关于R中Pearson r相关系数贝叶斯估计似然函数准确性的咨询

Hey there! Let's break down how to verify your likelihood function for Bayesian estimation of Pearson $r$ using Fisher's r-to-z transformation in R. I notice you didn't share your actual code snippet, but I can walk you through what a correct implementation should look like, and you can cross-reference it with your own work.

First, a quick recap: Fisher's r-to-z transform converts a sample Pearson correlation $r$ to a value $z$ where:
z = 0.5 * log((1 + r)/(1 - r))
This transformed $z$ follows an approximate normal distribution, with mean $\zeta$ (the population-level transformed correlation) and standard error SE = 1/sqrt(n - 3)—where $n$ is your sample size. This normality is what makes the likelihood straightforward to define.

There are two common ways to parameterize this likelihood: either directly in terms of the transformed population parameter $\zeta$, or in terms of the original Pearson $r$. Here's what each should look like:

1. Parameterized for $\zeta$ (Transformed Correlation) #

If you're working with $\zeta$ as your parameter of interest, the likelihood is just the normal density of your observed transformed $z$ value, centered at $\zeta$:

# Replace these with your actual data values
r_obs <- 0.35  # Your sample Pearson correlation
n <- 40        # Your sample size

# Compute observed z-transform and standard error
z_obs <- 0.5 * log((1 + r_obs)/(1 - r_obs))
se_z <- 1 / sqrt(n - 3)

# Likelihood function for population zeta (ζ)
likelihood_zeta <- function(zeta) {
  dnorm(z_obs, mean = zeta, sd = se_z)
}

2. Parameterized for $r$ (Original Correlation) #

If you want to work directly with the original $r$ (instead of $\zeta$), you need to include a Jacobian adjustment—this accounts for the change in variables when converting between $r$ and $\zeta$. The Jacobian for this transform is 1 - r^2, so you multiply the normal density by this term:

# Reuse observed values from above
r_obs <- 0.35
n <- 40
z_obs <- 0.5 * log((1 + r_obs)/(1 - r_obs))
se_z <- 1 / sqrt(n - 3)

# Likelihood function for population r
likelihood_r <- function(r) {
  # Convert r to population zeta
  zeta <- 0.5 * log((1 + r)/(1 - r))
  # Normal density * Jacobian adjustment
  dnorm(z_obs, mean = zeta, sd = se_z) * (1 - r^2)
}

To confirm your likelihood function is accurate:

• Double-check the Fisher transform formula—make sure you're using (1 + r)/(1 - r) inside the log, not the reverse.
• If you're parameterizing in $r$, don't skip the Jacobian term! This is a super common mistake that will throw off your posterior.
• Verify the standard error: it's always 1/sqrt(n - 3)—the "-3" comes from the degrees of freedom lost when estimating the mean and variance for the correlation.
• Remember Bayes' theorem as you noted: the posterior is proportional to prior(x) * likelihood(x), so your likelihood needs to correctly reflect the sampling distribution of your observed $r$ (after transformation).

内容的提问来源于stack exchange，提问作者rnorouzian