Probability and Likelihood Likelihood needed for many of
Probability and Likelihood
Likelihood needed for many of ADMB’s features • • • Standard deviation Variance-covariance matrix Profile likelihood Bayesian MCMC Random effects See Hilborn and Mangel 1997 for a simple introduction • See Pawitan 2001 for a comprehensive description
Probability distributions • Probability of an event given a probability distribution • Probability distribution defined by its form and the values of its parameters
Use of probability distributions • Gambling, working out what is the best bet in a game of cards
What we desire • The probability of a parameter given the information (data) we have (observed)
Likelihood: compare the probability of the observed data under different values of the parameter The outcome 3 is more probable if the true parameter value is 0. 6
Likelihood: a numerical quantity to express the order of preference of values of the parameter MLE
Normal distribution maximum likelihood (one data point) Likelihood -ln(Likelihood) without constants, σ known
Joint likelihood: Combining multiple data sets • Share the parameter values for each data set • Estimate the parameters while maximizing the combined likelihood (assuming independence) Think: Bernoulli → Binomial But, with the possibility of combining different likelihood functions
Using Likelihoods PARAMETER_SECTION. init_number sigma. PROCEDURE_SECTION pred_y=a+b*x; f=nobs*log(sigma) +0. 5*sum(square((pred_y-y)/sigma));
. pin file #a 4 #b 2 #init_number sigma 1. 5
Standard deviation file (*. std) index name 1 a 2 b 3 sigma value 4. 0782 e+00 1. 9091 e+00 1. 4122 e+00 std dev 7. 0394 e-01 1. 5547 e-01 3. 1577 e-01
Correlation Matrix (*. COR) index name value std dev 1 2 3 1 a 4. 0782 e+000 7. 0394 e-001 1. 0000 2 b 1. 9091 e+000 1. 5547 e-001 -0. 7730 1. 0000 3 sigma 1. 4122 e+000 3. 1577 e-001 -0. 0000 1. 0000
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