# Poisson autoregressive model PAR(1) gives E-BFMI warning

Hi,

I set up a first-order Poisson Auto-regressive model PAR(1) and tested it on simulated data. During iterating, I hit an example where I get E-BFMI<0.2, but no other warnings, which @betanalpha may be interested in academically.

I do get a compile-time warning, but I thought (perhaps wrongly?) it is okay because the parameters are combined in a linear fashion.

Here is the code, which includes generating the test data:

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats
from numpy import log, pi, cos, exp
import sys

np.random.seed(1)

N = 1000

W = np.random.normal(0, 1, size=N)

T = 1.
dt = T / N
# correlation: set to < 1, otherwise not stationary!
tau = 0.01
phi = exp(-1. / (tau / dt))
#phi = 0.999
print('AR phi:', phi)
print('decay time scale:', tau, 'time interval of sampling:', dt, 'number of steps until decay:', int(tau / dt), 'number of decay timescales sampled:', int(T / tau))
print('OU theta:', - phi * dt)
tau = -1. / log(phi)
gamma = 1. / tau

c = 50.0
sigma = 0.04 * c
y0 = W[0] * sigma + c
y = np.empty(N)
y[0] = y0
t = np.linspace(0, N * T, N)
for i in range(1, N):
# phi [AR]  corresponds to - theta * dt [OU]
y[i] = c + phi * (y[i-1] - c) + W[i] * sigma

z = np.random.poisson(y)

plt.figure(figsize=(20, 4))
plt.plot(t, y)
plt.plot(t, z)
plt.xlabel('Time')
plt.ylabel('Value')
plt.savefig('fracar1_t.pdf', bbox_inches='tight')
plt.close()

import stan_utility

model = stan_utility.compile_model_code("""
data {
int N;
real dt;
int<lower=0> z_obs[N];
}
parameters {
real phi;
real c;
real lognoise;
real<lower=0> y[N];
}
transformed parameters {
real tau;
real noise;
noise = exp(lognoise);
tau = - dt / log(phi);
}
model {
phi ~ normal(0, 10) T[0, 100];
lognoise ~ normal(0, 10);
for (i in 1:(N-1)) {
(y[i+1] - c) - phi * (y[i] - c) ~ normal(0, noise);
}
z_obs ~ poisson(y);
}
""")

data = dict(N=N, dt=dt, z_obs=z)

results = stan_utility.sample_model(model, data, outprefix="oustanfit", control=dict(max_treedepth=12))

stan_utility.plot_corner(results, outprefix="oustanfit")


And the output:

AR phi: 0.9048374180359595
decay time scale: 0.01 time interval of sampling: 0.001 number of steps until decay: 10 number of decay timescales sampled: 100
OU theta: -0.0009048374180359595
Compiling slimmed Stan code[anon_model_cfe6382326cee97ea72c3e4c8c4d188c]:
1: data {
2: int N;
3: real dt;
4: int<lower=0> z_obs[N];
5: }
6: parameters {
7: real phi;
8: real c;
9: real lognoise;
10: real<lower=0> y[N];
11: }
12: transformed parameters {
13: real tau;
14: real noise;
15: noise = exp(lognoise);
16: tau = - dt / log(phi);
17: }
18: model {
19: phi ~ normal(0, 10) T[0, 100];
20: lognoise ~ normal(0, 10);
21: for (i in 1:(N-1)) {
22: (y[i+1] - c) - phi * (y[i] - c) ~ normal(0, noise);
23: }
24: z_obs ~ poisson(y);
25: }
DIAGNOSTIC(S) FROM PARSER:
Info:
Left-hand side of sampling statement (~) may contain a non-linear transform of a parameter or local variable.
If it does, you need to include a target += statement with the log absolute determinant of the Jacobian of the transform.
Left-hand-side of sampling statement:
y[i + 1] - c - phi * y[i] - c ~ normal(...)

INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_cfe6382326cee97ea72c3e4c8c4d188c NOW.

Data
----
N         : 1000
dt        : 0.001
z_obs     : shape (1000,) [30 ... 84]

sampling from model ...
Rejecting initial value:
Log probability evaluates to log(0), i.e. negative infinity.
Stan can't start sampling from this initial value.

1000 transitions using 10 leapfrog steps per transition would take 1.84 seconds.

1000 transitions using 10 leapfrog steps per transition would take 1.49 seconds.

1000 transitions using 10 leapfrog steps per transition would take 1.69 seconds.

