The Mathematical Rescorla-Wagner Model - Report Example

Introduction The Rescorla-Wagner model (Rescorla and Wagner, 1972) is a theory developed by two psychologists named Robert A. Rescorla and Allan R. Wagner. It is a mathematical model explaining the amount of learning that which occurs on each trial episode of Pavlovian learning. The Pavlovian theory (Pavlov, 1927) originally theorized classical conditioning and learning, and has since been improved by other psychologists. In comparison to other prior models of associative learning, the Rescorla-Wagner model recognizes two new important aspects; I. That learning will only occur if the events of the trial are contrary to what is expected by the organism and, II. That the expectation of any trial outcome is based on a predictive value of the combination of all stimuli present. This means that surprise is necessary for learning to take place. If the US (unconditioned stimulus) does not surprise you, no learning will take place and vice versa. The second point means that if one has no experience with a certain CS (conditioned stimulus) that no learning will take place as there is no expectation. If the US takes event, then one is surprised, and thus they learn about what the CS predicts of the US. If one has many experiences with the CS where the US follows, then one is less and less surprised at each trial, and therefore learns little of the CS’s prediction of the US. The Rescorla-Wagner model states that the predictive value of the US is based on the sum total of all the associative strengths of the CS present within the trial. The Rescorla-Wagner equation may be simply represented as follows: ∆V = αβ (λ- ∑V) Where; ∆V is the change in the strength of association/ predictive value of a stimulus V α is the salience of the CS (bounded by 0 and 1) β is the rate parameter for the US (bounded by 0 and 1), sometimes called its association value λ is the maximum conditioning possible for the US ∑V is the total associative strength of all CS The overexpectation effect (Rescorla, 1970) is a decline in conditioned responding to each element of a pair of well-established conditioned stimuli (CSs), originally reinforced elementally (A–US/X–US), observed as a result of their being given further reinforcement in compound. In this experiment, two overexpectation designs were used. Food pictures were randomly assigned the cues; A, B, C etc. The first overexpectation design is summarized in the table below. The Rescorla-Wagner model predicts that AB→US pairings in phase 2 should reduce the associative strengths of A and B compared to C, the control cue which does not undergo training in phase 2. Phase 1 involved 6 A trials, 6 B trials and 6 C trials while phase 2 had 6 AB trials randomly intermixed with other trials. Table 1: table showing overexpectation design 1 Phase 1 Phase 2 Tests Rescorla-Wagner predictions Food A → US AB → US A Lower VA Food B → US B Lower VB Food C → US C Large VC In the second overexpectation design, food groups D, E, F, and G were involved. According to this design, F should become inhibitory in phase 2. A summation test (one which tests whether a stimulus functions as a conditioned inhibitor) was also performed in this design. In order to test whether F became inhibitory, a summation test was performed. If F is inhibitory, then it should reduce the hormone level when paired with G in the summation test. The table below summarizes the second overexpectation design. Table 2: table showing overexpectation design 2 Phase 1 Phase 2 Test Rescorla-Wagner predictions Food D → 50 DEF → 50 GF VF should be negative. Hence GF < G Food E → 50 Food F → 50 G → 50 G The experiment involved 156 participants completing a learning task in which they learned to associate different food pictures with hormone levels. Food = CS, hormone level = magnitude of US (i.e. λ). Sissons and Miller (2009) did an experiment on rats to examine the interaction of trial massing and overexpectation treatment using a Pavlovian fear conditioning. The overexpectation design used in the experiment was the same as that used in this experiment. In first-order conditioning, Experiment 1 found the trial spacing effect, the overexpectation effect (i.e., decreased responding to a cue when reinforced trials are massed), and a counteraction between trial massing and overexpectation treatment (i.e., an alleviation of the decrement in responding seen with trial massing or overexpectation treatment alone when the two treatments are administered together). Experiment 2 replicated Experiment 1 with the critical treatments embedded within a sensory preconditioning preparation. The trial spacing effect, the overexpectation effect, and the mutual counteraction of overexpectation