Probability learning: first-order markov structures of quarternary events

N-A SA TN D-5684

NASA TECHNICAL NOTE

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PROBABILITY LEARNING: FIRST-ORDER MARKOV STRUCTURES OF QUARTERNARY EVENTS

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by E d w a r d M I . Huff Ames Research Center M o f e t t Field, Calzy NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

WASHINGTON, D. C.

FEBRUARY 1970

TECH LIBRARY KAFB, NM

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PROBABILITY LEARNING:

FIRST- ORDER MARKOV STRUCTURES

OFQUARTERNARYEVENTS By E d w a r d M . Huff

Ames R e s e a r c h C e n t e r Moffett Field, Calif.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sole by t h e Cleoringhouse for Federal Scientific and T e c h n i c a l Information Springfield, Virginia 22151 C F S T I price $3.00

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PROBABILITY LEARNING:

FIRST-ORDER MARKOV STRUCTURES

OF QUARTERNARY EVENTS

By Edward M. Huff Ames Research C e n t e r SUMMARY I n many c o n t r o l t a s k s , man i s r e q u i r e d t o make d e c i s i o n s based on a s u b j e c t i v e evaluation of t h e likelihood of f u t u r e events. I t i s necessary t o examine, t h e r e f o r e , t h e degree t o which p r e d i c t i v e b e h a v i o r corresponds with t h e o b j e c t i v e p r o b a b i l i t i e s o f e v e n t s , and t h e degree t o which man can learn t o u s e i n f o r m a t i o n c o n t a i n e d i n e v e n t sequences. I n t h e p r e s e n t s t u d y , 15 groups of s u b j e c t s were exposed t o d i f f e r e n t s e q u e n t i a l l y dependent s t i m u l u s sequences i n a f o u r - a l t e r n a t i v e p r o b a b i l i t y l e a r n i n g paradigm. Groups were found t o d i f f e r i n b o t h l e a r n i n g r a t e and t e r m i n a l response l e v e l s as a j o i n t f u n c t i o n of t h e s t o c h a s t i c s t r u c t u r e and t h e degree of dependence i n t h e s t i m u l u s sequence. Events with common s t r u c ­ t u r a l p r o p e r t i e s i n d i f f e r e n t sequences were l e a r n e d i n a s i m i l a r f a s h i o n . INTRODUCTION

The c l a s s i c p r o b a b i l i t y l e a r n i n g paradigm i n which s u b j e c t s p r e d i c t suc­ c e s s i v e e v e n t s may, i n p r i n c i p l e , b e used with any f i n i t e number of s t i m u l i and any s t i m u l u s g e n e r a t o r . Most r e s e a r c h , however, h a s been r e s t r i c t e d t o b i n a r y s i t u a t i o n s i n which e v e n t s a r e s e q u e n t i a l l y independent ( r e f . 1 ) . The few s t u d i e s t h a t have been r e p o r t e d with Markovian b i n a r y s t r u c t u r e s ( r e f s . 2-5) have found t h a t humans l e a r n f i r s t - o r d e r event dependencies by responding t o s h o r t s t i m u l u s - r e s p o n s e sequences t h a t precede t h e i r p r e d i c t i o n s , although a l l such sequences are n o t l e a r n e d e q u a l l y w e l l . The manner i n which s u b j e c t s l e a r n t o p r e d i c t f o u r - a l t e r n a t i v e event sequences g e n e r a t e d by f i r s t o r d e r Markov p r o c e s s e s ( r e f . 6) i s r e p o r t e d h e r e . I n one of t h e few s t u d i e s u s i n g more t h a n two dependent e v e n t s , Bennett, F i t t s , and Noble ( r e f . 7) found t h a t a f i v e - a l t e r n a t i v e Markov p r o c e s s t h a t g e n e r a t e d diagrams "concordant" with p r e v i o u s l y determined response p r e f e r ­ ences was l e a r n e d more q u i c k l y t h a n a s i m i l a r s t i m u l u s p r o c e s s t h a t g e n e r a t e d "discordant" diagrams. These g e n e r a t o r s were of a s p e c i a l t y p e i n which o n l y two nonzero (and i n t h i s c a s e e q u a l ) t r a n s i t i o n p r o b a b i l i t i e s were allowed i n each row of t h e Markov m a t r i x . The nonzero e n t r i e s d e f i n e d two " a p p r o p r i a t e " c h o i c e s f o l l o w i n g each s t i m u l u s . Although t h e a u t h o r s concluded that. response p r e f e r e n c e s a r e important i n p r o b a b i l i t y l e a r n i n g , t h e i r u s e of concordant and d i s c o r d a n t diagrams was confounded with s t r u c t u r a l d i f f e r e n c e s between t h e g e n e r a t o r s ( r e f . 8 ) . Thus, t h e f a c t t h a t t h e concordant g e n e r a t o r c r e a t e d

