# To what extent should the posterior of prior parameters capture the posterior variance of parameters?

I am experimenting with mixtures of autoregressive models with the idea that the posterior on the mixture parameters should correspond roughly to “model selection” for the order of the autoregression. [aside: is “mixture” the right term here? It’s the same techniques but seems quite qualitatively different from e.g. a GMM since the purpose of the mixture is to choose just the size of the model]

For some background:

For an AR(p) model we have y(t) = \sum_{\tau = 1}^p b_p(t - \tau) y(t - \tau) + e_t. In a finite mixture I have basically that y \sim \theta_1 AR(1) + \cdots + \theta_{p_{max}} AR(p_{max}).

Each AR(p) model has a p-vector of coefficients AR(1): [b_1(1)], AR(2): [b_2(1), b_2(2)], \ldots. These can be parameterized in terms of reflection coefficients \Gamma_1, \ldots, \Gamma_p where \Gamma_1 determines b_1, the combination of \Gamma_1, \Gamma_2 determines (the 2-vector) b_2, etc.

It must hold that |\Gamma_\tau| < 1, so I’ve used a prior \frac{1}{2}(1 + \Gamma_\tau) \sim \beta(\mu_\tau, \nu_\tau), with various different experiments with the prior on \nu_\tau, e.g. \nu_\tau \sim \text{exp}(1) etc.

Fitting my model with p_{max} = 5 on data simulated from AR(2) actually seems to work reasonably well, except that it does not seem to do a good job of quantifying the certainty in the mixture coefficients \theta. In the below figure, \Gamma_1 and \Gamma_2 are very tightly concentrated on the correct value (I gave it a lot of data) and \Gamma_3, \ldots, \Gamma_5 have broad posteriors, which also makes sense since the data came from an AR(2) model.

My interpretation from this is that it “should” be “very confident” that the true model order is 2 – it seems clearly apparent in the relative disperson of \Gamma. However, this doesn’t seem to be reflected (at least to the extent that it is clear from the posterior on \Gamma) in the posteriors of \theta or especially of \nu_\tau.

After all that background explanation, my actual questions are:

If \Gamma_\tau \sim \beta(\mu_\tau, \nu_\tau) and the posterior of \Gamma_\tau is very concentrated [disperse] should that be clearly reflected in the posterior of \nu_\tau being (\nu_\tau is the pseudo-samples, which I understand should quantify uncertainty?) very large [small] ? And, to what extent could I correlate \nu with \theta in the prior (and would that be desirable)?

EDIT: There’s a labelling mistake – the plot shows the \log_{10}(\nu), which is why some are negative, but in any case, their posteriors are still very similar, despite the drastic differences in posteriors of \Gamma.

There’s a lot of code but here it is

/*
* A mixture of AR(p) models up to a maximum order p_max.
* Information on ragged datastructures in stan is relevant
*/

functions {
vector step_up(vector g){
/*
* Maps a vector of p reflection coefficients g (|g| < 1) to a
* stable sequence b of AR coefficients.
*/
int p = dims(g);  // Model order
vector[p] b;  // AR Coefficients
vector[p] b_cpy;  // Memory

b = g;
if(p == 1)
return -b;

for(k in 1:p - 1){
b_cpy = b;
for(tau in 1:k){
b[tau] = b_cpy[tau] + g[k + 1] * b_cpy[k - tau + 1];
}
b[k + 1] = g[k + 1];
}

return -b;
}

vector reverse(vector x){
int p = dims(x);
vector[p] x_rev;
for (tau in 1:p)
x_rev[tau] = x[p - tau + 1];
return x_rev;
}

vector ar_model_forecast(vector y, vector y0, vector b, real sigma){
int p = dims(b);
int T = dims(y);
vector[T] y_hat;
vector[p] b_rev = reverse(b);

