# Exercise 3

Exercise 3: Interior local minima.

To get the maximum likelihood estimates, we find the global maximum in that surface. But as we've seen, that global

minima is at process error equal the zero (essentially) even though we can be sure that process error is not equal to

zero. The problem is that the likelihood surface has multiple peaks, and the biologically realistic one (both process and non-process error zero is getting hidden by the peak with one of them zero). Sometimes (but not always) the local minima with both process and non-process non-zero is evident.

Type 'Lab2' at the matlab prompt.

When asked for data code, type 5 (Sharp-tailed grouse).

Questions for Exercise 3:

1. What is the ML estimate of process error, conditioned on process error being non-zero? I'll refer to this as s2p_new.

2. To get the ML estimate of non-process error, we would then hold process error at s2p_new while maximizing over the other parameters. However, we can get a rough estimate using: total variance = s2p + 2*s2np (process error + twice non-process error). For the first fit, s2p_old = 0, so we can calculate total variance as 2*s2np_old (the s2np_old estimate is in the title of the bottom panel). s2np_new = total variance - s2p_new

3. Write down {mu, s2p_old, s2np_old} and {mu, s2p_new, s2np_new}. At the matlab prompt, type Lab1, then type in data code 5, then at the prompt type in these two different parameter sets. Now you can see how these different estimates affect the ML estimate of the true population size.

To get the maximum likelihood estimates, we find the global maximum in that surface. But as we've seen, that global

minima is at process error equal the zero (essentially) even though we can be sure that process error is not equal to

zero. The problem is that the likelihood surface has multiple peaks, and the biologically realistic one (both process and non-process error zero is getting hidden by the peak with one of them zero). Sometimes (but not always) the local minima with both process and non-process non-zero is evident.

Type 'Lab2' at the matlab prompt.

When asked for data code, type 5 (Sharp-tailed grouse).

Questions for Exercise 3:

1. What is the ML estimate of process error, conditioned on process error being non-zero? I'll refer to this as s2p_new.

2. To get the ML estimate of non-process error, we would then hold process error at s2p_new while maximizing over the other parameters. However, we can get a rough estimate using: total variance = s2p + 2*s2np (process error + twice non-process error). For the first fit, s2p_old = 0, so we can calculate total variance as 2*s2np_old (the s2np_old estimate is in the title of the bottom panel). s2np_new = total variance - s2p_new

3. Write down {mu, s2p_old, s2np_old} and {mu, s2p_new, s2np_new}. At the matlab prompt, type Lab1, then type in data code 5, then at the prompt type in these two different parameter sets. Now you can see how these different estimates affect the ML estimate of the true population size.

## No comments