d i f f e r e n t kinds o f e v e n t sub-sequences t h a n t h e d i s c o r d a n t g e n e r a t o r ( e . g . , r e p e t i t i o n and a l t e r n a t i o n r u n s ) could have accounted f o r t h e i r r e s u l t s . I t i s d i f f i c u l t t o evaluate t h i s hypothesis s i n c e the response vector f o r t he two a p p r o p r i a t e a l t e r n a t i v e s was n o t r e p o r t e d ; however, t h e d i s c o r d a n t genera­ t o r d i d n o t s u p p r e s s t h e p r o p o r t i o n of a p p r o p r i a t e c h o i c e s below chance l e v e l d u r i n g t h e e a r l y s t a g e s of l e a r n i n g . As t h e a u t h o r s themselves n o t e , such s u p p r e s s i o n would be expected i f r e s p o n s e p r e f e r e n c e were t h e c o n t r o l l i n g v a r i a b l e . T h e i r s u g g e s t i o n t h a t between-group d i f f e r e n c e s r e s u l t e d i n t h e d i s c o r d a n t sequences b e i n g e f f e c t i v e l y " n e u t r a l " i s c l e a r l y s p e c u l a t i v e .

DESIGN AND PROCEDURE The q u a r t e r n a r y Markov m a t r i c e s used i n t h e p r e s e n t s t u d y were r e s t r i c t e d t o t h o s e t h a t a r e doubly s t o c h a s t i c and c o n t a i n one h i g h t r a n s i t i o n p r o b a b i l ­ i t y c t ( > O . 2 5 ) i n each row, t h e remainder of t h e p r o b a b i l i t i e s b e i n g e q u a l t o BC=(l - ct)/3l. I n such m a t r i c e s , each e v e n t h a s a high p r o b a b i l i t y [ a ) o f b e i n g followed by some p a r t i c u l a r e v e n t (perhaps i t s e l f ) and, provided t h a t ct < 1, each e v e n t w i l l a s y m p t o t i c a l l y appear with e q u a l r e l a t i v e frequency. P r e d i c t i n g t h e e v e n t s p e c i f i e d by t h e ct p r o b a b i l i t y w i l l be c a l l e d a profitable1 prediction. What w i l l be r e f e r r e d t o as t h e structure o f t h e p r o c e s s i s t h e g e n e r a l arrangement o f t h e ct p r o b a b i l i t i e s w i t h i n t h e 4 x 4 m a t r i x . S i n c e response p r e f e r e n c e s f o r i n d i v i d u a l s t i m u l u s diagrams were n o t c o n s i d e r e d , e x a c t l y f i v e s t r u c t u r e s were o f i n t e r e s t . These correspond with t h e unordered p a r t i t i o n s of f o u r i n d i s t i n g u i s h a b l e e v e n t s :

s5 = ( E i , E j , E k J E R >

where i # j # k # R J and t h e s u b s c r i p t e d S ' s and E's r e f e r t o t h e s t r u c ­ t u r e s and e v e n t s , r e s p e c t i v e l y . For each s t r u c t u r e , each s u b s e t of e v e n t s t e n d s t o g e n e r a t e a most probable sub-sequence i n which t h e elements c y c l e among themselves. For example

lBrunswikf s terminology ( r e f . 9 ) i s expanded h e r e . 2

....

