8 Basics of Hypothesis Testing

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1 8 Basics of Hypothesis Testing 4 Problems Problem : The stochastic signal S is either 0 or E with equal probability, for a known value E > 0. Consider an observation X = x of the stochastic variable X = S + where N(0, σ ) is independent Gaussian noise with zero mean and a known variance σ. a) Formulate a hypothesis test where H 0 assumes that s = 0 and H assumes that s = E. Derive the conditional probability density functions associated with each hypothesis. b) Determine the ML decision rule. hat is the decision boundary? c) Compute the probability of error, P e, when using the ML decision rule. d) Compare P e with the probability of error in the motivational example. For which value of E is P e the same in these two problems? hat is the geometric intuition behind this value? Problem : The distribution of the measurement noise impacts the accuracy of a hypothesis test. Among non-gaussian noise, the Laplace distribution is popular: Laplace(µ, σ/) has mean value µ, variance σ, and the probability density function f(x) = σ e σ x µ. Suppose that the stochastic signal S is either 0 or E with equal probability, for a known value E > 0. Consider an observation X = x of the stochastic variable X = S + where Laplace(0, σ /) is independent Laplace distributed noise with zero mean and a known variance σ. a) Formulate a hypothesis test where H 0 assumes that s = 0 and H assumes that s = E. Derive the conditional probability density functions associated with each hypothesis. b) Determine the ML decision rule. hat is the decision boundary? c) Compute the probability of error, P e, when using the ML decision rule. d) Compare P e with the probability of error in Problem for the same value of E/σ. hich type of noise is most severe and in which cases?

2 4. Problems 9 Problem 3: Consider a stochastic variable X with unknown statistics. There are two possible options for the statistics: H 0 : X is uniformly distributed between and +. H : X is exponentially distributed with variance. Based on an observation X = x, one can determine which of these distributions that best fits the observation. a) Find and sketch the ML decision regions. b) Find and sketch the MAP decision regions if PrH 0 } = /3 and PrH } = /3. c) For which values of x one can be completely sure to make the correct decision?

3 0 Basics of Hypothesis Testing 4. Hints Problem a) The conditional probability density functions can be computed in the same way as in the motivational example. b) Use the ML detection rule to compare the conditional probability density functions of the different hypotheses. c) Errors occur when we pick the wrong hypothesis. hat is the probability that this happens? d) here is the ML decision boundary located as compared to the two signal points? Problem a) Notice that the conditional distribution of X is a Laplace distribution with the realization of S as mean value. b) Use the ML detection rule to compare the conditional probability density functions of the different hypotheses. c) Errors occur when we pick the wrong hypothesis. hat is the probability that this happens? d) The error probabilities can be compared by testing different values; for example, E/σ = and E/σ = 8. Is one of the noise distributions always better? Problem 3 / x, f X H0 (x) = 0 x >, and hypothesis H has the conditional probability density function f X H (x) = e x u(x). b) Notice that the MAP decision rule can be expressed as a weighted version of the conditional probability density functions. c) e can only be completely sure when the probability of selecting the wrong hypothesis is zero.

4 4. Answer 4. Answer Problem f X S (x 0) = e x πσ σ. Hypothesis H has the conditional probability density function f X S (x E) = σ. e (x E) πσ b) The ML decision rule is Ĥ ML (x) = H 0 x < E, H x E, where the point x = E can be mapped to any of the hypotheses. The decision boundary is x = E. ( E c) The error probability is P e = Q. σ ) d) The error probabilities are the same at E =. The distance between the two signal points is the same in this case. Problem f X S (x 0) = σ e σ x. Hypothesis H has the conditional probability density function f X S (x E) = σ e σ x E. b) The ML decision rule is Ĥ ML (x) = H 0 x < E, H x E,

5 Basics of Hypothesis Testing where the point x = E is x = E. can be mapped to any of the hypotheses. The decision boundary c) The error probability is P e = e E σ. d) Laplacian noise has the smallest error probability when E/σ is small, while Gaussian noise has the smallest error probability when E/σ is large. Problem 3 a) The ML decision rule is H 0 x < 0, ln() < x, Ĥ ML (x) = H 0 x ln(), x >, none x <. b) The MAP decision rule is H 0 x, Ĥ MAP (x) = H x >, none x <. c) If we observe x between and 0, we can be completely sure that H 0 is true. If we observe x greater than, we can be completely sure that H is true.