R语言两种随机游走Metropolis-Hastings实现差异及校正项疑问

Hey there, let's tackle your two questions about Random Walk Metropolis-Hastings (RWMH) in R—great stuff to dive into!

Short answer: Yes, but the core RWMH logic stays consistent. Most differences come down to implementation choices that impact sampling efficiency or ease of use:

• Proposal distribution tweaks: Some built-in packages (like MCMCpack or rstan) might use adaptive proposal variances (adjusting σ² based on acceptance rates to hit the ~23% optimal rate), while a custom implementation might start with a fixed variance. This can lead to faster convergence in packaged tools vs. a basic custom script.
• Parameter transformation handling: Many packages automatically handle transformations (like logit for bounded parameters) and their associated Jacobian corrections, so you don't have to code them manually. If your custom implementation skips or messes up this correction, you'll see meaningful differences in sample outputs.
• Numerical stability optimizations: Packaged implementations often use log-probability calculations exclusively to avoid numerical underflow (critical when dealing with small likelihood values), while a naive custom script might work with raw probabilities and run into issues.
• Burn-in and initialization: Tools like rstan default to sensible burn-in periods and initial parameter values, whereas you'll have to set these manually in a custom implementation. Poor initialization or insufficient burn-in can make your custom samples look very different from package outputs.

Absolutely—you're spot on with this correction, and here's why:
When you do a random walk in the logit-transformed space (i.e., z_proposed ~ N(z_current, σ²) where z = logit(θ)), the proposal distribution is symmetric in the z-space. But when you transform back to the original θ-space (using invlogit), the symmetry breaks because the logit transformation compresses the tails of the θ-distribution (near 0 and 1).

To account for this, we need to include the log Jacobian determinant of the transformation in our acceptance probability. For the logit transform:

• The Jacobian of the inverse transform (θ from z) is dθ/dz = θ*(1-θ)
• The log of this Jacobian is log(θ*(1-θ))

In the MH acceptance probability formula, the ratio of proposal densities (q(θ_proposed | θ_current) / q(θ_current | θ_proposed)) becomes [q(z_proposed | z_current) / |J_proposed|] / [q(z_current | z_proposed) / |J_current|]. Since q is symmetric in z-space, those terms cancel out, leaving us with |J_current| / |J_proposed|. Taking the log gives exactly the correction term you used: log(xt*(1-xt)) - log(yt*(1-yt)) (where xt is current θ, yt is proposed θ).

A few quick tips to refine this:

• Avoid numerical underflow: When θ is very close to 0 or 1, θ*(1-θ) becomes tiny, and log() will return -Inf. Add a small epsilon (like 1e-8) to the product: log(max(xt*(1-xt), 1e-8)) to prevent this.
• Validate your implementation: Check if your posterior samples match expectations—for example, if you're sampling from a Beta posterior, compare the sample mean and variance to the theoretical Beta parameters. If they align, your correction is working.
• Alternative: Propose directly in θ-space: If you want to skip the Jacobian, you could use a symmetric proposal on θ (like θ_proposed = θ_current + rnorm(1, 0, σ), then clamp values to (0,1)), but this often leads to poor acceptance rates in the tails. The logit transform approach is generally more efficient.

内容的提问来源于stack exchange，提问作者The Pointer