Merge pull request #76 from skoghoern/main

removed line breaks from tmaze tutorial latex syntax for rendering on website

Author: bvdmitri

docs/src/how_to_contribute.md

@@ -48,10 +48,10 @@ If your example cannot be statically generated, put it inside the `interactive`
 
    Some other text
    ```
-   Do not add spaces before or after the `$$` or `$`
+   Do not add spaces nor line breaks before or after the `$$` or `$`
 
 2. **Equation Rules**
-   - No space after opening `$$` or `$`
+   - No space nor line breaks after opening `$$` or `$`
    - Separate display equations with empty lines
    - Inline equations use single `$...$`, e.g. `$$a + b$$` and not `$$ a + b $$`
 

examples/Basic Examples/T-Maze Active Inference/T-Maze.png

@@ -67,7 +67,7 @@
       "id": "c409c9b2",
       "metadata": {},
       "source": [
-        "<div align=\"center\"><img src=\"T-Maze.png\" width=\"200\"></div>"
+        "![](T-Maze.png)"
       ]
     },
     {
@@ -165,11 +165,9 @@
         "\n",
         "For a candidate policy $u$, the **Expected Free Energy** $G(u)$ can be decomposed to a form with preferences over states, defined as:\n",
         "\n",
-        "$$\n",
-        "G(u) \\;=\\;\n",
+        "$$G(u) \\;=\\;\n",
         "\\underbrace{D_{KL}\\!\\bigl[q(x\\!\\mid\\!u)\\;\\|\\;\\hat{p}(x)\\bigr]}_{\\text{risk}}\n",
-        "+\\underbrace{\\mathbb{E}_{q(x|u)}\\!\\bigl[H[q(y\\!\\mid\\!x)]\\bigr]}_{\\text{ambiguity}}\n",
-        "$$\n",
+        "+\\underbrace{\\mathbb{E}_{q(x|u)}\\!\\bigl[H[q(y\\!\\mid\\!x)]\\bigr]}_{\\text{ambiguity}}$$\n",
         "\n",
         "This decomposition shows that the cost function is composed of two primary drivers: risk, which measures the divergence between predicted outcomes and preferred states $\\hat{p}(x)$ to keep the agent goal-oriented, and ambiguity, which calculates the expected uncertainty of future observations to encourage states with clear, informative data. By minimizing $G(u)$, the agent naturally balances these terms to produce behavior that is simultaneously goal-directed and information-seeking—offering a principled solution to the classic exploration–exploitation trade-off.\n",
         "\n",
@@ -197,29 +195,23 @@
         "\n",
         "The central insight of [Paper 1 (Theorem 1)](https://arxiv.org/pdf/2504.14898#Theorem.1) is that EFE minimisation *arises naturally* from minimising a standard **Variational Free Energy (VFE)** functional if we *augment* the generative model with few prior terms:\n",
         "\n",
-        "$$\n",
-        "\\mathcal{F}[q] \\;\\triangleq\\;\n",
+        "$$\\mathcal{F}[q] \\;\\triangleq\\;\n",
         "\\mathbb{E}_{q(y,x,\\theta,u)}\\!\\left[\n",
         "  \\log\\frac{q(y,x,\\theta,u)}{p(y,x,\\theta,u)\\;\\hat{p}(x)\\;\\tilde{p}(u)\\;\\tilde{p}(x)}\n",
-        "\\right]\n",
-        "$$\n",
+        "\\right]$$\n",
         "\n",
         "The denominator is the ordinary generative model $p$ *augmented* by:\n",
         "- a **preference prior** $\\hat{p}(x)$ over desired future states, and\n",
         "- two **epistemic priors** $\\tilde{p}(u),\\tilde{p}(x)$ that encode ambiguity-seeking and novelty-seeking drives.\n",
         "\n",
         "**Theorem 1** states that with the specific choices\n",
         "\n",
-        "$$\n",
-        "\\tilde{p}(u)  \\;\\propto\\; \\exp\\!\\bigl(H[q(x\\!\\mid\\!u)]\\bigr) \\\\\n",
-        "\\tilde{p}(x)  \\;\\propto\\; \\exp\\!\\bigl(-H[q(y\\!\\mid\\!x)]\\bigr)\n",
-        "$$\n",
+        "$$\\tilde{p}(u)  \\;\\propto\\; \\exp\\!\\bigl(H[q(x\\!\\mid\\!u)]\\bigr) \\\\\n",
+        "\\tilde{p}(x)  \\;\\propto\\; \\exp\\!\\bigl(-H[q(y\\!\\mid\\!x)]\\bigr)$$\n",
         "\n",
         "the VFE decomposes exactly as\n",
         "\n",
-        "$$\n",
-        "\\boxed{\\mathcal{F}[q] \\;=\\; \\mathbb{E}_{q(u)}[G(u)] \\;+\\; \\underbrace{\\mathbb{E}_{q(y,x,\\theta,u)}\\!\\left[\\log\\tfrac{q(y,x,\\theta|u)}{p(y,x,\\theta|u)}\\right]}_{\\text{complexity } C(u)}\\;+\\; const.}\n",
-        "$$\n",
+        "$$\\boxed{\\mathcal{F}[q] \\;=\\; \\mathbb{E}_{q(u)}[G(u)] \\;+\\; \\underbrace{\\mathbb{E}_{q(y,x,\\theta,u)}\\!\\left[\\log\\tfrac{q(y,x,\\theta|u)}{p(y,x,\\theta|u)}\\right]}_{\\text{complexity } C(u)}\\;+\\; const.}$$\n",
