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Probability and Likelihood in Statistical Modeling

Explore the importance of likelihood in ADMB features like standard deviation, Bayesian MCMC, random effects, and more. Learn about probability distributions, likelihood comparisons, MLE, and joint likelihoods in statistical analysis.

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Probability and Likelihood in Statistical Modeling

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  1. Probability and Likelihood

  2. 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

  3. Probability distributions • Probability of an event given a probability distribution • Probability distribution defined by its form and the values of its parameters

  4. Use of probability distributions • Gambling, working out what is the best bet in a game of cards

  5. What we desire • The probability of a parameter given the information (data) we have (observed)

  6. 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

  7. Likelihood: a numerical quantity to express the order of preference of values of the parameter MLE

  8. Normal distribution maximum likelihood (one data point) Likelihood -ln(Likelihood) -ln(Likelihood) without constants -ln(Likelihood) without constants, σ known

  9. 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

  10. 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));

  11. .pin file #a 4 #b 2 #init_number sigma 1.5

  12. Standard deviation file (*.std) index name value std dev 1 a 4.0782e+00 7.0394e-01 2 b 1.9091e+00 1.5547e-01 3 sigma 1.4122e+00 3.1577e-01

  13. Correlation Matrix (*.COR) index name value std dev 1 2 3 1 a 4.0782e+000 7.0394e-001 1.0000 2 b 1.9091e+000 1.5547e-001 -0.7730 1.0000 3 sigma 1.4122e+000 3.1577e-001 -0.0000 -0.0000 1.0000

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