corresponds w i t h any m a t r i x i n which one of t h e e v e n t s , E i , h a s a h i g h proba­ b i l i t y of b e i n g followed by i t s e l f , and each of t h e remaining t h r e e e v e n t s , E j , E k , and E & , h a s a high p r o b a b i l i t y of b e i n g followed by a d i f f e r e n t e v e n t within the subset. S i n c e a l l t h e e v e n t s a r e members of some s t r u c t u r a l s u b s e t , each e v e n t may b e i d e n t i f i e d w i t h a class Cn (n = 1, 2 , 3 , 4) , where n i s t h e number o f elements i n t h e s u b s e t and, hence, t h e number of elements i n a most proba­ b l e sub-sequence c y c l e . S t r u c t u r e S 3 may a l t e r n a t i v e l y b e regarded, t h e n , as c o n t a i n i n g two classes o f e v e n t s , C 1 and C 3 , which t e n d t o c r e a t e t h e most EiEiEi and EjEkEaEjEk ..., r e s p e c t i v e l y . p r o b a b l e sub-sequences I n s p e c t i o n w i l l r e v e a l t h a t each S I , S 2 , and S 3 c o n t a i n s C 1 e v e n t s , and each S 2 and S4 c o n t a i n s C2 e v e n t s . S t r u c t u r e s S3 and S 5 a l o n e c o n t a i n e v e n t s i n classes C 3 and C 4 , r e s p e c t i v e l y .

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The advantage of u s i n g t h e s e p a r t i c u l a r r e s t r i c t i o n s ( r e f . 10) i s t h a t f o r f i x e d a, t h e redundancy ( r e f . 11) of a l l f i v e s t r u c t u r e s i s e q u a l . Fur­ t h e r , independent o f a , t h e marginal ( z e r o - o r d e r ) u n c e r t a i n t y o f a l l s t r u c ­ t u r e s i s maximal ( i . e . , two b i t s ) . Hence, n o t o n l y must t h e s u b j e c t l e a r n f i r s t - o r d e r dependencies f o r performance t o improve, b u t d i f f e r e n c e s i n behav­ i o r as a f u n c t i o n of t h e s t r u c t u r e v a r i a b l e a r e i s o l a t e d from t h e degree of redundancy i n t h e s t i m u l u s sequence. Three a v a l u e s , 0 . 4 , 0 . 5 5 , and 0 . 7 , were used i n t h e p r e s e n t s t u d y because t h e y a r e s u f f i c i e n t l y d i f f e r e n t from 0.25 and 1.0 t o allow d e p a r t u r e s from e i t h e r change o r maximizing2 b e h a v i o r t o be a s s e s s e d ; t h e y a l s o allow a r e a s o n a b l e range of s t i m u l u s dependency t o b e examined. A t each l e v e l of a a s i n g l e sequence of 500 e v e n t s was g e n e r a t e d according t o t h e r u l e s of each s t o c h a s t i c s t r u c t u r e . Sequences were r e q u i r e d t o have marginal e v e n t frequen­ cies and t r a n s i t i o n f r e q u e n c i e s governed by t h e a p r o b a b i l i t i e s , w i t h i n 4 a d 5 p e r c e n t of t h e i r expected v a l u e s , r e s p e c t i v e l y . T r a n s i t i o n f r e q u e n c i e s governed by t h e f? p r o b a b i l i t i e s were allowed t o vary from 10 t o 20 p e r c e n t as an i n v e r s e f u n c t i o n of a. I n a l l c a s e s , t h e e m p i r i c a l m a t r i c e s were judged t o b e h i g h l y r e p r e s e n t a t i v e of t h e corresponding t h e o r e t i c a l p r o c e s s . Each of t h e 15 e v e n t sequences ( f i v e s t r u c t u r e s f o r each of t h e t h r e e a v a l u e s ) was recorded on p a p e r t a p e and used t o c o n t r o l t h e temporal sequence of f o u r v i s u a l s t i m u l i (+, -, 0 , x) t o a d i f f e r e n t group of s u b j e c t s . Ten c o l l e g e s t u d e n t s between t h e ages of 17 and 28 were randomly a s s i g n e d t o each group and were p a i d a t a f i x e d h o u r l y r a t e . In o r d e r t o d i s t r i b u t e t h e e f f e c t s of p o s s i b l e symbol diagram p r e f e r e n c e s , t h e f o u r e v e n t s recorded on t h e c o n t r o l t a p e were randomly i d e n t i f i e d with a d i f f e r e n t symbol o r d e r i n g f o r each s u b j e c t . iSuch b e h a v i o r would b e o b t a i n e d i f s u b j e c t s optimized t h e number of p r o f i t a b l e p r e d i c t i o n s by i n v a r i a b l y choosing a l t e r n a t i v e s s p e c i f i e d by t h e a probabilities. 3 A s B d i m i n i s h e s t h e expected marginal frequency f o r each u n l i k e l y a l t e r n a t i v e approaches z e r o . The p r o p o r t i o n a l change due t o even a s i n g l e frequency i n v e r s i o n , t h e r e f o r e , becomes q u i t e s e v e r e and r e q u i r e s s p e c i a l t r e a t m e n t . The Anderson-Goodman contingency t e s t ( r e f . 1 2 ) , n e v e r t h e l e s s , d i d n o t r e v e a l s i g n i f i c a n t d e p a r t u r e s from t h e t h e o r e t i c a l m a t r i c e s a t t h e 0.01 l e v e l o f confidence. 3

