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“# Core Conceptn”, “popsynth core function is to create observed surveys from latent population models. n”, “n”, “First, let’s define what a population of objects is in terms of an”, “generative model. The two main ingredients are the objects’ spatialn”, “distribution ($\lambda(\vec{r}; \vec{\psi})$) and the distribution ofn”, “their inherent properties ($\pi(\vec{\phi} | \vec{\psi})$). Here,n”, “$\vec{\psi}$ are the latent population parameters, $\vec{r}$ are then”, “spatial locations of the objects, and $\vec{\phi}$ are the propertiesn”, “of the individual objects (luminosity, spin, viewing angle, mass,n”, “etc.). The spatial distribution is defined such that:n”, “n”, “$$\frac{d \Lambda}{dt}(\vec{\psi}) = \int d r \frac{dV}{dr} \lambda(\vec{r}; \vec{\psi})$$n”, “n”, “is the intensity of objects for a given set of populationn”, “parameters. With these definitions we can define the probability forn”, “an object to have position $\vec{r}$ and properties $\vec{\phi}$ asn”, “n”, “$$\pi(\vec{r}, \vec{\phi} | \vec{\psi}) = \frac{\lambda(\vec{r}; \vec{\psi}) \pi(\vec{\phi} | \vec{\psi})}{ \int d r \frac{dV}{dr} \lambda(\vec{r}; \vec{\psi})} $$n”, “n”, “popsynth allows you to specify these spatial and propertyn”, “distributions in an object-oriented way to create surveys. The finaln”, “ingredient to creating a sample for a survey is knowing how manyn”, “objects to sample from the population (before any selection effectsn”, “are applied). Often, we see this number in simulation frameworksn”, “presented as “we draw N objects to guarantee we have enough.” This isn”, “incorrect. A survey takes place over a given period of time ($\Deltan”, “t$) in which observed objects are counted. This is a description of an”, “Poisson process. Thus, the number of objects in a simulation of thisn”, “survey is a draw from a Poisson distribution:n”, “n”, “$$N \sim \mathrm{Poisson}\left(\Delta t \frac{d\Lambda}{dt}\right) \mathrm{.}$$n”, “n”, “Thus, popsynth first numerically integrates the spatialn”, “distribution to determine the Poisson rate parameter for the givenn”, “$\vec{\psi}$, then makes a Poisson draw for the number of objects inn”, “the population survey. For each object, positions and properties aren”, “drawn with arbitrary dependencies between them. Finally, selectionn”, “functions are applied to either latent or observed (with or withoutn”, “measurement error) properties.”

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“Note: If instead we draw a preset number of objects, as is done inn”, “many astrophysical population simulation frameworks, it is equivalentn”, “to running a survey up until that specific number of objects isn”, “detected. This process is distributed as a negative binomial process,n”, “i.e, wait for a number of successes and requires a differentn”, “statistical framework to compare models to data.n”, “n”, “In the following, the process for constructing distributions andn”, “populations is described.”

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