A Jump Back to Population Estimation
10th April 2016
The next project the MRes students in STOR-i are onto is researching and presenting aspects of one of the Masterclasses that
have occurred this term (excluding the one on simulation that I spoke about in Optimisation
and Error in Simul?ation). Emily Graham and I are looking into the Masterclass by
. In that post, I discussed
the basic framework of the problem. In this post, I wish to talk about some of this in more detail, particularly the
computational side.
As King used a . For MCMC, initial values are given to each of the
parameters, then the values are updated by simulating from either conditional distributions (if they are in a nice form) for from
distributions that approximate the posterior and accepting them with a certain probability. . Due to the property of detailed balance
(see later) the Markov chain tends to a stationary distribution equal to the posterior. Therefore, provided it has enough time to run
and the early stages are removed, an MCMC sample will be a good approximate sample from the posterior.
There is a problem that occurs in statistics which is involved in how to pick which model, $m$, to use. In the application to population
estimation, the different models correspond to different choices of main effect and interaction log-linear terms to include. The
different models gave quite a wide range of estimates for the size of the population, so which one is correct? Really, we don’t know
but just choosing one model and quoting its results will not give use the fact that we are uncertain about our choice. One way to
include this uncertainty is model averaging.
For model averaging, the model itself is chosen to be a discrete parameter, $m\in\mathcal{M}$, and is added to the posterior. The
parameters used for a particular model are $\theta_m$. This means that the posterior has form
$$\pi(n_0,\theta,m|\mathbf{x})\propto f(\mathbf{x}| n_0,\theta_m,m) \pi(n_0|\theta_m,m)\pi(\theta_m|m)\pi(m).$$
$f(\mathbf{x}| n_0,\theta_m,m) $ is the likelihood of the data, and the other terms are prior distributions that incorporate current
beliefs about the other parameters. This makes MCMC really hard. You can’t just add in parameters easily between steps in a chain!
Fortunately, a clever way of doing this has been suggested by ,
R. King and S. P. Brooks, Biometrika. Vol. 88, No. 2, pp. 317-336 (2001)
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