Regularising yearly prevalence estimates using hierarchical model (time series) Say I'm attempting to get the best possible estimates for the prevalence of drug use in the US in each of the past 5 years.  
I could build a model with either a Bayesian logistic regression model with either a time index or time trend variable included. However, each of these approaches seems sub-optimal to me:
Yearly estimates using the time index model doesn't give use any information from the underlying time trend, so it invariably overfits.
Yearly estimates from the time trend model ignores idiosyncratic differences in prevalence from year to year, so invariably underfits.
Is there a way to use some sort of hierarchical multi-level model where what we essentially do is set the prior of the coefficients associated with each year to what would be expected if prevalence followed a linear trend over time. That way, yearly estimates could be regularised towards the linear trend. 
Is there anything inherently wrong with this approach? 
And, would simply including both a time trend and time index in the same model accomplish the same task?  
Models
Time Index
used_drugs ~ binomial(1,p),
logit(p) = a[year],
a[year] ~ normal(0,1.5) #Note: numbers are just examples of priors 
Time Trend
used_drugs ~ binomial(1,p),
logit(p) = a + b*year,
a ~ normal(0,1.5), 
b ~ normal(0,0.5)
Time Index and Time Trend (non-hierarchical) 
used_drugs ~ binomial(1,p),
logit(p) = a[year] + b*year,
a[year] ~ normal(0,1.5), 
b ~ normal(0,0.5)
Time Index and Time Trend (hierarchical) Note: I'm not sure if this is correct.
used_drugs ~ binomial(1,p),
logit(p) = a[year],
a[year] ~ normal(mu,sigma),
mu = a + b*year,
a ~ normal(0.1.5)
b ~ normal(0,0.5)
 A: Interesting topic, I think the intuition that you'd need something hierarchical is very correct. There are various approaches from this idea forward I think. 
The yearly coefficients are not very useful, because a sudden change from December 31st to January 1st doesn't make sense in this case. Most approaches will use separate variables for the days of the week, allthough for you, you might just sum over the weeks because that timescale is probably not of interest to you. 
Which means you want to "explain" weekly totals with a trend over a period of five years, with presumably some yearly seasonal variation. 
For the long term trend, you are proposing a basically linear trend. This could be a good idea, but there are many alternatives. For example, you might want to use some logistic growth model https://facebook.github.io/prophet/docs/saturating_forecasts.html#forecasting-growth, or try and detect "changepoints", between which there is a linear trend that might be increasing or decreasing. Specifying and fitting such a model is not easy, particularly with the changepoints. A logistic growth model is actually not appropriate, because you are modelling probabilities already, and are kind of assuming a logistic growth model actually, but I'm just coming up with some examples here. 
For the yearly rhythm,


*

*The prophet algorithm / library / model uses partial Fourier sums, see https://facebook.github.io/prophet/docs/seasonality,_holiday_effects,_and_regressors.html#fourier-order-for-seasonalities

*You can use splines; see "Bayesian splines" on Google,

*A more theoretical approach is to use Gaussian Processes
I'd start with the approach the prophet library is using. 
Note that if you are only interested in the actual trend, you might even leave out the yearly seasonality because they might not influence the predictions of the overall trend coefficients much. 
