This wiki page says
From a geometric standpoint, if we consider $f(x|\theta)$ as a function of two variables then the family of probability distributions can be viewed as a family of curves parallel to the x-axis, while the family of likelihood functions is the orthogonal curves parallel to the $\theta$-axis.
I cannot imagine what this kinds of
parallel look like.
Consider this particular figure (fig_1), there are 3 distributions. I assume these are a family of probability distributions, if there is someone having different opinions about family of probability distributions, please discussed here.
there are 3 curves in the figure, it seems not to be reasonable to say the curves parallel to the x-axis.
Is my understanding correct?
this is the Python code to plot that figure (fig_1)
sigma_values = [0.5, 1.0, 2.0] linestyles = ['-', '--', ':'] mu = 0 x = np.linspace(-10, 10, 1000) fig, ax = plt.subplots(figsize=(5, 3.75)) for sigma, ls in zip(sigma_values, linestyles): # create a gaussian / normal distribution dist = norm(mu, sigma) plt.plot(x, dist.pdf(x), ls=ls, c='black', label=r'$\mu=%i,\ \sigma=%.1f$' % (mu, sigma))
the most confusing part is "curves parallel", according to my understanding a parallel of curves should looks like as follow (fig_2)
Is that implying a 3d surface where some contour lines are parallel? Is it reasonable to say the plotted contour lines are parallel to x-axis in this link?