testing by betting—simple vs composite

hypothesis-testingsequential-statisticsplug-in-methodmixture-method

In game-theoretic hypothesis testing, suppose we are testing a simple null vs a composite alternative. We receive samples $(X_t)$ are testing

$$H_0:(X_t)\sim P,H_1:(X_t)\sim Q\in {\Theta }_1.$$

In the simple vs simple case, testing by betting—simple vs simple, the optimal payoff function turned out to simply be the likelihood-ratio of $Q$ and $P$. But now there is no single $Q$, the likelihood-ratio isn’t well-defined. What do we do?

Since the observations are revealed sequentially (we can simulate this in a batch setting), it’s natural to try and “learn” some appropriate $Q\in {\Theta }_1$ that has good power (you could look at this as trying to learn the true distribution $Q^{\ast }\in {\Theta }_1$ using the data thus far).

There are two general methods.

Plug-in method

Since the methods are inherently sequential and the payoff function $S_t$ (see game-theoretic hypothesis testing:Payoff functions) need only be ${\mathcal{F}}_{t-1}$-measurable, we can employ the likelihood ratio test but change which $Q\in {\Theta }_1$ is used every time. That is, if $(X_t)\sim Q^{\ast }$, we can try and learn $Q^{\ast }$ over time.

In particular, we can consider a payoff function $S_t=q_t(X_t|{\mathcal{F}}_{t-1})/p(X_t|{\mathcal{F}}_{t-1})$. $Q_t$ can be chosen baed on $X_1,\dots ,X_{t-1}$. Regardless of how it’s chosen $S_t$ will remain a test-martingale for $P$. (We are assuming densities are well-defined, otherwise we resort to Radon-Nikodym derivatives.)

While it may be tempting to use MLE to choose $Q_t$, this is advised against when the data are discrete, otherwise it may assign 0 probability to an outcome that may in fact occur, and we will go broke.

The mixture method

Instead of choosing a particular $Q\in {\Theta }_1$ to use at each timestep, we mix over all such distributions by placing a distribution over ${\Theta }_1$. That is, we take

$$S_t(X_t)={\int }_{q\in {\Theta }_1}q(X_t|{\mathcal{F}}_{t-1}){\rho }_t(dq)/p(X_t|{\mathcal{F}}_{t-1}),$$

where ${\rho }_t$ is the mixing distribution, and it may depend on ${\mathcal{F}}_{t-1}$. Since averages of distributions remain distributions, the numerator remains a distribution and $S_t$ remains a test-martingale. But note there is no guarantee that the numerator remains a distribution in ${\Theta }_1$ (unless ${\Theta }_1$ happens to be fork-convex) .