{
  "type": "excalidraw",
  "version": 2,
  "source": "https://excalidraw.com",
  "elements": [
    {
      "id": "canvas",
      "type": "rectangle",
      "x": 0,
      "y": 0,
      "width": 900,
      "height": 1080,
      "angle": 0,
      "strokeColor": "#FFFFFF",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1455065940,
      "version": 2,
      "versionNonce": 846875164,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "a0"
    },
    {
      "id": "forward-title",
      "type": "text",
      "x": 40,
      "y": 32,
      "width": 820,
      "height": 37.5,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1623069722,
      "version": 2,
      "versionNonce": 12318116,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 30,
      "fontFamily": 2,
      "text": "Forward: collect incoming probability",
      "textAlign": "left",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "Forward: collect incoming probability",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "a1"
    },
    {
      "id": "forward-condition",
      "type": "text",
      "x": 40,
      "y": 78,
      "width": 820,
      "height": 27.5,
      "angle": 0,
      "strokeColor": "#666666",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 560753665,
      "version": 2,
      "versionNonce": 1902290588,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 22,
      "fontFamily": 2,
      "text": "Fix the start (y, t).   Symmetric step: 0 < α < 1/2.",
      "textAlign": "left",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "Fix the start (y, t).   Symmetric step: 0 < α < 1/2.",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "a2"
    },
    {
      "id": "forward-left-time",
      "type": "text",
      "x": 70,
      "y": 130,
      "width": 180,
      "height": 30,
      "angle": 0,
      "strokeColor": "#666666",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 652812421,
      "version": 2,
      "versionNonce": 1215668516,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 24,
      "fontFamily": 2,
      "text": "T − Δt",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "T − Δt",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "a3"
    },
    {
      "id": "forward-right-time",
      "type": "text",
      "x": 655,
      "y": 130,
      "width": 180,
      "height": 30,
      "angle": 0,
      "strokeColor": "#666666",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 776241042,
      "version": 2,
      "versionNonce": 1544254236,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 24,
      "fontFamily": 2,
      "text": "T",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "T",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "a4"
    },
    {
      "id": "edge-forward-upper",
      "type": "arrow",
      "x": 260,
      "y": 210,
      "width": 385,
      "height": 100,
      "angle": 0,
      "strokeColor": "#888888",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1281024122,
      "version": 2,
      "versionNonce": 1391320228,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "points": [
        [
          0,
          0
        ],
        [
          385,
          100
        ]
      ],
      "lastCommittedPoint": null,
      "startBinding": null,
      "endBinding": null,
      "startArrowhead": null,
      "endArrowhead": "arrow",
      "index": "a5"
    },
    {
      "id": "label-background-forward-upper",
      "type": "rectangle",
      "x": 428.5,
      "y": 244,
      "width": 48,
      "height": 32,
      "angle": 0,
      "strokeColor": "#FFFFFF",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 824585298,
      "version": 2,
      "versionNonce": 15118236,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "a6"
    },
    {
      "id": "probability-forward-upper",
      "type": "text",
      "x": 428.5,
      "y": 244,
      "width": 48,
      "height": 31.25,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1111789506,
      "version": 2,
      "versionNonce": 1775413284,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 25,
      "fontFamily": 2,
      "text": "α",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "α",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "a7"
    },
    {
      "id": "edge-forward-middle",
      "type": "arrow",
      "x": 260,
      "y": 310,
      "width": 385,
      "height": 0,
      "angle": 0,
      "strokeColor": "#888888",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 472352630,
      "version": 2,
      "versionNonce": 1344691228,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "points": [
        [
          0,
          0
        ],
        [
          385,
          0
        ]
      ],
      "lastCommittedPoint": null,
      "startBinding": null,
      "endBinding": null,
      "startArrowhead": null,
      "endArrowhead": "arrow",
      "index": "a8"
    },
    {
      "id": "label-background-forward-middle",
      "type": "rectangle",
      "x": 400.5,
      "y": 294,
      "width": 104,
      "height": 32,
      "angle": 0,
      "strokeColor": "#FFFFFF",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 348865472,
      "version": 2,
      "versionNonce": 1897005988,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "a9"
    },
    {
      "id": "probability-forward-middle",
      "type": "text",
      "x": 400.5,
      "y": 294,
      "width": 104,
      "height": 31.25,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 2022474820,
      "version": 2,
      "versionNonce": 1673296028,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 25,
      "fontFamily": 2,
      "text": "1 − 2α",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "1 − 2α",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aA"
    },
    {
      "id": "edge-forward-lower",
      "type": "arrow",
      "x": 260,
      "y": 410,
      "width": 385,
      "height": 100,
      "angle": 0,
      "strokeColor": "#888888",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1156935578,
      "version": 2,
      "versionNonce": 517599012,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "points": [
        [
          0,
          0
        ],
        [
          385,
          -100
        ]
      ],
      "lastCommittedPoint": null,
      "startBinding": null,
      "endBinding": null,
      "startArrowhead": null,
      "endArrowhead": "arrow",
      "index": "aB"
    },
    {
      "id": "label-background-forward-lower",
      "type": "rectangle",
      "x": 428.5,
      "y": 344,
      "width": 48,
      "height": 32,
      "angle": 0,
      "strokeColor": "#FFFFFF",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 911345586,
      "version": 2,
      "versionNonce": 1197147420,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aC"
    },
    {
      "id": "probability-forward-lower",
      "type": "text",
      "x": 428.5,
      "y": 344,
      "width": 48,
      "height": 31.25,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 551795739,
      "version": 2,
      "versionNonce": 865626788,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 25,
      "fontFamily": 2,
      "text": "α",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "α",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aD"
    },
    {
      "id": "node-forward-upper",
