Probability and Likelihood

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