treatment and trial massing all proved significant. In Experiment 3, either the excitatory context resulting from trial massing or the nontarget conditioned stimulus of overexpectation treatment was extinguished. Kimble (2010) also did an experiment on albino rats to examine two theoretical accounts of the overexpectation effect. She used the Rescorla-Wagner model and conducted three experiments in total. Experiment 1 was an attempt to obtain overexpectation in their preparation. Experiment 2 assessed whether preexposure of the companion stimulus can attenuate overexpectation. Experiment 3 assessed whether compound preexposure attenuated the preexposure effect and whether attenuation of overexpectation occurs when one of stimuli that underwent compound preexposure is extinguished (the summation test). The results of both the Kimble, and Sissons and Miller experiments confirmed the two hypotheses of overexpectation as all three experiments turned out as predicted. Method Part A: Participants The participants of this experiment were recruited at random. They were 156 in total. The participant composition was as follows; 111 were women, this figure corresponded to 70.7% of the total participant population. The rest of the 45 participant were the men; the minority corresponding to a percentage of 28.7 of the entire participant population. The average age range of the participants was 25, as the ages ranged from 19 to 44. The mean age encountered was 20, there were 56 of them, while the least frequently encountered ages were 26, 31, 36, 38, 39, and 44; only one of each participated in the experiment. The standard deviation of age was 4.651. Part B: Procedure The experiment sought to find out whether the hormone levels of foods will influence when the client eats certain foods. The experiment involved the participants completing a learning task in which they learned to associate different food pictures with hormone levels. There were 5 test trials: A, B, C, GF, and G. The training trial procedure was as follows: The first screen displayed the food cue(s) and a scale on which the participant could predict the hormone level. Although there were no numbers displayed on the scale, the computer recorded the participant’s hormone level prediction as a number ranging from 0 to 100. Once the participant had chosen a hormone level, he/she clicked on an OK button to see the actual, (correct) hormone level. The correct hormone level was displayed for 2 seconds. If the participant’s prediction was within 5 units of the correct hormone level, then an additional text that read ‘Your prediction was very good!’ was displayed. All trials were separated by a 1-second inter-trial interval that displayed a blank screen. The test trial procedure was identical to the training trials except that no feedback was provided after the participant made a prediction and clicked the OK button. The training test trials are summarized in table 3 below. Cue food assignment (A, B, C, etc) was done randomly by the computer. Table 3: table showing overexpectation designs 1 & 2 Phase 1 Phase 2 Test Rescorla-Wagner predictions Food A → 50 Foods AB → 50 A VA = 25 Food B → 50 B VB = 25 Food C → 50 C V C = 50 Food D → 50 Foods DEF → 50 GF VG + VF = 50 + (-17)= 33 Food E → 50 Food G → 50 Food G → 50 G VG = 50 Phase 1 of the training trials involved 6 A-50 trials, 6 B-50 trials, 6 C-50 trials, 6 D-50 trials, 6 E-50 trials, G B-50 trials. Phase 2 of the training trials involved 6 AB-50 trials, 6 DEF-50 trials, 6 G-50 trials. The number 50 used here refers to the hormone level shown on that trial. There were some additional training trials in which the foods were followed by a hormone level other than 50. Results The 2 hypotheses tested in this experiment were based on the predictions of the Rescorla-Wagner model. These were: In the first overexpectation design, A and B should become weaker predictors of the outcome than C; and that in the second overexpectation design, F should become inhibitory and hence should pass a summation test for inhibition when paired with G. The pattern of the data obtained was consistent with the Rescorla-Wagner predictions as the predictions of A and B were lower than that of C; A = B < C. For each participant, their A and B predictions at test were averaged, and then compared to their prediction on the C test trial. In the second design, VF was negative, hence GF < G; meaning that F was inhibitory. The results of the experiment are summarized in the table below Table 4: table showing the results of the experiment A and B C G GF Average 48.39 50.1 49.97 51.76 Standard deviation 9.046 12.208 7.481 11.546 A paired-samples t-test was conducted to compare the average A and B test trial hormone level predictions and