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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.

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  • $\begingroup$ Please explain what "numerical issues" are arising and (if possible) tell us why they happen. For instance, would it be because $\text{bf}_1$ is very large, very close to $0,$ very close to $1,$ or something else? $\endgroup$ – whuber Jan 9 at 16:28
  • $\begingroup$ Thanks. I have added a clarification to the question. $\endgroup$ – user3302113 Jan 9 at 16:41
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    $\begingroup$ Have a look at how other people have defined functions such as log1p_exp, that should help. $\endgroup$ – Björn Jan 9 at 16:44
  • $\begingroup$ Thank you, I'm currently using numerical work arounds like that. The aim of my question was check if there really aren't any mathematical tricks I could use instead. $\endgroup$ – user3302113 Jan 9 at 17:59
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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 $$x = s \exp(\eta) + 1.$$

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