Rejecting initial value:
Log probability evaluates to log(0), i.e. negative infinity.
Stan can't start sampling from this initial value.
Rejecting initial value:
Log probability evaluates to log(0), i.e. negative infinity.
Stan can't start sampling from this initial value.
Iteration:    1 / 2000 [  0%]  (Warmup)
Rejecting initial value:
Log probability evaluates to log(0), i.e. negative infinity.
Stan can't start sampling from this initial value.
Iteration:    1 / 2000 [  0%]  (Warmup)

1000 transitions using 10 leapfrog steps per transition would take 1.34 seconds.

Iteration:    1 / 2000 [  0%]  (Warmup)
Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
Exception: normal_lpdf: Scale parameter is 0, but must be > 0!  (in 'unknown file name' at line 22)

If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

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Elapsed Time: 21.246 seconds (Warm-up)
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WARNING:pystan:Maximum (flat) parameter count (1000) exceeded: skipping diagnostic tests for n_eff and Rhat.
To run all diagnostics call pystan.check_hmc_diagnostics(fit)
WARNING:pystan:Chain 1: E-BFMI = 0.0668
WARNING:pystan:Chain 2: E-BFMI = 0.0811
WARNING:pystan:Chain 3: E-BFMI = 0.0839
WARNING:pystan:Chain 4: E-BFMI = 0.0572
WARNING:pystan:E-BFMI below 0.2 indicates you may need to reparameterize your model
processing results ...
Inference for Stan model: anon_model_cfe6382326cee97ea72c3e4c8c4d188c.
4 chains, each with iter=2000; warmup=1000; thin=1;
post-warmup draws per chain=1000, total post-warmup draws=4000.

mean se_mean     sd   2.5%    25%    50%    75%  97.5%  n_eff   Rhat
phi        0.89  2.8e-3   0.03   0.83   0.87   0.89   0.91   0.94     99   1.04
c         50.34  8.7e-3   0.58  49.15  49.98  50.36  50.73  51.45   4520    1.0
lognoise   0.55    0.02   0.13   0.29   0.46   0.55   0.65   0.79     49   1.09
y[1]      48.65    0.06   4.02  40.78  45.93   48.6  51.32  56.75   4849    1.0
y[2]      48.63    0.05    3.4  42.26  46.27  48.62  50.87  55.43   4841    1.0
y[3]      48.73    0.05   2.99  43.15  46.68  48.68  50.76  54.76   4245    1.0
y[4]      48.92    0.04   2.77  43.83  47.01  48.84  50.73  54.53   4035    1.0
y[5]      48.77    0.04   2.63  43.67  46.94  48.71   50.5  54.13   3466    1.0
y[6]      48.05    0.04   2.59  42.93  46.28  48.03  49.78  53.29   4257    1.0
y[7]      47.95    0.04   2.49  43.13  46.26  47.91  49.59   53.0   4404    1.0
y[8]      47.43    0.04   2.47  42.56  45.78  47.44  49.07  52.23   4448    1.0
y[9]      46.93    0.04   2.43  42.21  45.26  46.96  48.53  51.69   4360    1.0
y[10]     46.71    0.04    2.4  41.88  45.13  46.69   48.3  51.45   4283    1.0
y[11]     46.36    0.04   2.41  41.63  44.74  46.28  47.98  50.95   4317    1.0
y[12]     46.26    0.04   2.38   41.7  44.66  46.28  47.86  50.87   4600    1.0
y[13]     46.98    0.04   2.41  42.36  45.33  46.96  48.65  51.69   4441    1.0
y[14]     47.25    0.04    2.4  42.59  45.65  47.22  48.84  51.94   4253    1.0
y[15]     47.27    0.03   2.41  42.64  45.59  47.21  48.87   52.2   5278    1.0
y[16]     46.79    0.04   2.41  42.06  45.18   46.8  48.35  51.65   4551    1.0
y[17]     46.68    0.03   2.45  41.83  44.99  46.74  48.36  51.41   5032    1.0
y[18]     47.01    0.04    2.4  42.47  45.36  47.04  48.61   51.8   4606    1.0
y[19]     47.64    0.04   2.38  42.99  46.08   47.6  49.24  52.53   4390    1.0
y[20]     48.28    0.04    2.4  43.61  46.73  48.28  49.83  53.11   4622    1.0
y[21]     48.95    0.04   2.38  44.36  47.39  48.91  50.48   53.7   4140    1.0
y[22]     49.38    0.04   2.38  44.77  47.74  49.38  50.97  54.12   4087    1.0
y[23]     49.75    0.04   2.41   45.1  48.15   49.7  51.31  54.47   4672    1.0
y[24]     50.05    0.04   2.39  45.49  48.45  49.97  51.67  54.72   4663    1.0
y[25]     50.45    0.04   2.39  45.98  48.81   50.4  52.01  55.27   3854    1.0
-- snip, they are all the same --
y[998]    45.93    0.04   2.54   41.0   44.2  45.88  47.68  50.87   4810    1.0
y[999]    45.57    0.04   2.66  40.37  43.74   45.5  47.45  50.81   5131    1.0
y[1000]   45.29    0.04   2.88  39.69  43.34  45.27  47.27  50.87   4731    1.0
tau      9.2e-3  2.8e-4 2.7e-3 5.4e-3 7.4e-3 8.8e-3   0.01   0.02     91   1.04
noise      1.75    0.03   0.23   1.33   1.58   1.74   1.91   2.21     52   1.09
lp__      1.5e5    17.9 123.44  1.5e5  1.5e5  1.5e5  1.5e5  1.5e5     48   1.09

