I always saw the algorithm transform its objective function by log. I wonder this is because logarithm function is monotonically increased (logx) to do maximization, or decreased (−logx) to do minimization? So for MM algorithm, any functions should be transformed into log?Are there also other reasons that log transformation is often used? I wonder log is always work or we have some mathematical operations that can replace log for this purpose?

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persstemc1 · Accepted Answer

The logarithm is used for the &quot;logarithmic derivative&quot; trick, which turns products into sums. This is useful because the functions to optimize are often products, often with exponentials, and detection of the extrema requires differentiation.  (  log  ⁡  a  b  c      )    ′    =            (      a      b      c              )        ′                    a      b      c        =            a      ′        a    +            b      ′        b    +            c      ′        c  is more manageable than       a    ′    b  c  +  a      b    ′    c  +  a  b      c    ′   when the factors are complex.

Crystal Wheeler · Accepted Answer

I think another reason is that by employing products, the floating-point numbers are a bit out of control for precise calculations; small numbers get smaller and underflow may occur.

I always saw the algorithm transform its objective function by log. I wonder this is because logarit

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