Problems in connecting two different distributions with each other for a regression of Markov model.

[Jan Münch]

Hey Stan Team and discussion participians,

one of my toy models for inferencing Markov Models of Ion channels Looks like this. (See below)

It works fine. (As long as I exclude probabilties = 0 or 1 for the binominal Distribution. The derivative of the log probabilty seems to be infinit there.?)

But in the real data I not going to see y_t but some y_gaus ~ normal(y_t, sigma). Are finitie mixtures here the right way to go? Are there better ways?

My data will be something like

p(t) = 1- exp(-theta*t)+....

n_open ~ Binomial(N_channel, p(t))

I ~ normal(Const * n_open, sigma)

I would like to infer p(t) via Theta  (or more complex Sums of exponentials)  and n_open and N_channel even though they are discrete. The const will probably come from different Data.

Thanks alot for any hints.

Jan Münch

data {

    int<lower=1> N_data;                                 // number of data points
    int<lower=0, upper = 1000> y_t[N_data];                // array observations ever element bigger als 0 to 20 binomainal draw(20, theta)
    real<lower=0, upper =  10> time[N_data];
    int N_channel;
}

transformed data{                                   // ... declarations ... statements ...
}

parameters {                                        // The parameters we want to inference by via Stan
    simplex[2] mu;
    real<lower = 0, upper = 5> theta[2];
}

transformed parameters {                            // ... declarations ... statements ...


     real<lower = 0, upper = 1> probabilty[N];       // fitted values

    for(i in 1:N)
        probabilty[i] = 1 - mu[1] * exp(-theta[1] * time[i])- mu[2]* exp(-theta[2]*time[i]);


}

model {
    theta[1] ~ uniform(0.0,3);
    theta[2] ~ uniform(2.0,5);
    y_t ~ binomial(N_channel, probabilty);
    //y_gaus    ~ normal(y_t,sigma);

}
generated quantities {                              // ... declarations ... statements ...


}
"""