S u b j e c t s were t e s t e d two a t a t i m e i n s e p a r a t e booths and i n s t r u c t e d t o p r e d i c t which symbol would appear n e x t on a small r e a r p r o j e c t i o n c o n s o l e s c r e e n . They made t h e i r p r e d i c t i o n s by p r e s s i n g one o f f o u r b u t t o n s , each i d e n t i f i e d w i t h one o f t h e symbols, d u r i n g a 1 . 5 second i n t e r s t i m u l u s i n t e r v a l i d e n t i f i e d by t h e o n s e t o f a yellow p a n e l l i g h t . Each symbol appeared f o r 4 seconds. The s e l e c t e d response b u t t o n l i t up f o l l o w i n g a p r e d i c t i o n and remained on u n t i l t h e n e x t symbol appeared.

RESULTS A s e p a r a t e l e a r n i n g curve was f i r s t computed f o r each e v e n t , E , i n each s t r u c t u r e by determining t h e average p e r c e n t a g e o f p r o f i t a b l e p r e d i c t i o n s made by t h e 10 s u b j e c t s d u r i n g each b l o c k of 1 2 s u c c e s s i v e E o c c u r r e n c e s . 5 S i n c e a l l e v e n t s o c c u r r e d a t l e a s t 120 times i n a given sequence, t h i s procedure g u a r a n t e e d a minimum o f 10 b l o c k s o f e q u a l s t a t i s t i c a l p r e c i s i o n f o r each curve.

F i g u r e 1 p r e s e n t s l e a r n i n g curves f o r s t i m u l u s c l a s s e s Cn (n = 1, 2 , 3 , Figure 2 s e p a r a t e s t h e l e a r n i n g curves by s t r u c t u r e f o r each o f t h e two e v e n t c l a s s e s which are com­ b i n e d over s t r u c t u r e s i n f i g u r e 1, t h a t i s , C1 and C 2 . The s t a t i s t i c a l p r e ­ c i s i o n c h a r a c t e r i s t i c a l l y v a r i e s from curve t o curve as a r e s u l t of t h e unequal class membership d i s t r i b u t i o n between t h e s t r u c t u r e s . I n a l l cases where two o r more e v e n t s w i t h i n a s t r u c t u r e belonged t o t h e same c l a s s , how­ e v e r , no meaningful d i f f e r e n c e s between t h e i n d i v i d u a l l e a r n i n g c u r v e s were noted.

4) averaged by block over s t r u c t u r e s and s u b j e c t s .

