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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what is the range of plausible observations, given the model, and what are the different models that could plausibly have generated the data?
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.
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):
Ratio of likelihood of any model to likelihood of ‘best’ model
Log-likelihood ratio ln = - ½ z2
z2 = -2ln
the specified model and
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):
The observation could come from ANY model in the support range. All models in the ‘support range’ are supported by the data.
leave out censored cases (conditional analysis)