Approximation of the normalization constant

Background

Let $\pi(x)$ denote a probability density called the target. In many problems, e.g. in Bayesian statistics, the density $\pi$ is typically known only up to a normalization constant,

\[\pi(x) = \frac{\gamma(x)}{Z},\]

where $\gamma$ can be evaluated pointwise, but $Z$ is unknown.

In many applications, it is useful to approximate the constant $Z$. For example, in Bayesian statistics, this corresponds to the marginal likelihood, and it is used for model selection.

Normalization constant approximation in Pigeons

As a side-product of parallel tempering, we automatically obtain an approximation of the logarithm of the normalization constant $\log Z$. This is done automatically using the stepping stone estimator computed in stepping_stone().

It is shown in the standard output report produced at each round:

using DynamicPPL
using Pigeons

# example target: Binomial likelihood with parameter p = p1 * p2
an_unidentifiable_model = Pigeons.toy_turing_unid_target(100, 50)

pt = pigeons(target = an_unidentifiable_model)

┌ Info: Neither traces, disk, nor online recorders included.
│    You may not have access to your samples (unless you are using a custom recorder, or maybe you just want log(Z)).
└    To add recorders, use e.g. pigeons(target = ..., record = [traces; record_default()])
──────────────────────────────────────────────────────────────────────────────────────────────────
  scans        Λ        time(s)    allc(B)  log(Z₁/Z₀)   min(α)     mean(α)    min(αₑ)   mean(αₑ)
────────── ────────── ────────── ────────── ────────── ────────── ────────── ────────── ──────────
        2       3.24    0.00125   6.99e+05      -8.14    0.00178       0.64          1          1
        4       1.64    0.00229   1.37e+06      -5.04     0.0352      0.818          1          1
        8       1.17    0.00413   2.67e+06      -4.42      0.708      0.871          1          1
       16        1.2    0.00851   5.61e+06      -4.03      0.549      0.867          1          1
       32       1.11     0.0168    1.1e+07      -4.77      0.754      0.877          1          1
       64       1.35     0.0288   2.19e+07      -4.79      0.698       0.85          1          1
      128        1.6     0.0514   4.34e+07      -4.97      0.725      0.823          1          1
      256       1.51      0.162   8.58e+07      -4.92      0.758      0.832          1          1
      512       1.46      0.258   1.73e+08         -5      0.806      0.838          1          1
 1.02e+03       1.49      0.581   3.47e+08      -4.92      0.798      0.834          1          1
──────────────────────────────────────────────────────────────────────────────────────────────────

and can also be accessed using:

stepping_stone(pt)

-4.9221510045598675

From ratios to normalization constants

To be more precise, the steppping stone estimator computes the log of the ratio, $\log (Z_1/ Z_0)$ where $Z_1$ and $Z_0$ are the normalization constants of the target and reference respectively.

Hence to estimate $\log Z_1$ the reference distribution $\pi_1$ should have a known normalization constant. In cases where the reference is a proper prior distribution, for example in Turing.jl models, this is typically the case.

In scenarios where the reference is specified manually, e.g. for black-box functions or Stan models, more care is needed. In such cases, one alternative is to use variational PT in which case the built-in variational distribution is constructed so that its normalization constant is one.