Hypothesis Testing

Hypothesis Testing For the purposes of hypothesis testing in the context of the sciences, there are two buckets of hypotheses. Here, the main goal of any good scientific experiment is to discern the likelihood measurements of the experiment are drawn from a statistical model generated by either of the two hypotheses.

These two hypotheses are 1. Null Hypothesis 2. Alternate Hypothesis

The Null Hypothesis is the supposition that the interrogated phenomenon does not occur or that the measurements resulting from the experiment are consistent with random fluctuations in the measurement apparatus.

The Alternate Hypothesis is the supposition that the interrogated phenomenon does occur.

The goal of any good science experiment is to discern the likelihood that measurements generated by the experiment are generated by one hypothesis or the other. In most cases, we can never truly *know*, but we can be *damn sure*.

In the context of photometry and other measures derived from photometric methods (e.g., variation in light curves), the Null Hypothesis is the supposition that the observer only observed noise and not any astrophysical source. The Alternate Hypothesis is therefore the supposition that the astrophysical source was observed.

For any given CCD-like detector, the light frame reported by the detector is the direct sum of a few disparate sources of counts.

$$L_{ij}(\lambda, t) = \biggl(\epsilon_{ij}(\lambda)I_{ij}(\lambda) + T_{ij}\biggr)t + B_{ij}$$

The Light frame ($L_{ij}(\lambda)$; measured in counts) is the sum of:

- The counts resulting from the counts generated by photoelectrons across a heterogeneous detector and optical system and over some integration time ($t$) $\epsilon_{ij}(\lambda)I_{ij}(\lambda)t$ - The counts resulting from the detector and optical system even when no light is incident on the optical system. With no light, the optical system is said to be in-the-dark and so any counts generated from such a system constitute the so-called Dark counts. $D_{ij}(t) = T_{ij}t + B_{ij}$. The Dark counts is the result of thermally agitated electrons finding themselves in the read-out circuitry as well as the relatively constant base-level of electronic noise owing to a variety of sources.