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tnich

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Random variables can be generated from probability distributions, but also from physical processes. If you first generate values of random variables from a probability distribution and then find likelihoods of distribution parameters based on those values, then yes, you have created a circular process. It is not circular, though, if you measure some physical process and use likelihood functions to help construct a mathematical model of the process.

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Thanks, that really cleared up all of the confusion.Random variables can be generated from probability distributions, but also from physical processes. If you first generate values of random variables from a probability distribution and then find likelihoods of distribution parameters based on those values, then yes, you have created a circular process. It is not circular, though, if you measure some physical process and use likelihood functions to help construct a mathematical model of the process.

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Stephen Tashi

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It isn't clear what line of reasoning you're thinking about when you say "the logic".So the logic becomes circular. Is there something I'm not seeing?

What is your definition of "plausibility"? The likihood function does not determine the "probability" of the parameters given the observed random variables - if that's what you're thinking. It also does not determine the "liklihood" of the parameters. It's better to think of the liklhood function as giving the liklihood of theSo a Likelihood function determines the plausibility of parameters given the observed random variables.

If we are considering a family of probability distributions and each member of the family is specified by giving specific values to some parameters then the liklihood function gives the "liklihood" of the data as a function of the parameters and the data. The phrase "liklihood of the data" is used instead of "probability of the data" because it is incorrect to say that evaluating a probability density function produces a probability. Evaluating a probability density function, in the case of a continuous distribution, gives a "probability density". not a "probability". For example, the probability density of a random variable U that is uniformly distributed on [0,1] is the constant function f(x) = 1. The fact that f(1/3) = 1 does imply that the probability that the value 1/3 occurs is 1. "Liklihood of" is a way to say "probability density of".

One procedure for estimating parameters from given values of data is to use the values of the parameters that maximize the value of the liklihood function. It should be emphasized that (like many things in statistics - e.g. hypothesis testing) this is a

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FactChecker

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The maximum likelihood estimator allows you to determine the model parameter that makes the given data most likely. There is no formal "logic" that will say that the parameter is correct. So you are wise to be cautious. If you also obtain a confidence interval for the parameter, you can see how unusual it would be to get that data if the true parameter was, in fact, outside of that interval. Even then, you would have no hard logic to say whether it is in or out of the confidence interval -- only hypothetical probabilities.

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