// Prepend the initial values y0
vector[p + T] y_full;
y_full[:p] = y0;
y_full[p + 1:] = y;

for(t in 1:T){
y_hat[t] = dot_product(b_rev, y_full[t:t + p - 1]);
}
return y_hat;
}

real ar_model_lpdf(vector y, vector y0, vector b, real sigma){
int T = dims(y);
vector[T] y_hat = ar_model_forecast(y, y0, b, sigma);
return normal_lpdf(y | y_hat, sigma);
}

vector ar_model_rng(vector y, vector y0, vector b, real sigma){
int T = dims(y);
vector[T] y_hat = ar_model_forecast(y, y0, b, sigma);
vector[T] y_rng;
for(t in 1:T)
y_rng[t] = normal_rng(y_hat[t], sigma);
return y_rng;
}
}

data {
int<lower=1> T;  // Number of examples
int<lower=1> p_max;  // The model order

simplex[p_max + 1] mu_th;  // Dirichlet mean for theta

vector[T] y;  // the data series y(t)
}

transformed data {
vector[p_max + T] t;  // "time"
int n_params = (p_max * (p_max + 1)) / 2;
int model_size[p_max];

for(p in 1:p_max)  // For a raged data structure
model_size[p] = p;

for(i in -p_max + 1:T)
t[i + p_max] = i;
t = t / T;  // Normalize to between [-p/T, 1]
}

parameters {
real mu;  // Mean value
real r;  // Linear trend coefficient
vector[p_max] y0;  // Initial values
simplex[p_max + 1] theta;  // Mixture parameters

// vector<lower=0, upper=1>[p_max] mu_gamma;  // Mean vector for gamma
real<lower=0> nu_th; // pseudo-samples for Dirichlet
vector<lower=0>[p_max] nu_gamma;  // pseudo-samples gamma
vector<lower=-1, upper=1>[p_max] gamma;  // Reflection coefficients

real<lower=0> sigma;  // noise level
}

transformed parameters {
vector[n_params] b;  // AR Coefficients

{int pos = 1;
// Compute the actual AR coefficients
for(p in 1:p_max){
b[pos:pos + model_size[p] - 1] = step_up(gamma[:p]);
pos += model_size[p];}}}

model {
vector[p_max] alpha;  // transformed gamma prior params
vector[p_max] beta;
vector[p_max + 1] alpha_th;  // transformed Dirichlet prior for theta

vector[p_max + 1] lpdfs;  // mixture pdfs
vector[p_max + T] trend;  // trend term

real mu_gamma = 0.5;

nu_th ~ inv_gamma(3, 3);
alpha_th = mu_th * nu_th;

theta ~ dirichlet(alpha_th);

// Noise level in the signal
sigma ~ normal(0, 5);

// Priors for the reflection coefficients
// mu_gamma ~ uniform(0, 1);  // A vector
// nu_gamma ~ inv_gamma(3, 3);
nu_gamma ~ exponential(0.1);
alpha = mu_gamma * nu_gamma;
beta = (1 - mu_gamma) * nu_gamma;

// trend parameters
r ~ normal(0, 2);  // The linear time coefficient
mu ~ normal(0, 2);  // A mean offset

// The simple trend term
trend = mu + r * t;

// Initial values
y0 ~ normal(trend[:p_max], sigma);

// Mixture AR(p), including "AR(0)" (pure noise)
{int pos = 1;
vector[T] detrend_y = y - trend[p_max + 1:];
for(p in 1:p_max){
lpdfs[p] = theta[p] + ar_model_lpdf(detrend_y | y0[:model_size[p]],
b[pos:pos + model_size[p] - 1],
sigma);
pos += model_size[p];
}
lpdfs[p_max + 1] = theta[p_max + 1] + normal_lpdf(detrend_y | 0, sigma);
}

target += beta_lpdf(0.5 * (1 + gamma) | alpha, beta);  // in (0, 1)
target += log_sum_exp(lpdfs);
}

generated quantities {
vector[T] y_ppc;
vector[T] trend;  // trend term
int p = categorical_rng(theta);
int pos;

trend = mu + r * t[p_max + 1:];
y_ppc = trend;

if(p == 1){  // AR(1)
y_ppc += ar_model_rng(y, y0[:1], b[:1], sigma);
}
else if(p == p_max + 1){  // AR(0)
for(i in 1:T)
y_ppc[i] += normal_rng(0, sigma);
}
else{  // AR(p), p > 1
pos = 1 + sum(model_size[:p - 1]);
y_ppc += ar_model_rng(y, y0[:model_size[p]], b[pos:pos + model_size[p] - 1], sigma);
}
}



Tough question!

