Baysian Approaches

1. Quantitative Methods 2011 Baysian Approaches Kun Guo, PhD Reader in Cognitive Neuroscience School of Psychology University of Lincoln

2. Example

3. Vision is an act of interpretation Visual system interprets incoming retinal signals in the context of existing knowledge of the world. Some issues within this process: (1) signals are corrupted by variability of noise; (2) uncertainity in computation. Brain represents sensory information probabilistically, in the form of probability distribution. Ideal Bayesian Inference Observer: assigning probabilities to any degree of belief abouth the state of the world. Posterior ≈ Prior × Likelihood : Visual perception (Bayesian posterior probability of a scene) ≈ Prior probability of the state × Current input from the eye

4. Bayes’ Theorem Thomas Bayes (1701 – 1761): English mathematician “An essay towards solving a problem in the doctrine of chances” was read to Royal Society in 1763 (doctrine of chance ~ theory of probability). Posterior ≈ Prior × Likelihood: a method of statistical inference to calculate the impact of evidence on beliefs. The probability is interpreted as a degree of belief (conditional probability distribution)rather than frequency. In application, the initial degree of belief is called prior and the updated degree of belief is called posterior. Widely applied in Science, Engineering, Medicine and Law, especially after 1950s.

5. Bayesian Inference P(A|B): posterior, the degree of belief in A after B is observed. P(A): prior, the degree of belief in A before B is observed. P(B|A)/P(B): likelihood, impact of B on the degree of belief in A.

6. Target Predictor 4 200 ms Predictor 3 Predictor 2 Time FP Predictor 1 Applying Bayesian Inference in Vision research Landing light Stimulus sequence comprised four collinear bars (predictors) which appeared successively towards the foveal region, followed by a target bar with same or different orientation. Guo et al. (2004) Effects on orientation perception of manipulating the spatiotemporal prior probability of stimuli. Vision Research 44: 2349-2358

7. Applying Bayesian Inference in Vision research Stimulus demonstration – non-collinear trial

8. Applying Bayesian Inference in Vision research Spatiotemporal structure of the priors

9. Applying Bayesian Inference in Vision research Spatiotemporal structure of the priors

10. Applying Bayesian Inference in Vision research Spatiotemporal structure of the priors

11. G(q-0;sp) r(q) = Prior distribution q r(qp|q) C(q;sl,sp,qt) G(qr-q;sl) r(qr|q)= Psychophysical function Real orientation difference Noisy brain Representation (Likelihood) Distribution of perceived orientation difference Applying Bayesian Inference in Vision research Free parameters: width of prior, width of likelihood, co-linearity threshold, fitting error normal sequence — higher expectation of co-linearity — sharper prior distribution Guo et al., (2004) Vision Res. 44: 2349-2358

12. Different spatiotemporal structure Different experimental frequency Applying Bayesian Inference in Vision research

13. Bayesian Applications in Psychology & Neuroscience Learning and reasoning Language processing and acquisition Memory Vision Sensorimotor control Reference: Trends in Cognitive Sciences, 2006, Vol 10(7), “Special issue: Probabilistic models of cognition”.