How to increase the effective sample size?

Questions
1

Hi,

I’m working on a small comparison between different variable selection/ shrinkage priors, namely Spike & Slab Priors (George & McCulloch 1993) and the Horseshoe prior (Carvalho, Polson & Scott 2008).
When sampling from these models, I receive warning messages regarding the acceptance probability and the effective sample size. The code can be found below.
How do I deal with these kinds of warnings? Just increase the tune and sample size? Or is there any other advise?

Thanks.

import numpy as np
import pymc3 as pm
import theano.tensor as tt

from arviz import summary, plot_trace
from theano.tensor import _shared

# Simulate data
k = 6
n = 200
sigma = 0.5
beta = np.array([-0.2, 3, 5, 0, -4, 0])
L = np.array(
    [[1., 0., 0., 0., 0., 0.],
    [0.5, 0.4, 0., 0., 0., 0.],
    [0.2, 0.5 , -0.7, 0., 0., 0.],
    [0., 0., 0., 1., 0., 0.],
    [0.7, 0.06, 0.03, 0., 0.8, 0.],
    [-0.7, 0., 0., 0., 0.08, .1]]
)
K = L@L.T

X = pm.MvNormal.dist(mu=np.zeros(6), cov=K, shape=[1, 6]).random(size=n)
x = _shared(X)
y = X.dot(beta) + np.random.randn(n) * sigma 


# Spike and Slab Priors - George & McCulloch 1993
with pm.Model() as model1:
    spike_prior = pm.Normal.dist(mu=0, sigma=0.01)
    slab_prior = pm.Normal.dist(mu=0, sigma=10)

    # w = pm.Bernoulli("w", p=0.5, shape=k)
    w = pm.Beta("w", alpha=0.5, beta=0.5, shape=k)
    weights = pm.Deterministic("weights", pm.math.stack([1.-w, w], axis=1))

    beta = pm.Mixture("beta", w=weights, comp_dists=[spike_prior, slab_prior], shape=k)
    sd = pm.Gamma("sd", alpha=0.05, beta=0.1, shape=1)

    # Likelihood
    mu = pm.Deterministic("mu", tt.dot(x, beta))
    y_ = pm.Normal("y_", mu=mu, sd=sd, observed=y)

    tr1 = pm.sample(cores=4)


# Horseshoe Prior - Carvalho, Polson and Scott 2009
with pm.Model() as model2:

    # Priors
    sd = pm.Gamma("sd", alpha=0.05, beta=0.1, shape=1)
    tau = pm.HalfCauchy("tau", beta=sd, shape=1) # global shrinkage
    lam = pm.HalfCauchy("lam", beta=1, shape=k) # local shrinkage

    scale = pm.Deterministic("scale", pm.math.sqr(lam*tau))
    beta = pm.Normal("beta", mu=0, sigma=scale, shape=k)

    kappa = pm.Deterministic("kappa", 1/(1+pm.math.sqr(lam)))

    # Likelihood
    mu = pm.Deterministic("mu", tt.dot(x, beta))
    y_ = pm.Normal("y_obs", mu=mu, sd=sd, observed=y)

    tr2 = pm.sample(cores=4)