Description

Consider a Bayesian model where the likelihood is a binomial distribution with probability parameter $p$. Let us consider an over-parameterized model where we write $p = p_1 p_2$. Assume that each $p_i$ has a uniform prior on the interval $[0, 1]$.

In summary:

\[\begin{aligned} p_1 &\sim \text{Unif}(0, 1) \\ p_2 &\sim \text{Unif}(0, 1) \\ y &\sim \text{Bin}(n, p_1 p_2) \end{aligned}\]

Here we use the values:

This is a toy example of an unidentifiable parameterization. In practice many popular Bayesian models are unidentifiable.

When there are many observations, the posterior of unidentifiable models concentrate on a sub-manifold, making sampling difficult, as shown in the following pair plots.

Pair plot

Diagonal entries show estimates of the marginal densities as well as the (0.16, 0.5, 0.84) quantiles (dotted lines). Off-diagonal entries show estimates of the pairwise densities.

Movie linked below (🍿) superimposes 100 iterations of MCMC.

Trace plots

Moments

Cumulative traces

For each iteration $i$, shows the running average up to $i$, $\frac{1}{i} \sum_{n = 1}^{i} x_n$.

Local communication barrier

When the global communication barrier is large, many chains may be required to obtain tempered restarts.

The local communication barrier can be used to visualize the cause of a high global communication barrier. For example, if there is a sharp peak close to a reference constructed from the prior, it may be useful to switch to a variational approximation.

GCB estimation progress

Estimate of the Global Communication Barrier (GCB) as a function of the adaptation round.

The global communication barrier can be used to set the number of chains. The theoretical framework of Syed et al., 2021 yields that under simplifying assumptions, it is optimal to set the number of chains (the argument n_chains in pigeons()) to roughly 2Λ.

Last round estimate: $3.5523351664264387$

Evidence estimation progress

Estimate of the log normalization (computed using the stepping stone estimator) as a function of the adaptation round.

Last round estimate: $-11.771971135020596$

Round trips

Number of tempered restarts as a function of the adaptation round.

A tempered restart happens when a sample from the reference percolates to the target. When the reference supports iid sampling, tempered restarts can enable large jumps in the state space.

Pigeons summary

Pigeons inputs

Reproducibility

run(`git clone https://github.com/Julia-Tempering/InferenceReport.jl`)
cd("InferenceReport.jl")
run(`git checkout 881ca9b392a4a59a37875104a8a5cf908ce155b3`)

using Pkg 
Pkg.activate(".")
Pkg.instantiate()
 

using Pigeons
inputs = Inputs(; target = Pigeons.toy_turing_unid_target(), n_rounds = 10, record = [traces; round_trip; record_default()])
 

pt = pigeons(inputs)