Drawing values from custom distribution

Questions
5

Many thanks again @ricardoV94 . The code now runs properly but does not converge to the correct values. First I generate some random values from my distribution:

from scipy.special import erf

L0 = 1e42
Lstar = 2.59e44

def schechter(logL, alpha, n42, a, b):
    
    L = 10. ** logL
    
    schechter = 10 ** n42 * (L / L0) ** (-alpha) * np.exp(- L / Lstar)  * (L / Lstar)
    
    selfunc = 0.5 + 0.5 * erf(a * (logL - b))
    
    totmod = selfunc * schechter
    
    return totmod

logLt = np.linspace(40., 45., 1000)

alpha = 1.8

n42 = np.log10(6e-3)

a = 5.

b = 41.

mod = schechter(logLt, alpha, n42, a, b)

prob_schechter = mod / np.sum(mod)

nsim = 1000

realiz = np.random.choice(logLt, nsim, p=prob_schechter)

Now here is how I coded it:

import aesara.tensor as T

def schechter_theano(logL, alpha, n42, a, b):
        
    L = 10. ** logL
    
    schechter = 10 ** n42 * (L / L0) ** (-alpha) * T.exp(- L / Lstar) * (L / Lstar)
    
    selfunc = 0.5 + 0.5 * T.erf(a * (logL - b))
    
    totmod = selfunc * schechter
    
    return T.log(totmod)

with pm.Model() as model:
    
    alpha = pm.Uniform('alpha', lower=0.5, upper=4.0)
    
    n42 = pm.Uniform('n42', lower=0, upper=10)
    
    a = pm.Uniform('a', lower=1, upper=8)
    
    b = pm.Uniform('b', lower=40, upper=43)
    
    obs = pm.DensityDist('obs', alpha, n42, a, b, logp=schechter_theano, observed=realiz)
    
    trace = pm.sample()

This code runs fine but the fitted parameter values are way off. In the way I am calling my theano function, where is it specified that the variable is logL and the other arguments are parameters?