• How are you breaking the symmetry between, for example, the AR(2) terms and all the lower order terms of the AR(3)?
• Can you generate data from this model with p_{\text{max}} = 5 and try and fit it to that?

Possibly? Concentrated / disperse are relative terms here, can you plot the posteriors against the prior? Perhaps they have concentrated some amount already? I would expect them to contract very slowly in this model given its complexity.

The notation starts to deviate from the plot a little here, which \mu and \nu are we talking about exactly? \nu_{\theta}? Or the various \nu_{\gamma}? If there’s an extra level in the hierarchy here then things are likely to stay very disperse for an awful lot of data.

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The notation starts to deviate from the plot a little here, which μ and ν are we talking about exactly? νθ ? Or the various νγ ? If there’s an extra level in the hierarchy here then things are likely to stay very disperse for an awful lot of data.

Here’s a more precise description of the model, excluding \mu_y and r which are just trend and intercept terms.

\nu_\theta \sim \texttt{inv_gamma}(3, 3) \texttt{ // pseudo-samples for } \theta \\ \mu_\theta = [1, 1, 1, 1, 1] / 5 \texttt{ // uniform prior on model order mean}\\ \theta \sim \texttt{Dirichlet}(\mu_\theta \nu_\theta) \texttt{ //mixture distribution}\\ \\ \sigma \sim \texttt{HalfNormal}(0, 1) \texttt{ // noise level} \\ \nu_\tau \sim \texttt{exp}(1) \texttt{ // pseudo samples for } \Gamma_\tau \\ \mu_\tau \sim \mathcal{U}(0, 1) \texttt{ // flat prior mean for } \Gamma_\tau = 0\\ \frac{1}{2}(1 + \Gamma_\tau) \sim \beta(\mu_\tau\nu_\tau, (1 - \mu_\tau)\nu_\tau) \texttt{ // } \Gamma_\tau \in (-1, 1)\\ y \sim \sum_{p = 1}^5 \theta_p \texttt{Normal}(y - AR(p, \Gamma_1, \ldots, \Gamma_p), \sigma)

Where AR(p, \Gamma_1, \ldots, \Gamma_p) is the AR model parameterized by \Gamma. In the normal distribution I mean that each sample y(t) is normal with std \sigma and mean generated from the AR coefficients and y(t - 1), \ldots, y(t - p). I eliminated any reference to b to avoid confusion since they’re deterministically generated by \Gamma.

How are you breaking the symmetry between, for example, the AR(2) terms and all the lower order terms of the AR(3) ?

I do not think the model has any symmetries. For component 1, the model is fully determined by \Gamma_1 and for component 2 by (\Gamma_1, \Gamma_2), the only set on which these components have the same likelihood is \Gamma_2 = 0. For AR(3), the only way it has the same likelihood as the AR(2) component is if \Gamma_3 = 0. The sequence of AR models is parameterized only by the sequence \Gamma_1, \ldots, \Gamma_5. This is why I’m unsure about the term “mixture” here, it’s more about “choosing” whether or not to add another coefficient.

Can you generate data from this model with pmax=5 and try and fit it to that?

Yes, in Figure 1. I generated data from p = 3 and in Figure 2 from p = 5. It actually appears to work quite well, I am in awe of stan’s capability. It just doesn’t seem to reflect the posterior certainty for \Gamma in the other parameters \theta and \nu in the way I expected. It looks as if it’s totally certain about the model order based on how concentrated \Gamma posteriors are, but then the values of \nu_1, \ldots, \nu_5 don’t reflect that certainty.

