The maximum likelihood method
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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

Plausible observations

and

plausible models

The maximum likelihood method

Likelihood = probability that an observation is predicted by the specified model


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.


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.


Observation from a normal distribution n 2
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):



Range of plausible models likelihood ratio
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


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).


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.


Observation from a binomial distribution with parameter p and index m
Observation from a binomial distribution with parameter p and index m

Likelihood function:

Log-likelihood function:



Analysis of young adults who left home and index m

leave out censored cases (conditional analysis)

20 obervations


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