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What is the conversion of the following equation into log space?
$bf2 = 1 + (p * (bf1 - 1))$
Given log.bf1 (log Bayes factor), how do I get to log.bf2 without having to compute bf1, but instead using log.bf1?
Context: Numerical issues are arising because bf1 is often very large or very small. Hence, the computations are performed in log space to avoid numerical over/underflow.
To simplify the notation, write $x$ for $\text{bf}_1$ and $x^\prime$ for $\text{bf}_2.$ By subtracting $1$ from each side, write their relation as
$$y^\prime = x^\prime - 1 = p(x - 1) = py.$$
Let $s = \operatorname{signum}(y)$ be the sign of $y$ (equal to $1$ when $y\gt 0,$ $-1$ when $y \lt 0,$ and $0$ when $y=0$). Multiplying by $s$ to make both sides nonnegative (assuming $p\ge 0$) allows us to take logarithms, giving
$$\log(s y^\prime) = \log(s y) + \log(p).$$
Thus,
By representing the information in terms of $s = \operatorname{signum}(x-1)$ together with $\eta = \log|x - 1|,$ the update step merely adds $\log(p)$ to $\eta.$ After a sequence of updates you can recover the final value of $x$ from the final value of $\eta$ as $$x = s \exp(\eta) + 1.$$
