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Perception as Bayesian Inference Many Bayesian inference problems involve the product of two or more Gaussians. A convenient property of Gaussians is that their product is also Gaussian. In this problem, we will lead you through an example to derive this property yourself. Consider an observer who infers a stimulus S from a measurement x. Suppose that the measurement distribution p(x|s) is a Gaussian distribution with standard deviation Ïƒ and the prior distribution is a Gaussian with mean Î¼prior and standard deviation Ïƒprior. (Hint: i. Multiply the Gaussian distributions for prior and likelihood; ii. Expand; iii. Rewrite or factorize these expressions by completing the square; iv. Express Upost, Gpost in terms of x, Ïƒ, Uprior, Ïƒprior.) ii. Recall that p(s|x) is the posterior distribution; i.e., its integral should be equal to 1. Write down the normalization constant. iii. What is the variance in the special case that Ïƒprior = 0? iv. Multisensory integration: Let's assume the observer is given auditory and visual spatial signals xA and xV, each sampled from conditionally independent Gaussian distributions p(xV|s) and p(xA|s). We now assume a so-called flat prior. Write down the posterior distribution: p(s|x, VxA).

          Perception as Bayesian Inference

Many Bayesian inference problems involve the product of two or more Gaussians. A convenient property of Gaussians is that their product is also Gaussian. In this problem, we will lead you through an example to derive this property yourself.

Consider an observer who infers a stimulus S from a measurement x. Suppose that the measurement distribution p(x|s) is a Gaussian distribution with standard deviation Ïƒ and the prior distribution is a Gaussian with mean Î¼prior and standard deviation Ïƒprior. (Hint: i. Multiply the Gaussian distributions for prior and likelihood; ii. Expand; iii. Rewrite or factorize these expressions by completing the square; iv. Express Upost, Gpost in terms of x, Ïƒ, Uprior, Ïƒprior.)

ii. Recall that p(s|x) is the posterior distribution; i.e., its integral should be equal to 1. Write down the normalization constant.

iii. What is the variance in the special case that Ïƒprior = 0?

iv. Multisensory integration: Let's assume the observer is given auditory and visual spatial signals xA and xV, each sampled from conditionally independent Gaussian distributions p(xV|s) and p(xA|s). We now assume a so-called flat prior. Write down the posterior distribution: p(s|x, VxA).

perception as bayesian inference many bayesian inference problems involve product of two or more gaussians convenient property of gaussians is that their product is also gaussian in this pro 60963

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