The p r e s e n t d a t a s u p p o r t t h e h y p o t h e s i s p r e s e n t e d e a r l i e r concerning t h e importance of t h e s t r u c t u r a l v a r i a b l e i n p r o b a b i l i t y l e a r n i n g . They show t h a t q u a r t e r n a r y s t r u c t u r e s , and i n a l l l i k e l i h o o d n - a r y s t r u c t u r e s i n gen­ e r a l , i n v o l v e m u l t i p l e l e a r n i n g r a t e s and asymptotic l e v e l s ( s e e f i g . 1 ) which a r e a j o i n t f u n c t i o n of t h e s t r u c t u r a l p r o p e r t i e s and t h e redundancy of t h e c o n t r o l l i n g s t i m u l u s g e n e r a t o r . Furthermore, although some degree of w i t h i n s t r u c t u r e i n t e r a c t i o n i s r e f l e c t e d i n t h e learning rates (see f i g . 2 ) , f o r a given c1 t h e t e r m i n a l response l e v e l f o r an e v e n t i s mainly determined by t h e sub-sequence class o f which it i s a member, r a t h e r t h a n t h e p a r t i c u l a r stimu­ l u s s t r u c t u r e i n which i t i s imbedded. Indeed, t h e s e asymptotic l e v e l s are,

4The p r e s e n t a t i o n i n t e r v a l was made q u i t e long r e l a t i v e t o t h e responsei n t e r v a l i n o r d e r t o induce t h e s u b j e c t t o p r e p a r e h i s response w h i l e examin­ i n g t h e p r i o r s t i m u l u s . The s u b j e c t s , t h e r e f o r e , may have found i t e a s i e r than s u b j e c t s i n p r e v i o u s s t u d i e s ( r e f . 13) t o l e a r n f i r s t - o r d e r a s s o c i a t i o n s . 5The common a n a l y t i c t e c h n i q u e with b i n a r y sequences i s t o examine b l o c k s o f a r b i t r a r y event o c c u r r e n c e s . This procedure i s most convenient when t h e diagram f r e q u e n c i e s have a l s o been f o r c e d i n t o t h e o r e t i c a l agreement f o r each b l o c k , and i t i s i d e n t i c a l w i t h t h e p r e s e n t procedure when marginal frequen­ c i e s are e q u a l . Here ( i n o r d e r t o avoid unnecessary h i g h e r - o r d e r dependen­ c i e s ) , diagram f r e q u e n c i e s were n o t r e s t r i c t e d w i t h i n b l o c k s .

