Add descr to NB2 section 4

Author: Hugo Bowne-Anderson

notebooks/2.Parameter_estimation_hypothesis_testing.ipynb

@@ -495,6 +495,19 @@
     "n_successes_b = np.sum(np.random.uniform(size=N) <= p_b)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Once again, we need to specify our models for $p_a$ and $p_b$. Each will be the same as the CTR example above\n",
+    "- Binomial likelihoods\n",
+    "- uniform priors on $p_a$ and $_p$.\n",
+    "\n",
+    "We also want to calculate the posterior of the difference $p_a-p_b$ and we do so using `pm.Deterministic()`, which specifies a deterministic random variable, i.e., one that is completely determined by the values it references, in the case $p_a$ & $p_b$.\n",
+    "\n",
+    "We'll now build the model:"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -511,6 +524,13 @@
     "    diff_clicks = pm.Deterministic('diff_clicks', prob_a-prob_b)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Sample from the posterior and plot them:"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -533,7 +553,15 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "**Task**: Determine whether the mean beak length of the Galapogas finches differs between species."
+    "**Task**: Determine whether the mean beak length of the Galapogas finches differs between species. For the mean of each species, use the same model as in previous hand-on section:\n",
+    "\n",
+    "- Gaussian likelihood;\n",
+    "- Normal prior for the means;\n",
+    "- Uniform prior for the variances.\n",
+    "\n",
+    "Also calculate the difference between the means and, for bonus points, the _effect size_, which is the difference between the means divided by the pooled standard deviations = $\\sqrt{(\\sigma_1^2+\\sigma_2^2)/2}$. Hugo will talk through the importance of the _effect size_.\n",
+    "\n",
+    "Don't forget to sample from the posteriors and plot them!"
    ]
   },
   {
@@ -545,9 +573,9 @@
     "with pm.Model() as model:\n",
     "    # Priors for means and variances\n",
     "    μ_1 = pm.Normal('μ_1', mu=10, sd=5)\n",
-    "    σ_1 = pm.Lognormal('σ_1', 0, 10)\n",
+    "    σ_1 = pm.Uniform('σ_1', 0, 10)\n",
     "    μ_2 = pm.Normal('μ_2', mu=10, sd=5)\n",
-    "    σ_2 = pm.Lognormal('σ_2', 0, 10)\n",
+    "    σ_2 = pm.Uniform('σ_2', 0, 10)\n",
     "    # Gaussian Likelihoods\n",
     "    y_1 = pm.Normal('y_1', mu=μ_1, sd=σ_1, observed=df_fortis['blength'])\n",
     "    y_2 = pm.Normal('y_2', mu=μ_2, sd=σ_2, observed=df_scandens['blength'])\n",