Note: based on suggestions in the comments, I have rewritten this question. Please refer to the history for the original version.

In general my question regards how to compute likelihoods in mixed cases with both probabilities and probability densities.

Here's how I collect the data. I have a behavioral experiment involving (human) participants, who in each of $N$ trials perceive a stimulus of intensity $i\in [-1;1]$. In response to this stimulus intensity, participants in every trial make a binary choice $c$ ($0$ or $1$) and a continuous rating $r\in[0;1]$.

To account for these data, I have a model, which in each trial takes $i$ as input and internally computes a hidden variable $x=f(i,\theta)$. (*)

Based on $x$ and some to-be-optimized model parameters, I can in each trial compute the likelihood for the actual participant's choice $c$ through a sigmoid/softmax function of $x$:

$p(c)=c-{2\cdot(c-0.5)\over1+e^{-\beta \cdot x}}$

The likelihood of the continuous rating is given by a normal distribution (probability density) with mean $\lambda\cdot|x|$ and standard deviation $\sigma$ for $0<r<1$. For the special cases $r=0/1$, the probability is the area under the normal in the range $]-\infty;0]$ for $r=0$, and $[1;\infty[$ for $r=1$, respectively:

\begin{equation} p(r)=\begin{cases} \mathcal{N}(r,\lambda|x|,\sigma), & \text{if}\hspace{5pt}0<r<1\\ \Phi(0,\lambda|x|,\sigma), & \text{if}\hspace{5pt}r=0\\ 1-\Phi(1,\lambda|x|,\sigma), & \text{if}\hspace{5pt}r=1 \end{cases} \end{equation}

with $\mathcal{N}($variable,mean,std$)$ being the normal distribution and $\Phi($variable,mean,std$)$ being the CDF of the normal distribution.

Note that $p(c)$ and $p(r)$ can be assumed independent conditional on $x$.

My question: what is the mathematically correct way to calculate the combined likelihood of $p(c)$ and $p(r)$? My goal is to use this likelihood to perform maximum likelihood estimation of the model parameters $\beta,\lambda,\sigma,\theta$, such that the model becomes maximally predictive of behavior.

(*) i don't know whether the specifics of the function $f$ are important. It's a linear function, however with a nonstationary slope (let me know if more information is required).

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    $\begingroup$ Hello. Not clear to me. Do you mean they are two type of observations in your data, either a $0/1$ variable assumed to be generated from a Bernoulli distribuion or a continuous variable assumed to be generated from a Gaussian distribution ? $\endgroup$ – Stéphane Laurent Dec 5 '14 at 9:58
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    $\begingroup$ Do you know in advance the type of an observation ? Or is it random ? $\endgroup$ – Stéphane Laurent Dec 5 '14 at 12:06
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    $\begingroup$ Ok, if I understand these are paired observations, one binary and the other one continuous ? Are they assumed to be independent ? $\endgroup$ – Stéphane Laurent Dec 5 '14 at 13:14
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    $\begingroup$ You should state your model, formally, with equations. Then I am sure we will spot your problem! $\endgroup$ – kjetil b halvorsen Dec 5 '14 at 17:49
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    $\begingroup$ A worked example of working with likelihoods involving both densities and probabilities is posted at stats.stackexchange.com/questions/49443/…. Your actual problem is unclear: what do you mean by "1st observation" and "2nd observation"? Why do they have no parameters in common? It sounds like you might be struggling with writing down the likelihood, but for us to help you with that you will need to indicate exactly what your probability model for the data is. $\endgroup$ – whuber Dec 5 '14 at 21:58

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