Samples were drawn using NUTS at Fr 05 Mär 2021 11:24:31 .
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
checking results ...
/home/user/.local/lib/python3.8/site-packages/stan_utility-0.1.2-py3.8.egg/stan_utility/utils.py:76: UserWarning: Chain 0: E-BFMI = 0.06682835264021626
warnings.warn("Chain {}: E-BFMI = {}".format(chain_num, numer / denom))
/home/user/.local/lib/python3.8/site-packages/stan_utility-0.1.2-py3.8.egg/stan_utility/utils.py:76: UserWarning: Chain 1: E-BFMI = 0.08107339340014912
warnings.warn("Chain {}: E-BFMI = {}".format(chain_num, numer / denom))
/home/user/.local/lib/python3.8/site-packages/stan_utility-0.1.2-py3.8.egg/stan_utility/utils.py:76: UserWarning: Chain 2: E-BFMI = 0.08391842399004969
warnings.warn("Chain {}: E-BFMI = {}".format(chain_num, numer / denom))
/home/user/.local/lib/python3.8/site-packages/stan_utility-0.1.2-py3.8.egg/stan_utility/utils.py:76: UserWarning: Chain 3: E-BFMI = 0.05720619539756567
warnings.warn("Chain {}: E-BFMI = {}".format(chain_num, numer / denom))
/home/user/.local/lib/python3.8/site-packages/stan_utility-0.1.2-py3.8.egg/stan_utility/utils.py:84: UserWarning:   E-BFMI below 0.2 indicates you may need to reparameterize your model


You may be glad that I iterated beyond the model written above.

My current setup is the more realistic case (for my problem) of a mixture of two PAR(1) processes, one of which can be observed directly:

import numpy as np
import matplotlib.pyplot as plt
from numpy import log, pi, cos, exp
import sys

np.random.seed(1)

N = int(sys.argv[1])

BW = np.random.normal(0, 1, size=N)
W = np.random.normal(0, 1, size=N)

T = 1.
dt = T / N
# correlation: set to < 1, otherwise not stationary!
tau = 0.01
phi = exp(-1. / (tau / dt))
#phi = 0.999
print('AR phi:', phi)
print('decay time scale:', tau, 'time interval of sampling:', dt, 'number of steps until decay:', int(tau / dt), 'number of decay timescales sampled:', int(T / tau))
print('OU theta:', - phi * dt)

Bc = 100.0
Btau = 0.6
Bsigma = 0.04 * Bc
By0 = BW[0] * Bsigma + Bc
By = np.empty(N)
Bphi = exp(-1. / (Btau / dt))
By[0] = By0
for i in range(1, N):
# phi [AR]  corresponds to - theta * dt [OU]
By[i] = Bc + Bphi * (By[i-1] - Bc) + BW[i] * Bsigma

print('phis:', Bphi, phi, Btau, tau)
Bz = np.random.poisson(By)

