Hello, I did reduce_sum before following the wonderful case study tutorial. In that example, the likelihood can be vectorized. Is it still possible to use reduce_sum when the likelihood cannot be vectorized? Specifically, I can use log_mix to run a 2-component finite mixture model. But the likelihood cannot be vectorized (as far as I know). Thanks.

Not being able to vectorize does not mean you canâ€™t use reduce sum. Can you write your log likelihood as a big sum? If yes, then things are promisingâ€¦if notâ€¦then maybe itâ€™s the sum over a few big sub sums?

Itâ€™s about big and costly sums.

Thank you for your reply. The finite mixture log-likelihood for 2 classes is like this:

```
for (n in 1:N) {
target += log_mix(lambda,
normal_lpdf(y[n] | mu[1], sigma[1]),
normal_lpdf(y[n] | mu[2], sigma[2]))
};
```

A for-loop is needed and canâ€™t be vectorized (according to the Stan manual). If there are more than 2 classes, there will be another for-loop instead of log_mix.

So I donâ€™t know how one can express this log-likelihood as a big sum or a few big sub sums.

This for-loop is a big sum over `N`

terms. Thatâ€™s what the `+=`

is doing; itâ€™s summing another term into the target for each iteration of the loop.

Oh. Thanks for pointing me to the right direction. Would this work?

```
functions {
real partial_sum_ll(real[] y_slice,
int start, int end,
vector mu,
vector sigma,
real lambda) {
sum=0;
for (i in start:end) {
sum += log_mix(lambda,
normal_lpdf(y_slice[i] | mu[1], sigma[1]),
normal_lpdf(y_slice[i] | mu[2], sigma[2]))
};
return sum;
}
```

Almost. The indexing is wrong. Maybe have a look at the within chain parallelization case study?

Could you tell me where I was wrong?

I read the tutorial again.

It seems to me I should just loop from *start* to *end*.

The y slice runs from 1 to end - start + 1.

Oh, I get it now. Thank you very much!