1 / 27

Probability Review

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

chloe-foley
Download Presentation

Probability Review

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Probability Review (many slides from Octavia Camps)

  2. 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

  3. 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

  4. 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

  5. Conditional Probability • One of the most useful concepts! W A B

  6. Bayes Theorem • Provides a way to convert a-priori probabilities to a-posteriori probabilities:

  7. Using Partitions: • If events Ai are mutually exclusive and partition W W B

  8. 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

  9. Random Variables Distributions • Cumulative Probability Distribution (CDF): • Probability Density Function (PDF):

  10. Random Distributions: • From the two previous equations:

  11. Uniform Distribution • A R.V. X that is uniformly distributed between x1 and x2 has density function: X1 X2

  12. Gaussian (Normal) Distribution • A R.V. X that is normally distributed has density function: m

  13. Statistical Characterizations • Expectation (Mean Value, First Moment): • Second Moment:

  14. Statistical Characterizations • Variance of X: • Standard Deviation of X:

  15. Mean Estimation from Samples • Given a set of N samples from a distribution, we can estimate the mean of the distribution by:

  16. Variance Estimation from Samples • Given a set of N samples from a distribution, we can estimate the variance of the distribution by:

  17. Image Noise Model • Additive noise: • Most commonly used

  18. Additive Noise Models • Gaussian • Usually, zero-mean, uncorrelated • Uniform

  19. Measuring Noise • Noise Amount: SNR = s/ n • Noise Estimation: • Given a sequence of images I0,I1, … IN-1

  20. 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

  21. Balance between bias and variance Mean squared error as performance criterion

  22. Least Squares (LS) If errors only in b Then LS is unbiased But if errors also in A (explanatory variables)

  23. Errors in Variable Model

  24. 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

  25. 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.

  26. Optical flow • One patch gives a system:

  27. Noise model • additive, identically, independently distributed, symmetric noise:

More Related