absolute functions

[Sebastian Weber]

Hi!

I understand that discontinuous functions are inadvertently bad news for HMC. But I wonder if the absolute value functions could be replaced with something which removes the sign, but remains fully continuous (or I am missing here something):

abs(a) := sqrt( a^2 )

If possible, a function log_diff_exp_abs would be great to have, i.e.

  real log_diff_exp_abs(real la, real lb) {

    return(0.5 * log_diff_exp(log_sum_exp(2*la, 2*lb), log(2) + la + lb));

  }

which uses

sqrt( (a - b)^2 ) = sqrt( a^2 - 2 * a * b + b^2 )

Anything wrong here?

Best,

Sebastian

[Bob Carpenter]

abs(a) is continuous, just not continously differentiable.
Same for sqrt(a^2).

You can get close with a "soft" absolute value that is
equal to abs(a) except for a small neighborhood around 0
where it's smoothed out.

But that doesn't help with the other issue that abs(a) creates,
namely that it introduces a reflection-based multimodality
if a is a parameter.

What's the purpose of that log_diff_exp_abs function you're
proposing?

- Bob

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[Michael Betancourt]

Just use x * coth(alpha * x) for some large alpha.
For details, http://arxiv.org/pdf/1212.4693.pdf.

[Sebastian Weber]

Hi!

Yes, I just calculated the diff myself and came to the same conclusion. I will look at the coth stuff, thanks.

The purpose of the log_diff_exp_abs function is to be able to calculate what is called the "Bateman" function, i.e. in PK systems you often have the following form:

1/(a-b) * ( exp(-a*t) - exp(-b*t))

If a > b, then everything is straightforward. If a < b the result is still always positive, but using log_diff_exp does not work, as a - b is negative and hence only the total expression remains positive.

Doing this squaring and root thing solves the issue.

Sebastian

PS: I just had to find out, that the Bateman function seems to be, in general, differently defined if you refer to mathworld: http://mathworld.wolfram.com/BatemanFunction.html