02-Chapter 2. Small Worlds and Large World.ipynb

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   "source": [
    "using Distributions\n",
    "using StatsPlots\n",
    "using Turing\n",
    "using Logging\n",
    "using LaTeXStrings\n",
    "\n",
    "default(labels=false)"
   ]
  },
  {
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  {
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   "id": "98d6af68",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.0, 0.15, 0.4, 0.45, 0.0]\n"
     ]
    }
   ],
   "source": [
    "ways = [0, 3, 8, 9, 0]\n",
    "ways = ways ./ sum(ways)\n",
    "println(ways)"
   ]
  },
  {
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   "source": [
    "Code 2.2"
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     "execution_count": 3,
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     "output_type": "execute_result"
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   ],
   "source": [
    "b = Binomial(9, 0.5)\n",
    "pdf(b, 6)"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "Code 2.3"
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   "source": [
    "# size of the grid\n",
    "size = 20\n",
    "\n",
    "# grid and prior\n",
    "p_grid = range(0, 1; length=size)\n",
    "prior = repeat([1.0], size)\n",
    "\n",
    "# compute likelihood at each value in grid\n",
    "likelyhood = [pdf(Binomial(9, p), 6) for p in p_grid]\n",
    "\n",
    "# compute product of likelihood and prior\n",
    "unstd_posterior = likelyhood .* prior\n",
    "\n",
    "# standardize the posterior, so it sums to 1\n",
    "posterior = unstd_posterior / sum(unstd_posterior);"
   ]
  },
  {
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   "source": [
    "Code 2.4"
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   "cell_type": "code",
   "execution_count": 5,
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      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(p_grid, posterior; \n",
    "    xlabel=\"probability of water\", \n",
    "    ylabel=\"posterior probability\",\n",
    "    title=\"$size points\",\n",
    "    markershape=:circle\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "901303b3",
   "metadata": {},
   "source": [
    "Code 2.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "7e765e97",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "size = 20\n",
    "p_grid = range(0, 1; length=size)\n",
    "\n",
    "# prior is different - 0 if p < 0.5, 1 if >= 0.5\n",
    "prior = convert(Vector{AbstractFloat}, p_grid .>= 0.5)\n",
    "\n",
    "# another prior to try (uncomment the line below)\n",
    "# prior = exp.(-5*abs.(p_grid .- 0.5))\n",
    "\n",
    "# the rest is the same\n",
    "likelyhood = [pdf(Binomial(9, p), 6) for p in p_grid]\n",
    "unstd_posterior = likelyhood .* prior\n",
    "posterior = unstd_posterior / sum(unstd_posterior);\n",
    "\n",
    "plot(p_grid, posterior; \n",
    "    xlabel=\"probability of water\", \n",
    "    ylabel=\"posterior probability\",\n",
    "    title=\"$size points\",\n",
    "    markershape=:circle\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "676955e6",
   "metadata": {},
   "source": [
    "Code 2.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "8f264c8e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Chains MCMC chain (1000×13×1 Array{Float64, 3}):\n",
       "\n",
       "Iterations        = 501:1:1500\n",
       "Number of chains  = 1\n",
       "Samples per chain = 1000\n",
       "Wall duration     = 6.05 seconds\n",
       "Compute duration  = 6.05 seconds\n",
       "parameters        = p\n",
       "internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size\n",
       "\n",
       "Summary Statistics\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m    mean \u001b[0m \u001b[1m     std \u001b[0m \u001b[1m naive_se \u001b[0m \u001b[1m    mcse \u001b[0m \u001b[1m      ess \u001b[0m \u001b[1m    rhat \u001b[0m \u001b[1m e\u001b[0m ⋯\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  \u001b[0m ⋯\n",
       "\n",
       "           p    0.6370    0.1436     0.0045    0.0071   407.9428    1.0023     ⋯\n",
       "\u001b[36m                                                                1 column omitted\u001b[0m\n",
       "\n",
       "Quantiles\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m    2.5% \u001b[0m \u001b[1m   25.0% \u001b[0m \u001b[1m   50.0% \u001b[0m \u001b[1m   75.0% \u001b[0m \u001b[1m   97.5% \u001b[0m\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m\n",
       "\n",
       "           p    0.3377    0.5444    0.6397    0.7456    0.8901\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@model function water_land(W, L)\n",
    "    p ~ Uniform(0, 1)\n",
    "    W ~ Binomial(W + L, p)\n",
    "end\n",
    "\n",
    "Logging.disable_logging(Logging.Warn)\n",
    "chain = sample(water_land(6, 3), NUTS(0.65), 1000)\n",
    "display(chain)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7eff1d6b",
   "metadata": {},
   "source": [
    "Code 2.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0ecc3511",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# analytical calculation\n",
    "W = 6\n",
    "L = 3\n",
    "x = range(0, 1; length=101)\n",
    "\n",
    "b = Beta(W+1, L+1)\n",
    "plot(x, pdf.(b, x); label = L\"\\mathcal{Beta}\")\n",
    "\n",
    "# quadratic approximation\n",
    "b = Normal(0.6374, 0.1414)\n",
    "plot!(x, pdf.(b, x); style=:dash, label=L\"\\mathcal{N}(0.64, 0.14)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20959b51",
   "metadata": {},
   "source": [
    "Code 2.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "9c23768b",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_samples = 1000\n",
    "p = Vector{Float64}(undef, n_samples)\n",
    "p[1] = 0.5\n",
    "W, L = 6, 3\n",
    "\n",
    "for i ∈ 2:n_samples\n",
    "    p_old = p[i-1]\n",
    "    p_new = rand(Normal(p_old, 0.1))\n",
    "    if p_new < 0\n",
    "        p_new = abs(p_new)\n",
    "    elseif p_new > 1\n",
    "        p_new = 2-p_new\n",
    "    end\n",
    "\n",
    "    q0 = pdf(Binomial(W+L, p_old), W)\n",
    "    q1 = pdf(Binomial(W+L, p_new), W)\n",
    "    u = rand(Uniform())\n",
    "    p[i] = (u < q1 / q0) ? p_new : p_old\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32a267cc",
   "metadata": {},
   "source": [
    "Code 2.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "0de04e77",
   "metadata": {},
   "outputs": [
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       "  1183.1,684.227 1203.28,638.587 1223.45,593.249 1243.63,548.53 1263.8,504.759 1283.98,462.277 1304.16,421.433 1324.33,382.581 1344.51,346.078 1364.68,312.279 \n",
       "  1384.86,281.533 1405.03,254.182 1425.21,230.553 1445.38,210.957 1465.56,195.684 1485.73,184.998 1505.91,179.132 1526.09,178.285 1546.26,182.618 1566.44,192.248 \n",
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      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "density(p; label = \"MCMC\")\n",
    "b = Beta(W+1, L+1)\n",
    "plot!(x, pdf.(b, x); label = L\"\\mathcal{Beta}\", style=:dash)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a8d34df9",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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