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The maximum likelihood method

Plausible observations and plausible models. The maximum likelihood method. Likelihood = probability that an observation is predicted by the specified model. MLE.

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The maximum likelihood method

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  1. Plausible observations and plausible models The maximum likelihood method Likelihood = probability that an observation is predicted by the specified model

  2. MLE • Observations are ‘outcomes of random experiments’: the outcome is represented by a random variable (e.g. Y). A representation of Y is yi (I = 1, 2, …. m) • The distribution of possible outcomes is given by probability distribution. • The same data (observations) can be generated by different models and the different observations may be generated by the same model.  what is the range of plausible observations, given the model, and what are the different models that could plausibly have generated the data? • Plausible observations and plausible models • A probability model predicts an outcome and associates a probability with each outcome.

  3. What is a plausible model? A model that predicts observations with a probability that exceeds a given minimum. What is the most plausible model? A model that most likely predicts observations, i.e. that predicts the observations with the largest probability most likely model, given the data.

  4. Observation from a normal distribution N(,2) Probability that an observation is predicted by N(,2): probability that 120 is predicted by N(100,100): Probability that 120 is predicted by N(120,100):

  5. Log-likelihood 0.3989

  6. Range of plausible modelsLikelihood ratio Ratio of likelihood of any model to likelihood of ‘best’ model Log-likelihood ratio ln  = - ½ z2 z2 = -2ln  With the specified model and the ‘best’model

  7. A plausible value of  is one for which the likelihood ratio exceeds a critical value (less negative), e.g. -1.9208, which corresponds to a 95% confidence interval, or -1.353 which corresponds to a 90% confidence interval. Values of  for which ln  > -1.9208 is the support range for . When  is outside the support range, we reject the claim that  does not differ significantly from b . We accept a risk of 5% of wrongly rejecting the claim (Type I error).

  8. To get support range, find * for which ln  = -1.9208 (given that ‘best’ value of  is 125 and 2 is fixed): Solution: The observation could come from ANY model in the support range. All models in the ‘support range’ are supported by the data.

  9. Observation from a binomial distribution with parameter p and index m Likelihood function: Log-likelihood function:

  10. Data: leaving parental home

  11. Analysis of young adults who left home leave out censored cases (conditional analysis) 20 obervations

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