Interpreting Jumps in fit.plot()

Modeling
#1

I am new here, but am working on a model of word frequencies (see model below for reference). For certain words, some rhat values in the fit object are high. When I inspect the plots of the fit object, I see “jumps” occurring (see below). Can anyone explain these jumps? What causes them, how do I interpret them, do they indicate a poorly specified model, etc?
Thanks!

fig15
fig15.png3000×2000 1.14 MB

model:
“”“STAN”"

data {
    int N_DAYS;
    int<lower=0> cnt[N_DAYS];
    int<lower=0> z[N_DAYS];
}

transformed data {
}

parameters {
   
   real<upper=0> daily_lp[N_DAYS]; // each day's probability
   
   real          alpha;
   real          mu;
   real<lower=0> stdev; 
   real<lower=0> drift;
   
   real          A;
   
   real<lower=0,upper=1> theta; // the weight between the two gaussians
}
    
transformed parameters {
}
    
model {    
    mu     ~ cauchy(0,1);
    stdev  ~ cauchy(0,1);
    alpha  ~ cauchy(0,1);
    drift ~ cauchy(0,1);
    theta ~ beta(1,1);
    
    A ~ normal(0,1);
    
    
    target += skew_normal_lpdf(daily_lp[1] | mu, stdev, alpha);
    target += skew_normal_lpdf(daily_lp[2] | mu, stdev, alpha);
    for(d in 3:N_DAYS) {
	real daymean = mu + A*(d/N_DAYS-0.5);
        target += log_sum_exp( log(theta)   + skew_normal_lpdf(daily_lp[d] | daymean, stdev, alpha),
                                       log(1-theta) + normal_lpdf(daily_lp[d] | daily_lp[d-1], drift) );
    }

    // treat as a binomial sample
    for(i in 1:N_DAYS){
        target += cnt[i]*(daily_lp[i]) + (z[i]-cnt[i])*log1m(exp(daily_lp[i]));
    }   
}
#2

It looks like the distributions of some parameters are multi-modal. The jumps show the Markov chain jumping between these modes. In general you don’t want multimodality, but sometimes it’s inevitable (i.e. it’s a legitimate feature of the model/posterior).

There are smarter people around that can perhaps suggest a reparametrisation – I have not analysed the model in any detail.

#3

That’s indeed the usual explanation. And indeed, this is a mixture model, the usual culprit.

What you want to do is check out @betanalpha’s case study on mixture models. It explains how to solve the problems you’re having by ordering the parameters. But it also introduces some other problems you might have to be careful with in a mixture model.