      "type": "rectangle",
      "x": 70,
      "y": 176,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#D0D0D0",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 1.5,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-upper"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 164115198,
      "version": 2,
      "versionNonce": 1081344412,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aE"
    },
    {
      "id": "state-forward-upper",
      "type": "text",
      "x": 80,
      "y": 191,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-upper"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 1681592984,
      "version": 2,
      "versionNonce": 838563364,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "z + h",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "z + h",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aF"
    },
    {
      "id": "node-forward-middle",
      "type": "rectangle",
      "x": 70,
      "y": 276,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#D0D0D0",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 1.5,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-middle"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 1616029194,
      "version": 2,
      "versionNonce": 372862492,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aG"
    },
    {
      "id": "state-forward-middle",
      "type": "text",
      "x": 80,
      "y": 291,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-middle"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 358171951,
      "version": 2,
      "versionNonce": 384897444,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "z",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "z",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aH"
    },
    {
      "id": "node-forward-lower",
      "type": "rectangle",
      "x": 70,
      "y": 376,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#D0D0D0",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 1.5,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-lower"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 1099054308,
      "version": 2,
      "versionNonce": 1503480476,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aI"
    },
    {
      "id": "state-forward-lower",
      "type": "text",
      "x": 80,
      "y": 391,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-lower"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 1945722962,
      "version": 2,
      "versionNonce": 296172836,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "z − h",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "z − h",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aJ"
    },
    {
      "id": "node-forward-terminal",
      "type": "rectangle",
      "x": 655,
      "y": 276,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#6200EE",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-terminal"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 530793154,
      "version": 2,
      "versionNonce": 760321820,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aK"
    },
    {
      "id": "state-forward-terminal",
      "type": "text",
      "x": 665,
      "y": 291,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#6200EE",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-forward-terminal"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 801439618,
      "version": 2,
      "versionNonce": 999028900,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "z",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "z",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aL"
    },
    {
      "id": "forward-footer",
      "type": "text",
      "x": 40,
      "y": 483,
      "width": 820,
      "height": 27.5,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1696699867,
      "version": 2,
      "versionNonce": 231232412,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 22,
      "fontFamily": 2,
      "text": "Differentiate in terminal state z and time T.",
      "textAlign": "left",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "Differentiate in terminal state z and time T.",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aM"
    },
    {
      "id": "panel-separator",
      "type": "line",
      "x": 40,
      "y": 538,
      "width": 820,
      "height": 0,
      "angle": 0,
      "strokeColor": "#E0E0E0",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 1,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 2136816773,
      "version": 2,
      "versionNonce": 1347240996,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "points": [
        [
          0,
          0
        ],
        [
          820,
          0
        ]
      ],
      "lastCommittedPoint": null,
      "startBinding": null,
      "endBinding": null,
      "startArrowhead": null,
      "endArrowhead": null,
      "index": "aN"
    },
    {
      "id": "backward-title",
      "type": "text",
      "x": 40,
      "y": 570,
      "width": 820,
      "height": 37.5,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 442490344,
      "version": 2,
      "versionNonce": 422805532,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 30,
      "fontFamily": 2,
      "text": "Backward: condition on the next step",
      "textAlign": "left",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "Backward: condition on the next step",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aO"
    },
    {
      "id": "backward-condition",
      "type": "text",
      "x": 40,
      "y": 616,
      "width": 820,
      "height": 27.5,
      "angle": 0,
      "strokeColor": "#666666",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1921483523,
      "version": 2,
      "versionNonce": 1708728228,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 22,
      "fontFamily": 2,
      "text": "Fix the target (z, T).",
      "textAlign": "left",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "Fix the target (z, T).",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aP"
    },
    {
      "id": "backward-left-time",
      "type": "text",
      "x": 70,
      "y": 665,
      "width": 180,
      "height": 30,
      "angle": 0,
      "strokeColor": "#666666",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 376519172,
      "version": 2,
      "versionNonce": 1468341404,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 24,
      "fontFamily": 2,
      "text": "t",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "t",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aQ"
    },
    {
      "id": "backward-right-time",
      "type": "text",
      "x": 655,
      "y": 665,
      "width": 180,
      "height": 30,
      "angle": 0,
      "strokeColor": "#666666",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1285276984,
      "version": 2,
      "versionNonce": 610110244,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 24,
      "fontFamily": 2,
      "text": "t + Δt",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "t + Δt",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aR"
    },
    {
      "id": "edge-backward-upper",
      "type": "arrow",
      "x": 260,
      "y": 845,
      "width": 385,
      "height": 100,
      "angle": 0,
