I am trying to go from a functional relationship between a predictor variable and a probability of an event occurring ($P = f(X)$) to an overall rate at which the event occurs. My scenario is estimating annual rates of fish escapement from a reservoir given a relationship between daily mean discharge $Q$ and probability of a fish escaping given said discharge, i.e. $P = f(Q)$. Let's assume that $P = f(Q)$ is a discrete relationship, i.e. I have discretized the continuous function $f(Q)$.
I am getting very low values for my annual escapement rate compared to what empirical data suggests (~3 % vs over 13%), so I am concerned that my methodology for upscaling from daily to annual escapement probability is flawed.
My original thinking for calculating the annual probability of escapement is the formula
$$
P_{annual} = \sum_i 1 – [1 – P(Q_i)]^{d_i}
$$
where $Q_i$ is a given discharge from the discretization of $f(Q)$ and $d_i$ is the total number of days in a year that reservoir discharge was held at that rate. Without any better information and given a fairly large fish population in the reservoir, I assume that the annual rate of escapement is equal to the annual probability of escapement, i.e.
$$
N_{escaped} = N*P{annual}
$$
where $N$ is the starting population of fish that year and $N_{escaped}$ is the number of fish that escaped that year.
Looking around the site, there seems to be some debate on whether my definition of $N_{escaped}$ is acceptable, and I'm not sure I am correctly defining $P_{annual}$ or that it doesn't translate to $N_{escaped}$ the way I have defined. Can any probability gurus clarify things for me?
EDIT
Thanks to @Creosote's answer the correct formulation of the annual escapement probability should actually be
$$
P_{annual} = 1 – \prod_i \left[1 – f(Q_i) \right]^{d_i}
$$
However, this changes my results on the order of 0.01%, so my problem is not solved (or perhaps the issue is not with the formulation). I obtained the discharge-escapement curve from this paper. The 13% annual escapement estimate for years 2009–2011 from the paper is based on mark-recapture studies, and the curve (with 95% confidence interval) is based on a multi-state model the authors generated from the same data (I have obtained the curve generated by the discharge-only model, which predicts slightly higher escapements that the figure in the paper). I use the reservoir release record to calculate the number of days that a release is within a given "flow band", e.g. the number of days in the year the flow is between 1000 cfs and 1250 cfs. The flow bands are actually defined such that the bounds of each band span a change in the escapement probability of 0.005%.
I don't get much above 4% for any of the years even when using the upper confidence limit of the curve. I recognize that there should be some discrepancy between the two results, but I would actually expect my escapement rates to be higher because the model accounts for tag loss and mortality. Any ideas?
EDIT 2
I'm accepting Creosote's answer because it provides the correct way to scale up from daily to annual probability. EdM's answer was also very helpful in pointing out that the issue probably has more to do with the underlying the escapement-discharge model than with the upscaling procedure. I'll post another edit if I get more information on my problem, but the daily–>annual probability question seems solved.
Best Answer
A randomly-chosen fish on a randomly-chosen day with discharge $Q_i$ will remain in place with probability $1-f(Q_i)$. So it'll remain in place for the whole year with probability $\prod_i \left( 1-f(Q_i) \right)^{d_i}$, and so your $P_\textrm{annual}$ is one minus that. Assuming I've understood your notation, that is.