a pass of hmc docs and comments (#60)

* separate hmc utils

* temporary save

* revise docs

* update docs for remaining utils

* change returns to return

* add missing dot

notebooks/time_series_forecasting.ipynb

@@ -118,27 +118,29 @@
    "source": [
     "The model we are going to use is called **Seasonal, Global Trend**, which when tested on 3003 time series of the [M-3 competition](https://forecasters.org/resources/time-series-data/m3-competition/), has been known to outperform other models originally participating in the competition:\n",
     "\n",
-    "$$\n",
     "\\begin{equation*}\n",
+    "\\begin{gathered}\n",
     "\\text{exp_val}_{t} = \\text{level}_{t-1} + \\text{coef_trend} \\times \\text{level}_{t-1}^{\\text{pow_trend}}\n",
     "+ \\text{s}_t \\times \\text{level}_{t-1}^{\\text{pow_season}}, \\\\\n",
     "\\sigma_{t} = \\sigma \\times \\text{exp_val}_{t}^{\\text{powx}} + \\text{offset}, \\\\\n",
-    "y_{t} \\sim \\text{StudentT}(\\nu, \\text{exp_val}_{t}, \\sigma_{t}), \\\\\n",
+    "y_{t} \\sim \\text{StudentT}(\\nu, \\text{exp_val}_{t}, \\sigma_{t}),\n",
+    "\\end{gathered}\n",
     "\\end{equation*}\n",
-    "$$\n",
+    "\n",
     "where `level` and `s` follows the following recursion rules:\n",
-    "$$\n",
+    "\n",
     "\\begin{equation*}\n",
+    "\\begin{gathered}\n",
     "\\text{level_p} =\n",
     "  \\begin{cases}\n",
     "  y_t - \\text{s}_t \\times \\text{level}_{t-1}^{\\text{pow_season}} & \\text{if } t \\le \\text{seasonality}, \\\\ \n",
     "  \\text{Average} \\left[y(t - \\text{seasonality} + 1), \\ldots, y(t)\\right] & \\text{otherwise},\n",
     "  \\end{cases} \\\\\n",
     "\\text{level}_{t} = \\text{level_sm} \\times \\text{level_p} + (1 - \\text{level_sm}) \\times \\text{level}_{t-1}, \\\\\n",
     "\\text{s}_{t + \\text{seasonality}} = \\text{s_sm} \\times \\frac{y_{t} - \\text{level}_{t}}{\\text{level}_{t-1}^{\\text{pow_trend}}}\n",
-    "+ (1 - \\text{s_sm}) \\times \\text{s}_{t}. \\\\\n",
-    "\\end{equation*}\n",
-    "$$"
+    "+ (1 - \\text{s_sm}) \\times \\text{s}_{t}.\n",
+    "\\end{gathered}\n",
+    "\\end{equation*}"
    ]
   },
   {
@@ -389,8 +391,8 @@
    "metadata": {},
    "source": [
     "Given `samples` from `mcmc`, we want to do forecasting for the testing dataset `y_test`. First, we will make some utilities to do forecasting given a sample. Note that to retrieve the last `level` and last `s` value, we substitute a sample to the model:\n",
-    "```\n",
-    "...    level, s = substitute(sgt, asample)(y, seasonality)`.\n",
+    "```python\n",
+    "...    level, s = substitute(sgt, asample)(y, seasonality)\n",
     "```"
    ]
   },
@@ -531,7 +533,7 @@
    "source": [
     "## References\n",
     "\n",
-    "[1] `Rlgt: Bayesian Exponential Smoothing Models with Trend Modifications`,</br>&nbsp;&nbsp;&nbsp;&nbsp;\n",
+    "[1] `Rlgt: Bayesian Exponential Smoothing Models with Trend Modifications`,<br>&nbsp;&nbsp;&nbsp;&nbsp;\n",
     "Slawek Smyl, Christoph Bergmeir, Erwin Wibowo, To Wang Ng, Trustees of Columbia University"
    ]
   }

numpyro/diagnostics.py

@@ -80,7 +80,7 @@ def autocorrelation(x, axis=0):
     """
     Computes the autocorrelation of samples at dimension ``axis``.
 
-    :param numpy.array x: the input array.
+    :param numpy.ndarray x: the input array.
     :param int axis: the dimension to calculate autocorrelation.
     :return: autocorrelation of ``x``.
     :rtype: numpy.ndarray