4

I

as a r u l e , d i f f e r e n t from a and s u p p o r t Hake and Hyman's c o n t e n t i o n ( r e f . 2 , pp. 72-73) t h a t p r o b a b i l i t y matching, when observed, i s t h e f o r t u i t o u s r e s u l t o f a complex l e a r n i n g p r o c e s s . No tendency f o r matching was found i n t h e p r e s e n t d a t a . There i s some i n d i c a t i o n , however, t h a t with s u f f i c i e n t stimu­ l u s dependency. The t e r m i n a l p r o p o r t i o n s of p r o f i t a b l e p r e d i c t i o n s would converge t o a common l e v e l f o r a l l e v e n t c l a s s e s ( s e e f i g . l ( c ) ) . DISCUSSION C e r t a i n i n t e r r e l a t e d f a c t o r s b e a r on t h e p r e s e n t f i n d i n g s . These a r e mentioned b r i e f l y i n o r d e r t o p o i n t up s i m i l a r i t i e s with o t h e r c l a s s i c a l l e a r n ­ i n g s i t u a t i o n s . F i r s t , as t h e number o f s t i m u l i i n an e v e n t class d i m i n i s h e s , t h e occurrences o f each component s t i m u l u s become c l u s t e r e d w i t h i n t h e sequence. The r e s u l t s , t h e r e f o r e , may r e f l e c t merely a r e l a t i v e advantage of massed v e r s u s d i s t r i b u t e d p r a c t i c e i n a complex t a s k ( r e f . 1 4 ) . Second, as t h e number of s t i m u l i involved i n a s t r u c t u r a l l y p r e f e r r e d sub-sequence i s reduced, t h e p o t e n t i a l of each component s t i m u l u s t o develop c o n f l i c t i n g remote a s s o c i a t i o n s i s diminished, provided t h a t a s s o c i a t i v e s t r e n g t h i s i n v e r s e l y r e l a t e d t o d i s t a n c e w i t h i n t h e sequence ( r e f . 1 5 ) . Both f a c t o r s a r e complicated, however, by t h e f a c t t h a t most p r o b a b l e sub-sequences i n v o l v i n g t h e g r e a t e s t number of e v e n t s a l s o have t h e g r e a t e s t number o f s t a r t i n g ( e n t r y ) p o i n t s , as w e l l as t h e g r e a t e s t p o t e n t i a l f o r a t y p i c a l r e e n t r y i n t o t h e i r own e v e n t s e t . Hence, t h e p r e s e n t r e s u l t s would b e expected i f one p o s i t s , as Hake and Hyman ( r e f . 2) s u g g e s t e d , t h a t s p e c i f i c s t i m u l u s n - t u p l e s , r a t h e r t h a n t h e p r o b a b i l i s t i c s t r u c t u r e of t h e s t i m u l u s g e n e r a t o r , a r e l e a r n e d . I n t h i s c a s e a d i f f e r e n t n - t u p l e would b e a s s o c i a t e d with each s t a r t i n g p o i n t i n t h e sub-sequence. I t i s a l s o t r u e , however, t h a t b e h a v i o r a l adjustment t o unexpected c o n t i n g e n c i e s i s o f t e n d i f f i c u l t because of mental o r motor s e t ( r e f s . 16, 17, 1 8 ) . Even i f s u b j e c t s d i d p e r c e i v e t h e p r o b a b i l i s t i c p r o p e r t i e s of t h e g e n e r a t o r , t h e n , one would expect a d i f f e r e n t i a l performance decrement between event c l a s s e s as a f u n c t i o n of t h e number o f a t y p i c a l sub-sequence r e e n t r i e s . These arguments may b e somewhat d i f f i c u l t t o u n t a n g l e , b u t i t i s c l e a r t h a t many o f t h e same v a r i a b l e s a r e o p e r a t i n g h e r e as i n t h e more determin­ i s t i c s e r i a l and p a i r e d - a s s o c i a t e s l e a r n i n g paradigms. I t may b e n o t e d , more­ o v e r , t h a t i n t h e extreme (a = 1 . 0 ) , each s t r u c t u r a l s u b s e t r e p r e s e n t s a s e r i a l l e a r n i n g t a s k . The p r e s e n t f i n d i n g s would appear t o b e , t h e r e f o r e , t h e p r o b a b i l i s t i c analog o f t h e w e l l documented i n v e r s e r e l a t i o n s h i p between s e r i a l l i s t l e n g t h and l e a r n i n g speed ( r e f . 1 4 ) . Ames Research C e n t e r N a t i o n a l Aeronautics and Space A d m i n i s t r a t i o n Moffett F i e l d , C a l i f . , 94035, A p r i l 8 , 1969 127- 5 1-09-01- 00- 2 1

5

REFERENCE S

1. Estes, W. K . : P r o b a b i l i t y Learning. C a t e g o r i e s of Human Learning, A. W. Melton, e d . , Academic Press (New York), 1964, p . 89.

2 . Hake, H. W . ; and Hyman, R . : P e r c e p t i o n o f t h e S t a t i s t i c a l S t r u c t u r e of a Random S e r i e s of Binary Symbols. J . Exp. P s y c h o l . , v o l . 45, 1953, pp. 64-74. E f f e c t of F i r s t - O r d e r C o n d i t i o n a l P r o b a b i l i t y i n a 3. Anderson, N . H . : Two-Choice Learning S i t u a t i o n . J . Exp. P s y c h o l . , v o l . 59, 1960, pp. 73-93.

