2. Testing Anticancer Drugs (70 points) Consider the following data from an imagined study. A population of cancer cells is cultured in a petri dish and treated with a drug candidate. The number of healthy cells in the dish is recorded at regular time intervals of 7 = 2 hr (Table 1). In addition to this assay, a control experiment is performed without addition of the drug candidate. To model the data, we assume that a healthy cell will survive the time interval 7 with a constant probability 0. Equivalently, there is a constant probability 1 0 that a healthy cell will die during the interval 7. Given the data, we aim to estimate the survival probabilities a and c for the assay and the control. n1 n2 assay 42 23 13 control 38 29 21 Table 1: Number of viable cells {n} at time points tk = k for k = 0, 1, 2. Cells in the assay are treated with the drug candidate; cells in the control are not. (a) Assuming a uniform prior for 0, derive an expression for the posterior distribution p(0 | {n}) to within a multiplicative constant. (b) Derive an expression for the mode 0 of the posterior distribution. (c) Derive an expression for the error-bar on your parameter estimate. (d) Using the data in Table 1, estimate the survival probability a and c for the assay the control; express your answers in the form 0 = 0 . Do you think that the drug has a significant effect on cell survival? Explain why or why not. and (e) The survival probability 0 for a time interval 7 is related to the cell death rate \ as 0 = e. Derive an expression for the posterior distribution p(\ | {n}) to within a multiplicative constant. (f) Based on your analysis of the assay data, predict how many healthy cells will remain at t = 8 hr (i.e., 4 hrs after the observation n2).