Objective: Understand Bayes Rule. Bayesian Perception. Priors and Perception. Structure

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1 Bayesian Perception Objective: Understand Bayes Rule If I hadn t believed it, I would never have seen it Anon. Reverend Thomas Bayes ( ) p(s j I) = p(i S j ) p(s j ) / p(i) or (almost) equivalently posterior = likelihood * prior What has this to do with perception? PSY305 Lecture 7 JV Stone 1 2 Structure Priors and perception Bayes theorem overview Conditional probability: Rain, frogs and flies Example: The Bayesian Doctor Bayesian object recognition Priors and Perception Perceptual inputs are incomplete in at least three respects 1 they do not explicitly have depth information 2 noise is always present 3 occluded parts of object missing from image In order to compensate, perceptual systems make use of assumptions, constraints or priors about the nature of the physical world. These assumptions are not always valid. 3 4

2 The light-from-above prior 1 The light-from-above prior 2 Invert this image If light comes from above then these buttons must be convex. Even though one image is just an inverted version of the other. 5 The light-from-above prior Appears convex if assume light comes from above Appears concave if assume light comes from above 6 The face-convexity prior Inverted version of left hand image. Convex p becomes concave d. 7 8

3 Conflicting Priors: The face-convexity prior but how convex? Convex face appears convex if assume light comes from above Rotating hollow face mask Concave face still appears convex, but only if assume light now comes from below 9 Shading information in identical images on right are consistent with both profiles on left. Our prior experience of faces allows us to correctly interpret how convex the imaged (right) face is. 10 The Rigidity Prior and Motion Assumes object is rigid Assumes object is locally rigid Shadow Priors 1 A moving shadow is cast by a moving object

4 Shadow Priors 2 A moving shadow is cast by a plane moving object. Shadow Priors 3 White trash - not sure what this implies Vision is ill-posed (the solid-object prior?) Bayes Theorem Overview There are two types of information: information you want and information you have. Bayes rule can be considered as a method for obtaining the information you want from the information you have

5 Bayes Theorem Overview Information you have Information you want Response Conditional probability If 75% (0.75) of psychology students are female then the probability of being a female given that you are a psychology student is We can write this as Also known as Bayes rule. p(female psychology student) = Conditional probability If 30% (0.3) of all females are psychology students then the probability of being a psychology student given that you are a female is 0.3. We can write this as Conditional probability If I told you one conditional probability, could you tell me the other one? That is, do you know how to get from here to here p(female psychology student) = 0.75 p(psychology student female) = 0.3? p(psychology student female) = No (nor do I). But if you know Bayes rule then you would have a strategy for obtaining one from the other. But why why would you want to do this 20

6 Conditional probability Because, the visual system (and most systems) can usually obtain one conditional probability but they almost always want the other one. 21 Conditional Probability: The frog s brain 1 The neurophysiologist wants to know the probability that a bug-detector neuron fired given that an image of a fly moved across the frog s retina: p(neuron fired fly present) (vertical bar reads as given that ). The frog wants to know the probability that there is a fly given that a bug-detector neuron fired: p(fly present neuron fired) These are two different conditional probabilities. The frog can get it wrong Fly p(fired fly)=0.8 p(fly fired)=0.9 Fire 22 Priors and the Bayesian Doctor Symptoms I I Likelihood Likelihood p(i S 1 )=1 p(i S 2 )=1 The Bayesian Doctor - Notes for later reading A doctor observes a set of symptoms I and needs to know which disease is most likely to have generated these symptoms. But there are two diseases S1 and S2 which produce identical symptoms: disease S1 implies symptoms I disease S2 implies symptoms I Frequency in population p(s 1 )=0.01 Prior probability S S 1 Disease S 1 Disease S 2 2 Frequency in population p(s 2 )=0.9 Prior probability Thus the conditional probability of the symptoms I given diseases S1 and S2 is the same (almost unity (1.0)) for both S1 and S2: p(i S1) = 1.0 and p(i S2) = 1.0 p(s 1 I) = p(i S 1 )p(s 1 ) = 1.0 x 0.01 = 0.01 Posterior probability p(s 2 I) = p(i S 2 )p(s 2 ) = 1.0 x 0.9 = 0.9 Posterior probability 23 Each of these conditional probabilities is known as a likelihood. 24

7 The Bayesian Doctor - Notes for later reading Because the two likelihoods are the same: Bayesian Object Recognition and Necker Cubes p(i S1) = 1.0 and p(i S2) = 1.0 the symptoms alone cannot be used to decide which disease is present. Fortunately one of these diseases (smallpox) S1 is very rare (0.01), whereas S2 (chicken pox) is common (0.9). Thus, in the absence of any evidence (symptoms), the prior probability of S1 is p(s1)=0.01, and the prior probability of S2 is p(s1)=0.9. The doctor can weight the likelihood of each disease according to its prior probability. This would lead him to diagnose disease S2. p(i S1)p(S1) = 1.0 x 0.01= 0.01 p(i S2)p(S2) = 1.0 x 0.9 = NEXT LECTURE Bayesian Object Recognition In a world containing three objects (S 1, S 2, S 3 ), which object produced the image? Each of N=3 possible objects yields the same image. Thus the probability of observing this image I is the same (e.g. 0.9) for all N=3 possible objects Sj: S 1 S 2 S 3 p(i S j ) = 0.9 for each object S j (j=1 3 implies S 1, S 2, S 3 ) Image plane 27