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Q1 Maximum Likelihood Basic 25 Points Q1.1 5 Points A coin is tossed 100 times and lands heads 82 times. Choose every correct option in the following. (select all applied, no partial credits for more or less) Q1.2 8 Points In the setting given at Q1.1, what is the probability of head that makes your overall observation most-likely? (autograded short answer: only the final result number, round to 2 decimal places,like 0.55) 0.82 This is a fair coin. Probability of the overall observation is . 0.5 82 Maximum possible likelihood of the overall observation is . 100 82 Minimum possible likelihood of the overall observation is . 100 18 None of the above

Q1.3 7 Points Assume that the coin used for Q1.1 turns out to be completely normal, which is supposed to generate almost equal numbers of heads and tails if randomly tossing many times. Choose every possible concern of using Maximum Likelihood Estimation. (select all applied, no partial credits for more or less) Q1.4 5 Points Maximum Likelihood Estimation gives us a distribution over the parameters as well as the best parameter itself. In other words, MLE provides not only the best parameter but also other parameters with the associated uncertainty of being "non-best". Does not fully use our observation. Does not incorporate our prior knowledge. Does fit tightly to the given observation. Does fit loosely to the given observation. None of the above. θ ^ θ p ( θ ) True False

Q2 Maximum Likelihood Estimation 30 Points Consider the following density function , ; , . This is a legal probability density (parametrized by ) because one can verify that the integral over is equal to 1. If necessary, mean and variance of this distributions can be verifed as and , respectively. f ( x ∣ θ ) = θ xe 2 − θx x ≥ 0 f ( x ∣ θ ) = 0 x < 0 f θ ∈ R x ∈ [−∞, ∞] 2/ θ 2/ θ 2

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Q2.1 10 Points Derive a likelihood of a dataset that consists of independent samples from this distribution. (Hint: The likelihood must be a function of the parameters ) Q2_1.pdf  Download 1 1 / 1 D = { x , x , ..., x } (1) (2) ( m ) m L ( θ ; D ) θ 

Q2.2 10 Points Derive the log-likelihood function of the dataset from Q2.1. (Hint: The log-likelihood ) Q2_2.pdf  Download 1 1 / 1 D l ( θ ; D ) = log L ( θ ; D ) 

Q2.3 10 Points Find the Maximum Likelihood Estimator . (Hint: Set the derivative equal to zero, and then solve the formula) Q2_3.pdf  Download 1 1 / 1 θ ^ 

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Q3 Logistic Regression 45 Points Q3.1 7 Points Suppose that you have trained a logistic regression classifier, and it outputs a prediction on a new example . Choose every correct interpretation in the following. (select all applied, no partial credits for more or less) Q3.2 7 Points Which of the following ways can we train a logistic regression model? (select all correct, no partial credits for more or less) h ( x ) = θ 0.2 x Our estimate for is 0.2. P ( y = 0∣ x ; θ ) Our estimate for is 0.2. P ( y = 1∣ x ; θ ) Our estimate for is 0.8. P ( y = 0∣ x ; θ ) Our estimate for is 0.8. P ( y = 1∣ x ; θ ) Minimize least-square error Maximize likelihood Solve normal equation Minimimize negative log-likelihood

Q3.3 5 Points In logistic regression, what do we estimate for one each unit’s change in ? Q3.4 6 Points Choose every option that correctly describes properties of the logistic function. (select all applied, no partial credits for more or less) X The change in multiplied with . Y Y The change in from its mean. Y How much changes. Y How much the natural logarithm of the odds (i.e., ) changes. log p ( y =0) p ( y =1) It maps a real-valued confidence value into a probability value. It is essentially an identity function between . [0, 1] It always maps zero confidence exactly into the probability 0.5. It is a continuous function that is differentiable everywhere. Its derivative can be easily evaluated with itself.

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