机器学习题库一、极大似然1、 ML estimation of exponential model (10)A Gaussian distribution is often used to model data on the real line, but is sometimes inappropriate when the data are often close to zero but constrained to be nonnegative. In such cases one can fit an exponential distribution, whose probability density function is given byGiven N observations xi drawn from such a distribution:(a) Write down the likelihood as a function of the scale parameter b.(b) Write down the derivative of the log likelihood.(c) Give a simple expression for the ML estimate for b.2、换成 Poisson 分布：3、二、贝叶斯假设在考试的多选中，考生懂得对的答案的概率为 p，猜测答案的概率为 1-p，并且假设考生懂得对的答案答对题的概率为 1，猜中对的答案的概率为，其中 m 为多选项的数目。那么已知考生答对题目，求他懂得对的答案的概率。1、Conjugate priorsThe readings for this week include discussion of conjugate priors. Given a likelihood for a class models with parameters θ, a conjugate prior is a distribution with hyperparameters γ, such that the posterior distribution与先验的分布族相似(a) Suppose that the likelihood is given by the exponential distribution with rate parameter λ:Show that the gamma distribution _is a conjugate prior for the exponential. Derive the parameter update given observations and the prediction distribution .(b) Show that the beta distribution is a conjugate prior for the geometric distributionwhich describes the number of time a coin is tossed until the first heads appears, when the probability of heads on each toss is θ. Derive the parameter update rule and prediction distribution.(c) Suppose is a conjugate prior for the likelihood ; show that the mixture prioris also conjugate for the same likelihood, assuming the mixture weights wm s...