the average C test trial hormone level predictions. The results of this test are shown in tables 4 and 5 below. There was not much of a significant difference in the scores for A and B (M= 48.39, SD= 9.046) and C (M= 50.10, SD= 12.208) predictions; t(155)= -1.539, p = 0.126. These results suggest that the Rescorla-Wagner prediction that A and B lose associative strength in phase 2 in comparison to C is correct. Table 4: Paired Samples Statistics Mean N Standard deviation Std. Error Mean Pair 1 A&B - C 48.39 156 9.046 0.724 50.10 156 12.208 0.977 Table 5: Paired Samples Test Pair 1 A&B -C Paired Differences t df Sig. (2-tailed) Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference Lower Upper -1.702 13.810 1.106 -3.886 0.482 -1.539 155 0.126 Another paired-samples t-test was conducted to test the hypothesis that F became inhibitory in phase 2 for which a summation test with G was done. The results of this test are shown in the tables 6 and 7 below. There was a significant difference in the scores for G (M= 49.97, SD= 7.481) and GF (M= 51.76, SD= 11.546) predictions; t(155)= -1.688, p = 0.094. These results suggest that the Rescorla-Wagner prediction that F becomes inhibitory in the summation test with GF did was incorrect as the results indicated higher values for GF than G alone, in contrast to the expectation that GF values would be lower than G. Table 6: Paired Samples Statistics Mean N Standard deviation Std. Error Mean Pair 1 G - GF 49.97 156 7.481 0.599 51.76 156 11.546 0.924 Table 7: Paired Samples Test Pair 1 G - GF Paired Differences t df Sig. (2-tailed) Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference Lower Upper -1.782 13.190 1.056 -3.868 0.304 -1.688 155 0.094 Discussion This experiment tested two hypotheses: In the first overexpectation design, A and B should become weaker predictors of the outcome than C; and in the second overexpectation design, F should become inhibitory and hence should pass a summation test for inhibition when paired with G. The expectations of the first overexpectation design were met. Foods A and B became weaker predictors of the outcome in phase 2 than food C. the general expectation was met i.e. A= B < C, but the expectation that A = B = 25 was not. The results of the experiment revealed that A and B= 48.39 while C = 50.10 (which was as expected). The second overexpectation design, however, did not turn out as expected. It was expected that if F was inhibitory, it would pass the summation test when it was paired with G. F did not seem to be inhibitory as the GF outcome values were 51.7 which were higher than those of G, 49.97 (F did not pass the summation test). This meant that F was excitatory rather than inhibitory. The results of this experiment were similar to the results of the two experiments done by Kimble, and Sissons and Miller described in the introduction section of this paper. the difference however, is that food F did failed to pass the summation test done when it was paired with G. the discrepancies may be attributed to the fact that this experiment involved human beings, while the others involved rats; and also, not all food groups had a hormone level of 50, which may have caused the results to turnout different than what was expected. The study was also limited to those who had access to and knowledge to operating a computer as the tests involved the use of a computer. There may also have been too many food groups involved in the experiment, such that the leaning process was not as clear and straight forward as the other experiments described before had. In summation, the test was generally successful even though the results were not exactly as expected. The Rescorla-Wagner model is able to predict the effect that stimuli have on learning outcomes. References Bouton, M.E. (2007). Learning and behavior: A contemporary synthesis. Sunderland, MA. pp 102-142 Kimble W. (2010). Acquisition and Performance Accounts of the Overexpectation Effect. Retrieved from http://etd.auburn.edu/etd/bitstream/handle/10415/2451/Kimble,%20Whintey%20Thesis.pdf?sequence=2 Rescorla RA. (1970). Reduction in the effectiveness of reinforcement after prior excitatory conditioning. Learning and Motivation. pp 372–381. Rescorla, R. A., & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and non-reinforcement. In A. H. Black & W.F. Prokasy (Eds.), Classicalconditioning II: Current theory and research (pp. 64-99). New York: Appleton-Century-Crofts. Pavlov, I.P. (1927/1960). Conditional Reflexes. New York: Dover Publications. Retrieved from http://psychclassics.yorku.ca/Pavlov/ Sissons H.T, Miller R. R. (2009). Overexpectation and trial massing. Journal of Experimental Psychology Animal Behavior Processes. (2):186-96. doi: 10.1037/a0013426 Read More