1. Estimation of Random Variables Two types of estimation: 1) Estimating parameters/statistics of a random variable (or several) from data. 2)Estimating the value of an inaccessible random variable X based on observations of another random variable,Y. e.g. Estimate the future price of a stock based on its present (and past) price.

2. Two conditional estimators : 1) Maximum A Posteriori Probability (MAP) Estimator: So we need to know the probabilities on the right hand side to do this estimate, especially P(X=x) which may not be available ( Remember, X is hard to observe) If X and Y are jointly continuous :

3. 2) Maximum Likelihood (ML) Estimator: i.e. Find the likeliest X value based on the observation. This is useful when P( Y = y | X ) is available, i.e. The likelihood of observing a Y value given the value of X is known. e.g. Probability of receiving a 0 on a communication channel given that a 0 or 1 was sent. If X and Y are jointly continuous :

4. Example 6.26 : X, Y are jointly Gaussian

6. Minimum Mean Square Estimator (MMSE) Estimate X given observations of Y

7. i.e. The best constant MMSE of X, is its mean. The estimation error in this case is Case 2: Linear Estimator g(Y) = aY + b

9. Error

10. high variance X is harder to estimate Best linear estimator is the constant estimator in this case.

11. Heuristic explanation of linear MMSE (standardized version of Y )

12. Case 3: Nonlinear estimator g(Y) Using conditional expectation:

13. is called the regression curve. The error achieved by the regression curve is

14. which is same as the linear MMSE.  for jointly Gaussian X, Y, linear MMSE is optimal and the same as MAP estimator. For jointly Gaussian X, Y Thus, the optimal nonlinear MMSE is