It's a Friday night, so a good time to derive the likelihood function of a simple model to estimate the death rate from an illness that kills quickly but takes a long time to cure. Buckle up!
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log_sum_exp(a, b) = log(exp(a) + exp(b)). So log(θ*(1 - F(d|t))+ (1 -θ)*(1-F(c|t))) = log_sum_exp(log(θ) + log(1 - F(d|t)), log(1-θ) + log(1 - F(c|t))). Beautiful! Two more challenges: what distributions for f(d|t) and f(c|t); and what about low reporting?
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On dist choice: one nice fact is that a mixture of normal densities has a mixture of normal CDF. So you can use mixtures of normals for your f()s to capture most of the weirdness going on in unusual arrival time distributions, while staying analytically convenient.
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On reporting rates--this is a real challenge! I'm sure there are clever people who do this for a living, but I'd use plug-in rates for sensitivity checking. It just doesn't seem like a easy thing to estimate. Have a great weekend!
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