{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["# Homework 2.3: Nonidentifiability in HIV modeling (35 pts)\n", "\n", "[Data set download](https://s3.amazonaws.com/bebi103.caltech.edu/data/hiv_data.csv)\n", "\n", "<hr>"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Last term, we discussed the nonidentifiability of a model for HIV clearance with a single patient's data. You should refresh yourself about the model by reading that lecture up to the \"The statistical model\" section of [that lecture from last term](https://bebi103a.github.io/lessons/27/nonidentifiable.html#The-statistical-model). You can access the data set here: [https://s3.amazonaws.com/bebi103.caltech.edu/data/hiv_data.csv](https://s3.amazonaws.com/bebi103.caltech.edu/data/hiv_data.csv). It consists of $(t_i, V_i)$ pairs, where $t_i$ is the time point and $V_i$ is the viral load in the patient's blood.\n", "\n", "Similarly to what we did last term, we will take the likelihood to be a Normal distribution with a location given by the theoretical curve.\n", "\n", "\\begin{align}\n", "&\\mu_i = V(t_i;V_0,c,\\delta) = V_0e^{-ct_i} + \\frac{cV_0}{c-\\delta}\\left[\\frac{c}{c-\\delta}(e^{-{\\delta}t_i} - e^{-ct_i}) - {\\delta}t_ie^{-ct_i}\\right] \\;\\forall i, \\\\[1em]\n", "&V_i \\sim \\text{Norm}(\\mu_i, \\sigma) \\;\\forall i.\n", "\\end{align}\n", "\n", "There are four parameters in this model: the initial viral load, $V_0$; the clearance rate due to cell death, $\\delta$; the clearance rate due to the Ritonavir drug $c$, and the scale parameter of the Normal distribution $\\sigma$. We will specify priors for all, and will be completely uninformative for our purposes here.\n", "\n", "\\begin{align}\n", "&g(V_0, c, \\delta) = \\text{constant},\\\\[1em]\n", "&g(\\sigma) = 1/\\sigma.\n", "\\end{align}\n", "\n", "**a)** Write down an expression proportional to the marginalized posterior $g(V_0, c, \\delta \\mid \\{t_i, V_i\\})$. You can do this analytically using the fact that\n", "\n", "\\begin{align}\n", "\\int_0^\\infty \\frac{\\mathrm{d}\\sigma}{\\sigma^{n+1}}\\,\\mathrm{e}^{-x/2\\sigma^2} \\propto x^{-n/2}.\n", "\\end{align}\n", "\n", "**b)** Compute the marginalized log posterior $g(c, \\delta)$ on the domain $c, \\delta \\in [0, 10]$ days\u207b\u00b9. You will have to use numerical quadrature to marginalize out $V_0$. You will need to perform this numerical quadrature for each *pair* of $(c, \\delta)$ values for which you wish to evaluate the marginalized posterior.\n", "\n", "**c)** Plot the marginalized posterior $g(c, \\delta)$ and comment on how the plot of the posterior makes clear the nonidentifiability of the model.\n", "\n", "**d)** What additional piece of prior information might make the model identifiable?\n", "\n", "<!--I also did this analysis where the variance scaled with the signal, but the posterior gives a much thinner range, and the students might be confused by this. -->"]}, {"cell_type": "markdown", "metadata": {}, "source": ["<br />"]}], "metadata": {"anaconda-cloud": {}, "kernelspec": {"display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3"}, "language_info": {"codemirror_mode": {"name": "ipython", "version": 3}, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.7"}}, "nbformat": 4, "nbformat_minor": 4}