Note09.md

Notes on Chapter 5

Loss Functions

squared-error loss

asymmetric squared-error loss function

This represents that estimating a value larger than the true estimate is preferable to estimating a value below.

absolute-loss

zero-one loss

log-loss

other loss functions:

This function emphasizes an estimate closer to 0 or 1 since if the true value θ is near 0 or 1, the loss will be very large unless θ^ is similarly close to 0 or 1.

This function is bounded between 0 and 1 and reflects that the user is indifferent to sufficiently-far-away estimates. It is similar to the zero-one loss above, but not quite as penalizing to estimates that are close to the true parameter.

The risk of estimate θ^

• If using the mean-squared loss, the Bayes action is the mean the posterior distribution
• Whereas the median of the posterior distribution minimizes the expected absolute-loss.
• In fact, it is possible to show that the MAP estimate is the solution to using a loss function that shrinks to the zero-one loss.

For a specific trading signal, call it $x$, the distribution of possible returns has the form:

$$R_i(x) = \alpha_i + \beta_ix + \epsilon $$

where $\epsilon \sim \text{Normal}(0, 1/\tau_i) $ and $i$ indexes our posterior samples. We wish to find the solution to

$$ \arg \min_{r} ;;E_{R(x)}\left[ ; L(R(x), r) ; \right] $$

according to the loss given above. This $r$ is our Bayes action for trading signal $x$. Below we plot the Bayes action over different trading signals.