4 . Chapman, J e a n P . : The Spacing o f S e q u e n t i a l l y Dependent T r i a s i n P r o b a b i l i t y Learning. J . Exp. P s y c h o l . , v o l . 62, 1961, pp. 545-551. 5 . Anderson, N . H . : An E v a l u a t i o n o f Stimulus Sampling Theory: Comments on P r o f e s s o r E s t e s ' Paper: C a t e g o r i e s o f Human Learning, A W. Melton, e d . , Academic P r e s s (New York), 1964, p . 129. 6. Feller, W.: An I n t r o d u c t i o n t o P r o b a b i l i t y Theory and I t s A p p l i c a t i o n s . John Wiley & Sons, I n c . , 1957, p . 338. The Learning of S e q u e n t i a l 7. Bennett, W. F . ; F i t t s , P. M . ; and Noble, M . : Dependencies. J . Exp. P s y c h o l . , v o l . 48, 1954, pp. 303-312. 8. Garner, W. R . : U n c e r t a i n t y and S t r u c t u r e as Psychological Concepts, John Wiley & Sons, I n c . , 1962, p . 287. 9. Brunswik, E . : P r o b a b i l i t y as a Determiner o f R a t Behavior. Psychol. , V O ~ . 25, 1939, pp. 175-197.

J . Exp.

10. Evans, S . H . : VARGUS 7: Computed P a t t e r n s From Markov P r o c e s s e s . Behav. S c i . , v o l . 1 2 , 1967, pp. 323-328.

11. Shannon, C . E . ; and Weaver, W . : The Mathematical Theory of Communication. Univ. of I l l i n o i s Press, Urbana, 1949. 1 2 . Anderson, R . W . ; and Goodwin, L . A . : S t a t i s t i c a l I n f e r e n c e About Markov Chains. Ann. Math. S t a t . , v o l . 28, 1957, pp. 89-110.

13. Tune, G . S . : Response P r e f e r e n c e s : A Review o f Some Relevant L i t e r a t u r e . Psychol. B u l l . , V O ~ . 61, 1964, pp. 286-302. 14. ffovland, C. I . : Human Learning and R e t e n t i o n . Handbook of Experimental Psychology, S. S. Stevens, e d . , John Wiley G Sons, 1951, Ch. 17. 15. Woodworth, R. S . ; and Schlosberg, H . : Experimental Psychology. R i n e h a r t , and Winston, N . Y . , 1954, p . 695.

6

Holt,

16. D a s h i e l l , . J . F . : A Neglected Fourth Dimension i n Psychological Research.

Psychol. Rev., v o l . 47, 1940, pp. 289-305.

17. Gibson, J . J . : A C r i t i c a l Review of t h e Concept of S e t i n Contemporary

Environmental Psychology. Psychol. B u l l . , v o l . 38, 1941, pp. 781-817.

A Note on I n c r e a s i n g t h e E f f i c i e n c y o f

18. Bugelski, B . R . ; and Huff, E. M . : Luchins' Mental S e t s . Am. J . P s y c h o l . , v o l . 75, 1962, pp. 665-667.

7

IO0

cn 80

z

0

k-

0

2

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I

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I

4

6

8

IO

BLOCKS

(a)

ct

= 0.4

IO0

cn 80

z 0 I­

o 60

a

w J m

3-

40

LL

0 LT

a

2?

20

0

I

J

I

I

I

2

4

6

8

IO

BLOCKS

(b)

ci

= 0.55

Figure 1.- Learning curves f o r event c l a s s e s Cn (n = 1, 2 , 3 , A ) averaged over s t r u c t u r e s and s u b j e c t s f o r each of t h r e e ci l e v e l s . 9

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4

6

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F i g u r e 1 . - Concluded.

10

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s2 s3 1

I

I

I

I

2

4

6

8

10

BLOCKS

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e v e n t s , a = 0.4

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0 I­

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2

4

6

8

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Figure 2 . -

events, a = 0.55

Learning curves f o r e v e n t classes by s t r u c t u r e s f o r each

C,

a

and C 2 level.

averaged s e p a r a t e l y

11

100

v,

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a

8

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100 -

v,

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I

I

I

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2

4

6

8

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events,

c1 =

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Figure 2 . - Continued.

12

I

I

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4

6

8

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Figure 2. - Concluded.

NASA-Langley, 1970

I

-5 A-3047

13

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