Fix Slice sampler bugs introduced in b988ba9

1. The previous interval was of the form `[q0 - uniform(0, self.w), q0 + self.w]` which is not symmetric around q0 in expectation, breaking detailed balance.
2. The left/right bound values used for updating earlier variables were only set to the accepted values during tuning. This might have made bug 1 worse.

Author: ricardoV94

RELEASE-NOTES.md

@@ -148,16 +148,16 @@ This includes API changes we did not warn about since at least `3.11.0` (2021-01
 - Added the low level `compile_forward_sampling_function` method to compile the aesara function responsible for generating forward samples (see [#5759](https://github.com/pymc-devs/pymc/pull/5759)).
 - ...
 
-
-## Documentation
+### Documentation
 - Switched to the [pydata-sphinx-theme](https://pydata-sphinx-theme.readthedocs.io/en/latest/)
 - Updated our documentation tooling to use [MyST](https://myst-parser.readthedocs.io/en/latest/), [MyST-NB](https://myst-nb.readthedocs.io/en/latest/), sphinx-design, notfound.extension,
   sphinx-copybutton and sphinx-remove-toctrees.
 - Separated the builds of the example notebooks and of the versioned docs.
 - Restructured the documentation to facilitate learning paths
 - Updated API docs to document objects at the path users should use to import them
 
-### Internal changes
+### Maintenance
+- ⚠ Fixed old-time bug in Slice sampler that resulted in biased samples (see [#5816](https://github.com/pymc-devs/pymc/pull/5816)).
 - ⚠ PyMC now requires Scipy version `>= 1.4.1` (see [4857](https://github.com/pymc-devs/pymc/pull/4857)).
 - Removed float128 dtype support (see [#4514](https://github.com/pymc-devs/pymc/pull/4514)).
 - Logp method of `Uniform` and `DiscreteUniform` no longer depends on `pymc.distributions.dist_math.bound` for proper evaluation (see [#4541](https://github.com/pymc-devs/pymc/pull/4541)).

pymc/step_methods/slicer.py

@@ -73,8 +73,11 @@ def astep(self, q0, logp):
             # uniformly sample from 0 to p(q), but in log space
             q_ra = RaveledVars(q, q0.point_map_info)
             y = logp(q_ra) - nr.standard_exponential()
-            ql[i] = q[i] - nr.uniform(0, self.w[i])
-            qr[i] = q[i] + self.w[i]
+
+            # Create initial interval
+            ql[i] = q[i] - nr.uniform() * self.w[i]  # q[i] + r * w
+            qr[i] = ql[i] + self.w[i]  # Equivalent to q[i] + (1-r) * w
+
             # Stepping out procedure
             cnt = 0
             while y <= logp(
@@ -104,16 +107,16 @@ def astep(self, q0, logp):
                 if cnt > self.iter_limit:
                     raise RuntimeError(LOOP_ERR_MSG % self.iter_limit)
 
-            if (
-                self.tune
-            ):  # I was under impression from MacKays lectures that slice width can be tuned without
+            if self.tune:
+                # I was under impression from MacKays lectures that slice width can be tuned without
                 # breaking markovianness. Can we do it regardless of self.tune?(@madanh)
                 self.w[i] = self.w[i] * (self.n_tunes / (self.n_tunes + 1)) + (qr[i] - ql[i]) / (
                     self.n_tunes + 1
-                )  # same as before
-                # unobvious and important: return qr and ql to the same point
-                qr[i] = q[i]
-                ql[i] = q[i]
+                )
+
+            # Set qr and ql to the accepted points (they matter for subsequent iterations)
+            qr[i] = ql[i] = q[i]
+
         if self.tune:
             self.n_tunes += 1
         return q