Slide 1Frequentist approachBayesian inferenceBayesian inferencePROBABILITY AND STATISTICS IN COMPUTER SCIENCE AND SOFTWARE ENGINEERING Chapter 10: Statistical Inference – part 21FREQUENTIST APPROACHUp to now, we have considered the random sample we collected from the population to be the only source of uncertainty in our problemThis is the frequentist approach – it assumes that the population parameters are fixed, and the statistical procedures were then based on distribution of the data given these parameters, Using this approach, we can only talk about probabilities in the long runThe estimator is unbiased in the long run (i.e., over many samples)If we construct confidence intervals using many samples, in the long run of them will include the parameter we are estimatingThe Bayesian Approach is different – it assumes uncertainty of the population parameter too•92BAYESIAN INFERENCEThe Bayesian approach starts with the observation that usually we have some a priori knowledge of the population parameter we are interested inFor example, if we are collecting a sample to determine the expected age of a person in a town, we can rule out negative numbers and really large numbersWhat we are really doing here is assuming a prior distribution for the parameter we are seeking, We can then use the data we observe in the our sample to see how this changes our assumptionWe will use the Bayes formula we saw earlier to estimate the distribution for , given the observed sample … this is the posterior distribution•93BAYESIAN INFERENCEIn certain situations, the prior distribution and the posterior distribution will belong to the same family of distributionsIn this case, we can use the information we gathered from the data we sampled to modify our prior distribution assumptions, i.e. the original estimates for the population parameter(s)First, recall Bayes formula:The denominator term was often computed using the Law of Total Probability
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