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Probability Review. (many slides from Octavia Camps). Intuitive Development. Intuitively, the probability of an event a could be defined as:. Where N(a) is the number that event a happens in n trials. More Formal:. W is the Sample Space: Contains all possible outcomes of an experiment

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## Probability Review

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**Probability Review**(many slides from Octavia Camps)**Intuitive Development**• Intuitively, the probability of an event a could be defined as: Where N(a) is the number that event a happens in n trials**More Formal:**• W is the Sample Space: • Contains all possible outcomes of an experiment • w2W is a single outcome • A 2W is a set of outcomes of interest**Independence**• The probability of independent events A, B and C is given by: P(ABC) = P(A)P(B)P(C) A and B are independent, if knowing that A has happened does not say anything about B happening**Conditional Probability**• One of the most useful concepts! W A B**Bayes Theorem**• Provides a way to convert a-priori probabilities to a-posteriori probabilities:**Using Partitions:**• If events Ai are mutually exclusive and partition W W B**Random Variables**• A (scalar) random variable X is a function that maps the outcome of a random event into real scalar values W X(w) w**Random Variables Distributions**• Cumulative Probability Distribution (CDF): • Probability Density Function (PDF):**Random Distributions:**• From the two previous equations:**Uniform Distribution**• A R.V. X that is uniformly distributed between x1 and x2 has density function: X1 X2**Gaussian (Normal) Distribution**• A R.V. X that is normally distributed has density function: m**Statistical Characterizations**• Expectation (Mean Value, First Moment): • Second Moment:**Statistical Characterizations**• Variance of X: • Standard Deviation of X:**Mean Estimation from Samples**• Given a set of N samples from a distribution, we can estimate the mean of the distribution by:**Variance Estimation from Samples**• Given a set of N samples from a distribution, we can estimate the variance of the distribution by:**Image Noise Model**• Additive noise: • Most commonly used**Additive Noise Models**• Gaussian • Usually, zero-mean, uncorrelated • Uniform**Measuring Noise**• Noise Amount: SNR = s/ n • Noise Estimation: • Given a sequence of images I0,I1, … IN-1**Good estimators**Data values z are random variables A parameter q describes the distribution We have an estimator j (z) of the unknown parameter q. If E(j (z) - q ) = 0or E(j (z) ) = E(q)the estimator j (z) is unbiased**Balance between bias and variance**Mean squared error as performance criterion**Least Squares (LS)**If errors only in b Then LS is unbiased But if errors also in A (explanatory variables)**Least Squares (LS)**bias Larger variance in dA,,ill-conditioned A, u oriented close to the eigenvector of the smallest eigenvalue increase the bias Generally underestimation**Estimation of optical flow**(a) (b) • Local information determines the component of flow perpendicular to edges • The optical flow as best intersection of the flow constraints is biased.**Optical flow**• One patch gives a system:**Noise model**• additive, identically, independently distributed, symmetric noise:

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