The Shape of Probability Distribution Functions https://youtu.be/v-j0UmWf3Us
@37:48
Here, the presenter indicates that that the "negative log" of a single dimensional Gaussian distribution makes a "parabola bowl shape". This "bowl shape" is apparently very useful for the Hamiltonian MCMC algorithm, as the Hamiltonian MCMC moves in the direction of the derivative of the "negative log" of the probability distribution function - ultimately moving towards the bottom of the bowl, the bottom of the bowl being the highest probability density region, (giving the most valuable information about the probability distribution function).
Is this analogy correct in general? Is it believed that the "negative log" of many high dimensional probability distributions will somewhat form a "bowl shape"?
Thanks

This relates a lot to Pearson distributions. Pearson described a large range of distributions by considering a general form of this 'bowl shape' by means of the equation:
$$-\left( \log \left[ p(x) \right] \right) ^\prime = - \frac{p^\prime(x)}{p(x)} = \frac{a + (x-\lambda)}{ b_0 + b_1(x-\lambda)+b_2(x-\lambda)^2}$$
So yes, many distributions have a bowl shape.
