Unbiased PopulationSimpleSliceSampler

[JohannesBuchner]

Hi @njzifjoiez,

I was informed (by ChatGPT) that stepping out is actually not necessary in slice sampling, and not doing it does not cause a bias. However, one needs to make the proposal reversible. Neal2003's slice sampling does this by randomizing the location of the current point on the slice (not set it to be at zero.

This small modification should let us have the cake and eat it too -- avoiding stepping out, but without a bias.

Would you be able to make the change and test it?

Here are the instructions I got:

• Compute the line’s allowable t-range from the unit cube:
  (tmin, tmax) = unitcube_line_intersection(u0, v)

• Choose the working width w:
  w = min(2*scale*factor_scale, tmax - tmin)

• Randomize the initial bracket so that it contains t0 = 0, has width w, and lies within [tmin, tmax]:

  u = Uniform(0,1)
  L = max(tmin, -u*w)
  R = L + w
  if R > tmax:
      R = tmax
      L = R - w

Now ensure t0=0 is inside [L, R]. Given tmin <= 0 <= tmax and w <= tmax - tmin, this holds.

• Initialize both worker and point bounds to that randomized [L, R]:

tleft_worker = tleft = L
tright_worker = tright = R

• Proceed with the usual shrinkage on rejections with shrink_factor = 1.0.

related to: #118

[njzifjoiez]

Hi there,

Yes, sure, I will have a look at that!

[JohannesBuchner]

Sorry. I no longer think slice sampling like described here with an incomplete slice is unbiased. I was misled by ChatGPT.

However, I think something that could be tried is to use a Metropolis proposal on the line, for example as follows:

1. Project the live points onto the line, obtaining a coordinate t for each live point, defined so that t=0 is the current location.
2. set sigma_max = max{t} - min{t}, the range on the line
3. set sigma_min = the median distance between live points on the line
4. set sigma = sigma_min
5. create a Gaussian proposal with sigma, and propose on the slice -> t_proposed.

• test likelihood at the proposed point (if not rejected already).
• If accepted, go there.
• If rejected and t_proposed>0: mark all t > t_proposed as inaccessible, i.e., setting the upper slice limit.
• If rejected and t_proposed<0: mark all t < t_proposed as inaccessible, i.e., setting the lower slice limit.

6. double sigma, and go to 5 unless sigma > sigma_max (or more simply: use a fixed number of loops).

This should be unbiased, because we are doing a sequence of Metropolis random walk steps that

• is independent of the local position,
• it retains some nice features from slice sampling (lower dimension simplifies efficiency).
• The support of the proposal is not pre-truncated (which could be biased), but it retains efficiency by assuming a monomodal shape is probed by the slice -- more precisely that the geometry is convex. I think this is a reasonable assumption even in multimodal posteriors, because the slice is unlikely to enter another mode and we only need to manage to walk within the same mode.