Turing Poisson Example.ipynb

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   "source": [
    "using Turing, Distributions, Plots, Random"
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     "execution_count": 65,
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   ],
   "source": [
    "n=100 # the number of observations\n",
    "\n",
    "# Generate a data set for testing\n",
    "dist = Poisson(3)\n",
    "x = [rand(dist) for i in 1:n]\n",
    "\n",
    "time = plot(1:n, x)\n",
    "hist = histogram(x, normed=true)\n",
    "plot(time, hist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analytic Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Gamma{Float64}(α=3.0, θ=1.0)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Our prior belief about the probability of heads in a coin toss.\n",
    "prior_belief = Gamma(3, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "# Import StatsPlots for animating purposes.\n",
    "using StatsPlots\n",
    "\n",
    "# Animate posterior distribution as number of observations increases.\n",
    "default(display=:inline)\n",
    "for i in 1:n\n",
    "\n",
    "    # Count the number of heads and tails.\n",
    "    count = sum(x[1:i-1])\n",
    "    \n",
    "    # Update our prior belief in closed form (this is possible because we use a conjugate prior).\n",
    "    updated_belief = Gamma(prior_belief.α + count, 1/((1/prior_belief.θ) + i))\n",
    "\n",
    "    # Plotting\n",
    "    plot(updated_belief, \n",
    "        size = (500, 250), \n",
    "        title = \"Updated belief after $i observations\",\n",
    "        xlabel = \"λ\", \n",
    "        ylabel = \"p(λ)\", \n",
    "        legend = nothing,\n",
    "        xlim = (0,10),\n",
    "        fill=0, α=0.3, w=3)\n",
    "    vline!([3.0])\n",
    "end\n",
    "default(display=:none)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Turing Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "poisson_model (generic function with 3 methods)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@model poisson_model(x, n) = begin\n",
    "    \n",
    "    λ ~ Gamma(1,2)\n",
    "    \n",
    "    for i in 1:n\n",
    "        x[i] ~ Poisson(λ)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Found initial step size\n",
      "│   ϵ = 0.2\n",
      "└ @ Turing.Inference /Users/john/.julia/packages/Turing/xGrY5/src/inference/hmc.jl:547\n",
      "\u001b[32mSampling 100%|███████████████████████████████| Time: 0:00:00\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "model = poisson_model(x, n)\n",
    "chain = sample(model, NUTS(0.65), 1000);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{ChainDataFrame,1}\n",
       "\n",
       "Summary Statistics\n",
       ". Omitted printing of 1 columns\n",
       "│ Row │ parameters │ mean    │ std      │ naive_se   │ mcse       │ ess     │\n",
       "│     │ \u001b[90mSymbol\u001b[39m     │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m  │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mFloat64\u001b[39m    │ \u001b[90mAny\u001b[39m     │\n",
       "├─────┼────────────┼─────────┼──────────┼────────────┼────────────┼─────────┤\n",
       "│ 1   │ λ          │ 3.23552 │ 0.172132 │ 0.00769796 │ 0.00206103 │ 362.874 │\n",
       "\n",
       "Quantiles\n",
       "\n",
       "│ Row │ parameters │ 2.5%    │ 25.0%   │ 50.0%   │ 75.0%   │ 97.5%   │\n",
       "│     │ \u001b[90mSymbol\u001b[39m     │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m │\n",
       "├─────┼────────────┼─────────┼─────────┼─────────┼─────────┼─────────┤\n",
       "│ 1   │ λ          │ 2.88478 │ 3.11605 │ 3.24064 │ 3.34866 │ 3.58392 │\n"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "describe(chain)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
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     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(chain[:λ], seriestype = :histogram)"
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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