Barearatio = 1. / 40.0

c = 50.0
sigma = 0.04 * c
y0 = W[0] * sigma + c
y = np.empty(N)
y[0] = y0
t = np.linspace(0, N * T, N)
for i in range(1, N):
# phi [AR]  corresponds to - theta * dt [OU]
y[i] = c + phi * (y[i-1] - c) + W[i] * sigma

z = np.random.poisson(y + By * Barearatio)

plt.figure(figsize=(20, 4))
plt.plot(t, y)
plt.plot(t, z)

plt.plot(t, By * (Barearatio if Barearatio > 0 else 1))
plt.plot(t, Bz * (Barearatio if Barearatio > 0 else 1))
plt.yscale('log')
plt.xlabel('Time')
plt.ylabel('Value')
plt.savefig('par1b_t.pdf', bbox_inches='tight')
plt.close()

import stan_utility

model = stan_utility.compile_model_code("""
data {
int N;
real dt;
int<lower=0> z_obs[N];
int<lower=0> B_obs[N];
real Barearatio;
}
parameters {
real logtau;
real logc;
real lognoise;
vector<lower=0>[N] y;

real logBtau;
real logBc;
real logBnoise;
vector<lower=0>[N] By;
}
transformed parameters {
real phi;
real tau;
real noise;
real c;

real Bphi;
real Btau;
real Bnoise;
real Bc;

tau = exp(logtau);
c = exp(logc);
noise = exp(lognoise);
phi = exp(- 1.0 / (tau / dt));

Btau = exp(logBtau);
Bc = exp(logBc);
Bnoise = exp(logBnoise);
Bphi = exp(- 1.0 / (Btau / dt));
}
model {
logBtau ~ normal(-10, 10) T[-100, 0];
logBc ~ normal(0, 10);
logBnoise ~ normal(0, 10);
By[2:N] ~ normal(Bphi * (By[1:(N-1)] - Bc) + Bc, Bc * Bnoise);
B_obs ~ poisson(By);

logtau ~ normal(-10, 10) T[-100, 0];
logc ~ normal(0, 10);
lognoise ~ normal(0, 10);
y[2:N] ~ normal(phi * (y[1:(N-1)] - c) + c, c * noise);
z_obs ~ poisson(y + By * Barearatio);
}
""")

data = dict(N=N, dt=dt, z_obs=z, B_obs=Bz, Barearatio=Barearatio)

results = stan_utility.sample_model(model, data, outprefix="oustanfit", control=dict(adapt_delta=0.99))

stan_utility.plot_corner(results, outprefix="oustanfit")


Here I get no compiler warnings, but a bunch of divergences:

WARNING:pystan:729 of 4000 iterations ended with a divergence (18.2 %).
WARNING:pystan:Try running with adapt_delta larger than 0.99 to remove the divergences.
WARNING:pystan:2604 of 4000 iterations saturated the maximum tree depth of 10 (65.1 %)
WARNING:pystan:Run again with max_treedepth larger than 10 to avoid saturation
WARNING:pystan:Chain 1: E-BFMI = 0.0937
WARNING:pystan:Chain 2: E-BFMI = 0.127
WARNING:pystan:Chain 3: E-BFMI = 0.16
WARNING:pystan:Chain 4: E-BFMI = 0.138
WARNING:pystan:E-BFMI below 0.2 indicates you may need to reparameterize your model


Advice on how to reparametrize is appreciated :-)

I guess one could use a inverse gamma function to derive the By distributions – perhaps they can be marginalised out with some conjugate prior? Not sure. I do want to keep the y-states though, because I want to fourier transform them later – I guess I cannot do that in Stan(?).

1 Like
           mean se_mean     sd   2.5%    25%    50%    75%  97.5%  n_eff   Rhat
phi        0.89  2.8e-3   0.03   0.83   0.87   0.89   0.91   0.94     99   1.04
c         50.34  8.7e-3   0.58  49.15  49.98  50.36  50.73  51.45   4520    1.0
lognoise   0.55    0.02   0.13   0.29   0.46   0.55   0.65   0.79     49   1.09
...
tau      9.2e-3  2.8e-4 2.7e-3 5.4e-3 7.4e-3 8.8e-3   0.01   0.02     91   1.04
noise      1.75    0.03   0.23   1.33   1.58   1.74   1.91   2.21     52   1.09
lp__      1.5e5    17.9 123.44  1.5e5  1.5e5  1.5e5  1.5e5  1.5e5     48   1.09


Some of the output in the first model aren’t mixing well. Large Rhat and low ESS. Rhat < 1.01 is the rule of thumb from [1903.08008] Rank-normalization, folding, and localization: An improved $\widehat{R}$ for assessing convergence of MCMC (which is a cool paper for the plots).