         "\n",
         "**What this buys us**: minimising $\\mathcal{F}[q]$ over the variational posterior $q$ simultaneously\n",
         "\n",
@@ -248,19 +240,13 @@
       "source": [
         "Theorem 1 gives the priors in terms of global quantities $H[q(x|u)]$ and $H[q(y|x)]$. To take advantage of local computations, we **factorize** the state-space model into\n",
         "\n",
-        "$$\n",
-        "p(y,x,u) \\;=\\; p(x_0)\\prod_{t=1}^{T} p(y_t|x_t)\\,p(x_t|x_{t-1},u_t)\\,p(u_t)\n",
-        "$$\n",
+        "$$p(y,x,u) \\;=\\; p(x_0)\\prod_{t=1}^{T} p(y_t|x_t)\\,p(x_t|x_{t-1},u_t)\\,p(u_t)$$\n",
         "\n",
         "With this factorized SSM [**Corollary 1** (Paper 2)](https://arxiv.org/pdf/2508.02197#corollary.1.1) reduces the priors to *per-timestep, local* expressions:\n",
         "\n",
-        "$$\n",
-        "\\tilde{p}(u_t) \\;\\propto\\; \\exp\\!\\bigl(H[q(x_t, x_{t-1}\\!\\mid\\!u_t)] - H[q(x_{t-1}\\!\\mid\\!u_t)]\\bigr)\n",
-        "$$\n",
+        "$$\\tilde{p}(u_t) \\;\\propto\\; \\exp\\!\\bigl(H[q(x_t, x_{t-1}\\!\\mid\\!u_t)] - H[q(x_{t-1}\\!\\mid\\!u_t)]\\bigr)$$\n",
         "\n",
-        "$$\n",
-        "\\tilde{p}(x_t) \\;\\propto\\; \\exp\\!\\bigl(-H[q(y_t\\!\\mid\\!x_t)]\\bigr)\n",
-        "$$\n",
+        "$$\\tilde{p}(x_t) \\;\\propto\\; \\exp\\!\\bigl(-H[q(y_t\\!\\mid\\!x_t)]\\bigr)$$\n",
         "\n",
         "These are exactly the two prior nodes we add to the factor graph:\n",
         "\n",
@@ -628,16 +614,16 @@
         "\n",
         "Since the epistemic priors depend on the current posterior, inference is run iteratively as explained in [Algorithm 1 (Paper 2)](https://arxiv.org/pdf/2508.02197#algorithm.1):\n",
         "\n",
-        "> **Input**:  generative model $p(y,x,u)$, preference prior $\\hat{p}(x)$, $\\tau_{max}$ iterations <br>\n",
-        "> **Output**: policy posterior $q(u)$ <br>\n",
-        "> $q_0(y,x,u) ←$ uninformative <br>\n",
-        "> **for** $\\tau = 1$ … $\\tau_{max}$:<br>\n",
-        "> &nbsp;&nbsp;&nbsp;&nbsp;**for** each timestep t:<br>\n",
-        "> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$p̃_τ(u_t) ← σ( H[q_{τ-1}(x_t, x_{t-1} | u_t)] − H[q_{τ-1}(x_{t-1} | u_t)] )$<br>\n",
-        "> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$p̃_τ(x_t) ← σ( −H[q_{τ-1}(y_t | x_t)] )$<br>\n",
-        "> &nbsp;&nbsp;&nbsp;&nbsp;**end** <br>\n",
-        "> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$q_\\tau(y,x,u) ←$ infer( $p(y,x,u)$ with updated priors )<br>\n",
-        "> **end** <br>\n",
+        "> **Input**:  generative model $p(y,x,u)$, preference prior $\\hat{p}(x)$, $\\tau_{max}$ iterations  \n",
+        "> **Output**: policy posterior $q(u)$  \n",
+        "> $q_0(y,x,u) ←$ uninformative  \n",
+        "> **for** $\\tau = 1$ … $\\tau_{max}$:  \n",
+        "> &nbsp;&nbsp;&nbsp;&nbsp;**for** each timestep t:  \n",
+        "> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$p̃_τ(u_t) ← σ( H[q_{τ-1}(x_t, x_{t-1} | u_t)] − H[q_{τ-1}(x_{t-1} | u_t)] )$  \n",
+        "> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$p̃_τ(x_t) ← σ( −H[q_{τ-1}(y_t | x_t)] )$  \n",
+        "> &nbsp;&nbsp;&nbsp;&nbsp;**end**  \n",
+        "> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$q_\\tau(y,x,u) ←$ infer( $p(y,x,u)$ with updated priors )  \n",
+        "> **end**  \n",
         "> **return** $q_{τ_{max}}(u)$\n",
         "\n",
         "In the RxInfer implementation:\n",
@@ -755,7 +741,7 @@
     },
     {
       "cell_type": "code",
-      "execution_count": 14,
+      "execution_count": null,
       "id": "7221d3b5",
       "metadata": {},
       "outputs": [],
@@ -770,7 +756,7 @@
         "function plot_tmaze(env::TMaze)\n",
         "    p = Plots.plot(\n",
         "        aspect_ratio=:equal, legend=false, axis=false, grid=false, ticks=false,\n",
-        "        background_color=MAZE_THEME.background, size=(600, 600), frame=:none, margin=0Plots.mm\n",
+        "        background_color=MAZE_THEME.background, size=(300, 300), frame=:none, margin=0Plots.mm\n",
         "    )\n",
         "    scale = 20\n",
         "    Plots.plot!(p, [1, 2, 2, 1], [1, 1, 4, 4], seriestype=:shape, c=MAZE_THEME.corridor, lw=0)\n",