  The vertical distribution of cold neutral hydrogen (HI) clouds is a
constraint on models of the structure, dynamics, and hydrostatic balance of the
interstellar medium. In 1978, Crovisier pioneered a method to infer the
vertical distribution of HI absorbing clouds in the solar neighborhood. Using
data from the Nan\c{c}ay 21-cm absorption survey, they determine the mean
vertical displacement of cold HI clouds, $\langle|z|\rangle$. We revisit
Crovisier's analysis and explore the consequences of truncating the HI
absorption sample in Galactic latitude. For any non-zero latitude limit, we
find that the quantity inferred by Crovisier is not the mean vertical
displacement but rather a ratio involving higher moments of the vertical
distribution. The resultant distribution scale heights are thus ${\sim}1.5$ to
${\sim}3$ times smaller than previously determined. In light of this discovery,
we develop a Bayesian Monte Carlo Markov Chain method to infer the vertical
distribution of HI absorbing clouds. We fit our model to the original
Nan\c{c}ay data and find a vertical distribution moment ratio
$\langle|z|^3\rangle/\langle|z|^2\rangle = 97 \pm 15\,\text{pc}$, which
corresponds to a Gaussian scale height $\sigma_z = 61 \pm 9\,\text{pc}$, an
exponential scale height $\lambda_z = 32 \pm 5\,\text{pc}$, and a rectangular
half-width $W_{z, 1/2} = 129 \pm 20\,\text{pc}$. Consistent with recent
simulations, the vertical scale height of cold HI clouds appears to remain
constant between the inner-Galaxy and the Galactocentric distance of the solar
neighborhood. Local fluctuations might explain the large scale height observed
at the same Galactocentric distance on the far side of the Galaxy.
