# puzzled by harmony [not!]

**R – Xi'an's Og**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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**I**n answering yet another question on X validated about the numerical approximation of the marginal likelihood, I suggested using an harmonic mean estimate as a simple but worthless solution based on an MCMC posterior sample. This was on a toy example with a uniform prior on (0,π) and a “likelihood” equal to sin(θ) [really a toy problem!]. Simulating an MCMC chain by a random walk Metropolis-Hastings algorithm is straightforward, as is returning the harmonic mean of the sin(θ)’s.

```
f <- function(x){
if ((0
```

`However, the outcome looks remarkably stable and close to the expected value 2/π, despite 1/sin(θ) having an infinite integral on (0,π). Meaning that the average of the 1/sin(θ)’s has no variance. Hence I wonder why this specific example does not lead to an unreliable output… But re-running the chain with a smaller scale σ starts producing values of sin(θ) regularly closer to zero, which leads to an estimate of I both farther away from 2 and much more variable. No miracle, in the end!`

`Filed under: Books, Kids, Mountains, pictures, R, Running, Statistics, Travel Tagged: Gaussian random walk, harmonic mean estimator, Metropolis-Hastings algorithm, Monte Carlo Statistical Methods, numerical integration, simulation `

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