# Nested hypothesis vs conditional statement, (likelihood ratio tests)

I am performing likelihood ratio tests (LRTs).

The data set is not critical its the purity of the statistics that are important, but this is (phylo)genetics data.

I am examining (tree) hypothesis '1' (for example virus 1 and virus 2 share a common ancestor, $$H_{1}$$) and tree hypothesis '2' (virus 1 and virus 3 share a common ancestor, $$H_{2}$$ ) using the same data set. My 'global' hypothesis is that there is only one solution H1 or H2 for the data set, so H1 is not equal to $$H_{2}$$.

Two independent likelihood tests are performed comparing each hypothesis against the unconstrained solution $$H_{0}$$ (maximum likelihood). Each LRT comprises a null and alternative hypothesis, giving two separate results at 0.05% threshold. $$H_{1}$$ and $$H_{2}$$ are different constrained models (phylogenetic trees).

The 'global' hypothesis is true, if,:

$$H_{1}$$ is rejected and $$H_{2}$$ is accepted

OR

$$H_{2}$$ is rejected and $$H_{1}$$ is accepted

If both $$H_{1}$$ and $$H_{2}$$ are rejected the 'global hypothesis' is unresolved, if both are accepted the result is not significant, i.e. either not enough data or the difference is spurious.

Questions

1. Is this a conditional statement or a nested hypothesis?
2. How do I write the statement in notation form?

Could I write the above statement as,

$$P(H_{1} | H_{2}) < 0.01$$ or $$P(H_{2} | H_{1}) < 0.01$$

Is there a more succinct/alternative method?