Sample from the product of the normal distribution and the cumulative normal distribution

Modeling
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1

Dear Stan developers,

Thanks for developing this fantastic program.
I am implementing MCMC for computing the posterior distribution of parameters of a cognitive model.

The data structure is like follows:

trial stimulus choice estimation
1     1        1      0.9
2     -2       0      -1.3
3     -3       0      -3.1
4     2        0      2.2

On each trial, a stimulus is shown and subjects were required to make a binary choice to decide whether the stimulus is smaller (choice=0) or larger than 0 (choice=1).
Subsequently, subjects should estimate the magnitude of the stimulus in the continuous dimension.

The likelihood of parameters given the data is the product of the normal pdf and the normal cdf:
Likelihood[trial] = normal(...)*normal_pdf(...).

An error occurred when I just coded the sampling process by the product of normal and normal_pdf.
Can Stan implement MCMC with that likelihood?

Thanks!

2

If the likelihood is the product of the normal pdf and cdf, then you can use the sum of the normal log pdf and log cdf:

 target += normal_lpdf(y | mu, sigma) + normal_lcdf(y | mu, sigma)
3

Thanks for the reply!
I am sorry that I might mislead the problem.

My target is a continuous estimation (e.g., -3, 0, or 1).
And the target is sampled from the product of the normal pdf and cdf.
So, I want to implement a code that samples the target from the product of the normal pdf and cdf:
target[t] ~ normal(mu1,sigma1)*normal_pdf(mu2,sigma2)

By the way, should I use += instead of ~?
I am unfamiliar with +=.
Does += indicate sampling?

4

Because you are specifying the likelihood as a function, rather than a distribution, then you need to use the target += notation.

The target += notation allows you specify the log-likelihood directly, rather than one of the built-in distributions. The Functions Reference for each distribution shows the log-probability function that is used. More information is in this section.

For example, when you call:

y ~ normal(mu, sigma);

This is the same as:

target += normal_lupdf(mu, sigma)

Because you want the likelihood to be:

y ~ normal(mu1,sigma1)*normal_cdf(mu2,sigma2)

This can be specified using the log-probability functions directly:

target += normal_lpdf(y|mu1, sigma1) + normal_lcdf(y|mu2, sigma2)
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5

What a kind reply!
I understood the situation.
Very thanks!!!

2 Likes
7

This is proportional to a skew normal distribution with shape parameter α = 1 if the location and scale parameters are the same. Would it be preferable to use skew_normal_lpdf or one of the variants directly?

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