lecture_03 - Probability distributions for qualitative data...

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    Probability distributions for qualitative data ILRST 411-Lecture 03
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       Maximum Likelihood Estimation The aim of maximum likelihood estimation is to find the  parameter value(s) that makes the observed data most  likely. This is because the likelihood of the parameters given  the data is defined to be equal to the probability of the data  given the parameters. In other words, if the probability of an event X dependent on  model parameters  p  is written  P ( X | p )  then we would talk about the likelihood  L ( p | X ) that is, the likelihood of  the parameters given the  data .  
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         Maximum Likelihood Estimation Normally , we are always interested in knowing the    Probability of a political party winning the election,     Probability of passing an exam,    Probability of getting a head in a single toss and so on. However, in the case of  data analysis , we have already  observed all the data: once they have been observed they  are fixed, there is no 'probabilistic' part to them anymore. We  are much more interested in the likelihood of the model  parameters that underly the fixed data. 
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        Maximum Likelihood Estimation Probability Knowing parameters -> Prediction of outcome Likelihood Observation of data -> Estimation of parameters   An example of MLE(Maximum likelihood estimate):    Lets toss a coin once, and we want to find the parameter values  that make the observed data most likely.Lets assume that p is  certain value (0.5), we might wish to find the  maximum  likelihood estimate  (MLE) of  p , given a specific dataset.      Beyond parameter estimation, the likelihood framework allows  us to make  tests  of parameter values. For example, we might  want to ask whether or not the estimated  p  differs  significantly   from 0.5 or not. 
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    Maximum Likelihood Estimation    With the help of an example below, we will see how the tests  can be performed when we introduce the concept of a  likelihood ratio test. Suppose we toss a coin 100 times, and observe 56 heads  and 44 tails. Instead of assuming that p is 0.5, we want to  find the MLE for p. We want to know whether this value  differs significantly from 0.50. We find the value for  p  that makes the observed data most  likely.
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    Maximum Likelihood Estimation Here given below: n = 100 (total number of tosses)  h = 56 (total number of heads)  Where the observed data is given and we plug into our  binomial probability model.
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