Learning the parameters

Learning the model's own numbers

Until now the numbers of the model, such as $q$, $r$, $\mu$, $\varphi$ and $\sigma$, were given. In practice you must learn them from the same data. A particle filter helps in two ways. First, it estimates the likelihood of the data for any choice $\theta$ of the numbers: on each day, the average of the particle weights estimates how probable that day's return was, and the sum of the logs of these averages is the log-likelihood. Run the filter for several choices of $\theta$ on a grid, always with the same random numbers so the comparison is smooth, and keep the best. The estimate is noisy, with a spread of about $3$ on a log-likelihood near $17{,}750$, so only gaps larger than that mean something.

The second way puts the unknown number inside the state: every particle carries its own value of $\theta$. This fails if $\theta$ is never moved, because resampling then copies a few values and the cloud of $\theta$ collapses to one point. The Liu-West fix moves each particle's $\theta$ a little after each resampling, but in a way that keeps the cloud's mean and spread. Choose a discount $\delta$ near $1$ (such as $0.95$). Then $h^{2}=1-\big((3\delta-1)/(2\delta)\big)^{2}$ and $a=\sqrt{1-h^{2}}$. Each value is first pulled toward the cloud mean $\bar\theta$ by the factor $a$, and then jittered by noise with variance $h^{2}V$, where $V$ is the cloud variance. A smaller $\delta$ means more jitter.

• Likelihood by filtering: the sum of the logs of the average weights, with the same random numbers for every choice.
• Parameters in the state: needs a small jitter after resampling, or the parameter cloud collapses.
• Compare models fairly: a log-likelihood needs a penalty for the number of parameters, such as the BIC.

Lab · what's graded

• The log-likelihood gain10 pts
• The Liu-West constant h squared15 pts
• The shrinkage factor a10 pts
• A shrunk particle15 pts
• The jitter size10 pts
• A BIC15 pts
• Parameters without jitter10 pts
• Why compare with a gap of about 315 pts