There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow this rule:
$a_i=1-F(i)$ while $F(i)=e^{-\lambda*i}$. $F(i)$ is the cdf of the exponential distributions.
There are some restriction for the three coefficients:$1>a_1>a_2>a_3>0$.
Now the story is: we have a dataset $Z_t$ which is a survival dataset or time series data that can be used to estiamte the three coefficients. Let's assume that we can estimate them and obtain the prior: $\hat{a}_i=1-\hat{F}(i)=e^{-\hat{\lambda}*i}$ . When we estimate those coefficients, we use the data which is discrete time data. For example, i=1,2, and 3. So we just need to estimate $a_i$ with $i=1,2,3$ by using the discrete time data. So $\hat{a}_1,\hat{a}_2,\hat{a}_3$ are estimators and each of them has their own variance (you can thing $\hat{a}_1$ with i=1,2,3 follow normal distributions asymptotically with mean=E(1-F(i)), $Var=\sigma_i^2$. We can also use the discrete time datasect $Z_t$ to calculate $\sigma_i^2$ with i=1,2,3)
After that, we define some new estimators say $\hat{\beta}_i=\frac{\hat{a}_i}{\hat{a}_1+\hat{a}_2+\hat{a}_3}$ with i=1,2,3. We have another four time series dataset:$Y_{t}, X_{1,t},X_{2,t},X_{3,t}$.
$Z_t$ is different from $Y_{t}, X_{1,t},X_{2,t},X_{3,t}$.
We also assume that there is a theorem (we call it as equation (1.1)):
$Y_t=\beta_1 X_{1,t}+\beta_2 X_{2,t}+\beta_3 X_{3,t}$
Where ${\beta}_1+{\beta}_2+{\beta}_3=1$ and $1>{\beta}_1>{\beta}_2>{\beta}_3>0$.
Our propuse is to draw $\hat{\beta}_{i,bayes}$ from equation (1.1) by appliyng the prior of $\hat{\beta}_i$. After that, we compare $\hat{\beta}_{i,bayes}$ with $\hat{\beta}_{i}$ which is directly calculated from the data $Z_t$. If equation (1,1) is correct, $\hat{\beta}_{i,bayes}$ should be close to $\hat{\beta}_{i}$.
Note that $\hat{\beta}_{i,bayes}$ must satisfy the two restrictions (sum to 1 and decrese with i).
That is my question. Any ideas?
For myself, I thought Dirchlet distribution may satisfy one condition (summation equals to 1). But Dirchlet distribution assume that $\hat{\beta}_{i,bayes}$ follows Gamma disribution. I also thought I can add some constraint to the bayesian method or use lasso method. But I think they may not work(?).