Looking at the model, the truncation on the phi prior should match the constraints on the phi variable itself. So like this:

real<lower = 0.0, upper = 100.0> phi;


I think you should do a non-centering on the AR(1) process. I’m not sure how to do that offhand cause of the lower = 0 constraint, but you can try following along here: Non-centered parameterisation with boundaries - #5 by Bob_Carpenter .

It would probably be easiest to not have the constraint though.

1 Like

I tried non-centering as in the code below (same concept as the generating python code btw), but I get rhats of 1e15. I guess this is because the y parameters are actually pinned down pretty well by the data.

data {
int N;
real dt;
int<lower=0> z_obs[N];
}
parameters {
real phi;
real c;
real lognoise;
real W[N];
}
transformed parameters {
real tau;
real noise;
vector[N] y;
noise = exp(lognoise);
tau = - dt / log(phi);

y[1] = c + W[1] * noise;
for (i in 2:N) {
y[i] = c + phi * (y[i-1] - c) + W[i] * noise;
}
}
model {
phi ~ normal(0, 10) T[0, 100];
lognoise ~ normal(0, 10);
c ~ normal(0, 10) T[0, 1000];
W ~ std_normal();
z_obs ~ poisson(y);
}

I would expect that this line in the model above would cause problems:

z_obs ~ poisson(y);


The rate parameter to a poisson must be positive (here). y looks unconstrained, and so the typical way to handle an unconstrained rate would be a log link function, which can be done with poisson_log (docs here).

Were there parameters in particular that weren’t mixing well with the non-centered version?

Also with the truncated distributions, also do the constrains phi and c.

real<lower = 0.0, upper = 100.0> phi;
real<lower = 0.0, upper = 1000.0> c;


The truncation does not apply a constraint automatically, so those parameters could wander out side of the [0, 100] truncation (and at that point I’m not sure what would happen – I think that’s undefined behavior-land).

Now I found a form that works:

data {
int N;
real dt;
int<lower=0> z_obs[N];
}
parameters {
real logtau;
real logc;
real lognoise;
real W[N];
}
transformed parameters {
real tau;
real phi;
real noise;
real c;
vector[N] y;
c = exp(logc);
noise = exp(lognoise);
tau = exp(logtau);
phi = exp(-dt / tau);

y[1] = c + W[1] * noise;
for (i in 2:N) {
y[i] = c + phi * (y[i-1] - c) + W[i] * noise;
}
}
model {
logtau ~ normal(log(dt), log(N));
lognoise ~ normal(0, 10);
logc ~ normal(0, 10);
W ~ std_normal();
z_obs ~ poisson(y);
}


But automatic initialisation did not work, so here is my init function:

def init_chain():
return dict(logtau=np.log(dt), logc=np.log(z.mean()), lognoise=np.log(z.std()), W=np.zeros(N))


Thanks for the pointer on the truncation, I did not know that it has to be specified twice.

Cool beans, good to hear. Watch out with stuff like this:

z_obs ~ poisson(y);


That y is not constrained and can be non-positive. If it is not positive then it won’t be possible to evaluate the likelihood and stuff will blow up (this might be why the default inits were not working).

The truncation is usually used for measurement models and the constraints and used to control where parameters can go or not, so the usual uses are a bit different.

Thanks. Autoregressive models in the state parameterization indeed exhibit a sequence of funnels between the neighboring states, hence the energy fraction of missing information warnings (for more see Hierarchical Modeling). The innovation parameterization in terms of the state differences is equivalent to a non-centered parameterization.

As with hierarchal models if the individual likelihood functions are uniformly strong informed that centering all of the states is optimal and if the likelihood functions are uniformly weakly informed then non-centering all of the states is optimal. When the behavior of the likelihood functions varies a mixed state/innovation parameterization is necessary.

What’s the innovation “innovation parameterization”? The link and googling didn’t reveal anything.

The innovation parameterization is when you write one of these time series models in terms of the difference between time points.

Like if we have a random walk we can express that by saying:

y_{n + 1} \sim N(y_{n}, \sigma)

Or we can write it in terms of innovations:

y_{n + 1} = y_{n + 1} + \epsilon_n\\ \epsilon_n \sim N(0, \sigma)

And then we can say \epsilon_n is gonna do a funnel thing there and instead do:

y_{n + 1} = y_{n + 1} + \sigma z_n\\ z_n \sim N(0, 1)

And so the last thing is like a non-centered parameterization. So you have a choice when you write this model and that links into the centered/non-centered discussion in the hierarchical modeling link.

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