) (16) (17) θ P(H) g(θ) E(θ i D) = = θ i f (θ i ) dθ i θ i L(D θ i ) g(θ i ) L(D θ i ) g(θ i ) dθ i = θ i L(D θ i ) g(θ i ) L(D θ i ) g(θ i ) dθ i (18) (18) D (2011, p. 85) EAP (5.19) 34

11 N P(X) (maximum likelihood estimation) (2002, pp ) 3 1 n 2 n θ i 3 n 1 I(θ) I(θ) V[ˆθ i θ i ] = 1 I(θ i ) V[ˆθ i θ i ] θ i ˆθ i 3 (2002, p.65) I(θ i ) = E ( ) 2 θ log L (u i θ) θ=θ i (19) (19) I(θ) 1 I(θ) 1 I(θ) I(θ) I(θ) I(θ) 35

12 3.2 X(x i = x 1, x 2, x 3,..., x N ) θ No. x i 1 x 1 P(X = x 1 ; θ) 2 x 2 P(X = x 2 ; θ) 3 x 3 P(X = x 3 ; θ)... N x N P(X = x N ; θ) x 1, x 2, x 3,..., x N P(x i ; θ) P(x i ; θ) = P(X = x 1 ; θ) P(X = x 2 ; θ) P(X = x 3 ; θ) P(X = x N ; θ) (20) (20) θ L(θ ; X = x 1, x 2, x 3,..., x N ) = P(X = x 1, x 2, x 3,..., x N ; θ) (21) L(θ ; X = x 1, x 2, x 3,..., x N ) θ, 2010 (2010, p. 62) (21) L(θ ; x i ) P(x i ; θ) x i θ L(θ ; x i ) N x 1, x 2, x 3,..., x N θ L( x i ) x θ L(θ ; x i ) (20) N log L(θ ; x 1, x 2, x 3,..., x N ) (22) 36

13 L(θ ; x 1, x 2, x 3,..., x N ) θ (, 2007, pp ) U(θ ; x i ) = θ log L (θ ; x i ) (23) E [U(θ ; x i )] (24) (24) 0 (, 2011, p.14) = [ 2 ] [ 2 ] Var [U(θ ; x i )] = E [U(θ ; x i ) 2 ] (E [U(θ ; x i )]) 2 0 (E [U(θ ; x)]) 2 = 0 Var [U(θ ; x i )] = E [U(θ ; x i ) 2 ] (25) U(θ ; x i ) (25) (26) I(θ) (2002, p. 65) (18) (2002) x i θ Var [U(θ ; x i )] = E [U(θ ; x i ) 2 ] = E I(θ) = E ( ( ) 2 θ log L (θ ; x i) ) 2 θ log L (θ ; x i) (26) 37

14 (2002, p. 66) j θ i 2002, pp I j (θ i ) = a 2 p j (θ i ) q j (θ i ) (27) 2 I j (θ i ) = a 2 j p j (θ i ) q j (θ i ) (28) 3 I j (θ i ) = a2 j (p j(θ i ) c j ) 2 q j (θ i ) p j (θ i ) (1 c j ) 2 (29) 1 2 3, 2002, p. 69, 2007, p. 272 I(θ) = b 4. EAP 38

15 (2010).. Khan Academy. (2009, February 24). Expected Value: E(x). [Video file]. Retrieved from Kredt7vY (2011). 8. (1996).. (2007). (2013) Retrieved from Sumi.pdf (2014). 1PLM, 2PLM, 3PLM Retrieved from Sumi.pdf (2015). PROX Retrieved from Sumi.pdf (2002).. (2009). e. (2010). Excel. (2012).!. 39

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