Possibly? Concentrated / disperse are relative terms here, can you plot the posteriors against the prior? Perhaps they have concentrated some amount already? I would expect them to contract very slowly in this model given its complexity.

I see. In Figure 0 I ran the model with only T = 6 samples (as opposed to 1000 from the previous models), which should be similar to sampling from the prior? I’m not sure how to properly setup my code to make it easier to sample straight from the prior without writing an entirely separate model :/

The posteriors on \nu are definitely moving in the direction I would expect, just not with the magnitude that seems reasonable. My thinking is that if \Gamma \sim \beta(\mu, \nu) and \nu \sim \mathrm{exp}(1) then if in the posterior \mathrm{Var}\ \Gamma \approx 0 then we should have \nu very large because the variance of \beta(\mu, \nu) is inversely proportional to \nu. But, this isn’t quite correct, is it? The posterior distributions don’t have to be anything like the prior?

I apologize for making this such a complicated question now. Can you suggest anything that focuses on the core issue of concentration in the posterior (non-hyper)parameters and the posterior of hyperparameters like \nu? I may be getting caught up too much in this particular model.

Figure 0 (prior samples)

Figure 1 (p = 3, T = 1000)

Figure 2 (p = 5, T = 1000)

Are you sure that’s a reasonable prior? Exponential puts a lot of probability mass near zero. When the parameters of Beta are near zero the distribution strongly favors extreme values, in this case \Gamma is likely to be near -1 or 1 even if the mu is near the center. Beta(1,1) (i.e. uniform) or beta(2,2) seems to me more plausible.

There’s a conceptual mistake here. The mixture weights \theta are not posterior probabilities of the components. They are prior probabilities. The posterior probabilities are softmax(lpdfs).
In other words, a priori, the order of the model is a dice roll with probabilities given by \theta. A posteriori, the model is confident that the outcome of that dice roll was 2 – but that doesn’t mean it’s confident the dice is biased! The value of \theta is still uncertain.

By “pseudo-samples” you mean, something like, your a priori confidence in value of \Gamma is as if you had observed \nu_\tau outcomes going certain way? I guess that’s the usual way of explaining beta priors but note that usually \nu_\tau is known (or rather, assumed). In your model the information flows in the opposite direction: you’re using the relatively precisely known \Gamma to estimate the value of \nu_\tau. “Pseudo-samples” doesn’t quite make sense here.

Hyper-parameters are to parameters as parameters are to data. If you have few data points then parameters are uncertain. If you have few parameters then the hyperparameters remain uncertain no matter how certain the non-hyper-parameters are.
Hyperparameters are most useful for hierarchical models where they model the between-group variation. Your model has only one group/time series so there’s no hierarchical structure. That makes the interpretation of the hyperparameters difficult.

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No, I had been trying a few different priors, but I couldn’t see any differences manifested in the posterior. inv-gamma seems preferable for the reason you suggest – avoiding the U shape in the beta distribution.

Your explanations are very helpful. The hierarchy I have is actually kind of pointless if I only have one realization of y(t) right? If there were multiple realizations of a signal, say y^{(1)}(t), \ldots, y^{(N)}(t) (maybe observing the same phenomenon every day for N days) and I modeled each as a mixture of AR(p) models with common hyperparameters, then generating data with the same order would result in the hyperparameters concentrating in the way I’ve been expecting.

Right I see. In my predictive sampling I drew p = categorical_rng(theta) and then sampled data from AR models of those randomly sampled orders, which is totally wrong then. How do you save the evaluation of lpdfs though, the compiler complains about assigning to non-local variables?

You can’t save anything in the model block. Either move it to transformed parameters or duplicate the code in generated quantitites. “Computing lpdfs twice” implies less overhead than you might think because the generated quantitites block

1. doesn’t use automatic differentiation and
2. only executes once per posterior draw, rather than every target evaluation like the model block.
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