1. Bayesian Brain Probabilistic Approaches to Neural Coding 1.1 A Probability Primer Bayesian Brain Probabilistic Approaches to Neural Coding 1.1 A Probability Primer Jinsan Yang Biointelligence Lab School of Computer Sci. & Eng. Seoul National University

2. (c) 2000-2008 SNU CSE Biointelligence Lab2 I A Probability Primer 1.1 What is Probability 1.2 Bayes Theorem 1.3 Measuring Information 1.4 Making an Inferance 1.5 Learning from Data 1.6 Graphical Models and Other Bayesian Algorithms

3. (c) 2000-2008 SNU CSE Biointelligence Lab3 1.1 What is Probability Two views of interpretation for probability  Frequentist view  Bayesian view Probability distribution and density  Random variable, vector from sample space  Probability mass function (pmf), probability density function (pdf)  Cumulative distribution function (cdf), distribution function Expectation and Statistics  mean, variance, covariance, correlation

4. (c) 2000-2008 SNU CSE Biointelligence Lab4 1.1 What is Probability Joint, conditional and marginal probability Independence and correlation  Independent rv’s:  Uncorrelated rv’s: If two variables are independent, they are uncorrelated but the reverse is not always true

5. (c) 2000-2008 SNU CSE Biointelligence Lab5 1.2 Bayes Theorem How to update the belief of a hypothesis based on how well the acquired data were predicted from the hypothesis  Prior, posterior probability  Generative model  Marginal data likelihood

6. (c) 2000-2008 SNU CSE Biointelligence Lab6 1.3 Measuring information Smaller P(X) means more informative Entropy  Average information  Measure of randomness Mutual information  Decrease of uncertainty about the world X by observing Y

7. (c) 2000-2008 SNU CSE Biointelligence Lab7 1.3 Measuring information Kullback-Leibler Divergence  Measure the difference in two probability distributions by information  Difference in information between P(X) and Q(X) when X follow the distribution P(X)  Does not satisfy the symmetry condition

8. (c) 2000-2008 SNU CSE Biointelligence Lab8 1.4 Making an inference Maximum likelihood estimate  Finding the state of the world X by maximizing the likelihood P(Y|X) of sensory input y to the brain  Point estimate may not be enough Maximum a posteriori estimate  Combining sensory information and prior probability of world states Bayesian estimate  Full probability distribution instead of a point estimate

9. (c) 2000-2008 SNU CSE Biointelligence Lab9 1.4 Making an inference Bayes filtering  Using the posterior probability as the prior probability in the next step  When the state changes by a state transition probability  Examples : Kalman filter, particle filter

10. (c) 2000-2008 SNU CSE Biointelligence Lab10 1.5 Learning from data Learning sensory transformation or state transition from experience by parameter estimation Fisher information: measure of steepness of the likelihood (How good is the estimation) Bayesian Learning: principled way of adding regularization terms

11. (c) 2000-2008 SNU CSE Biointelligence Lab11 1.5 Learning from data Bayesian Learning: principled way of adding regularization terms Maximizing wrt. w is the same as minimizing the least mean squared error with a penalty term

12. (c) 2000-2008 SNU CSE Biointelligence Lab12 1.5 Learning from data Marginal likelihood is a good criterion to see whether the prior is consistent with the observed data (called evidence) Represent dependency relations by graphical models  Bayesian network: directed acyclic graph (DAG) 1.6 Graphical models and other Bayesian algorithms