      "strokeColor": "#888888",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1943360809,
      "version": 2,
      "versionNonce": 40839452,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "points": [
        [
          0,
          0
        ],
        [
          385,
          -100
        ]
      ],
      "lastCommittedPoint": null,
      "startBinding": null,
      "endBinding": null,
      "startArrowhead": null,
      "endArrowhead": "arrow",
      "index": "aS"
    },
    {
      "id": "label-background-backward-upper",
      "type": "rectangle",
      "x": 428.5,
      "y": 779,
      "width": 48,
      "height": 32,
      "angle": 0,
      "strokeColor": "#FFFFFF",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1424802081,
      "version": 2,
      "versionNonce": 2118682276,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aT"
    },
    {
      "id": "probability-backward-upper",
      "type": "text",
      "x": 428.5,
      "y": 779,
      "width": 48,
      "height": 31.25,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 908802524,
      "version": 2,
      "versionNonce": 1298376092,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 25,
      "fontFamily": 2,
      "text": "α",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "α",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aU"
    },
    {
      "id": "edge-backward-middle",
      "type": "arrow",
      "x": 260,
      "y": 845,
      "width": 385,
      "height": 0,
      "angle": 0,
      "strokeColor": "#888888",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 344268586,
      "version": 2,
      "versionNonce": 1749553700,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "points": [
        [
          0,
          0
        ],
        [
          385,
          0
        ]
      ],
      "lastCommittedPoint": null,
      "startBinding": null,
      "endBinding": null,
      "startArrowhead": null,
      "endArrowhead": "arrow",
      "index": "aV"
    },
    {
      "id": "label-background-backward-middle",
      "type": "rectangle",
      "x": 400.5,
      "y": 829,
      "width": 104,
      "height": 32,
      "angle": 0,
      "strokeColor": "#FFFFFF",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 675466394,
      "version": 2,
      "versionNonce": 764711452,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aW"
    },
    {
      "id": "probability-backward-middle",
      "type": "text",
      "x": 400.5,
      "y": 829,
      "width": 104,
      "height": 31.25,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1388411285,
      "version": 2,
      "versionNonce": 290074020,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 25,
      "fontFamily": 2,
      "text": "1 − 2α",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "1 − 2α",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aX"
    },
    {
      "id": "edge-backward-lower",
      "type": "arrow",
      "x": 260,
      "y": 845,
      "width": 385,
      "height": 100,
      "angle": 0,
      "strokeColor": "#888888",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 2136097507,
      "version": 2,
      "versionNonce": 569634460,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "points": [
        [
          0,
          0
        ],
        [
          385,
          100
        ]
      ],
      "lastCommittedPoint": null,
      "startBinding": null,
      "endBinding": null,
      "startArrowhead": null,
      "endArrowhead": "arrow",
      "index": "aY"
    },
    {
      "id": "label-background-backward-lower",
      "type": "rectangle",
      "x": 428.5,
      "y": 879,
      "width": 48,
      "height": 32,
      "angle": 0,
      "strokeColor": "#FFFFFF",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1988274733,
      "version": 2,
      "versionNonce": 444389668,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "aZ"
    },
    {
      "id": "probability-backward-lower",
      "type": "text",
      "x": 428.5,
      "y": 879,
      "width": 48,
      "height": 31.25,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 1639987499,
      "version": 2,
      "versionNonce": 2066987804,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 25,
      "fontFamily": 2,
      "text": "α",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "α",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aa"
    },
    {
      "id": "node-backward-initial",
      "type": "rectangle",
      "x": 70,
      "y": 811,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#6200EE",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-initial"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 2130998123,
      "version": 2,
      "versionNonce": 1330517156,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "ab"
    },
    {
      "id": "state-backward-initial",
      "type": "text",
      "x": 80,
      "y": 826,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#6200EE",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-initial"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 1093086256,
      "version": 2,
      "versionNonce": 600176540,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "y",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "y",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "ac"
    },
    {
      "id": "node-backward-upper",
      "type": "rectangle",
      "x": 655,
      "y": 711,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#D0D0D0",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 1.5,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-upper"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 1172554497,
      "version": 2,
      "versionNonce": 1567350820,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "ad"
    },
    {
      "id": "state-backward-upper",
      "type": "text",
      "x": 665,
      "y": 726,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-upper"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 380127630,
      "version": 2,
      "versionNonce": 1742513180,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "y + h",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "y + h",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "ae"
    },
    {
      "id": "node-backward-middle",
      "type": "rectangle",
      "x": 655,
      "y": 811,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#D0D0D0",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 1.5,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-middle"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 217493308,
      "version": 2,
      "versionNonce": 214186916,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "af"
    },
    {
      "id": "state-backward-middle",
      "type": "text",
      "x": 665,
      "y": 826,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-middle"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 1606599281,
      "version": 2,
      "versionNonce": 1030340764,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "y",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "y",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "ag"
    },
    {
      "id": "node-backward-lower",
      "type": "rectangle",
      "x": 655,
      "y": 911,
      "width": 180,
      "height": 68,
      "angle": 0,
      "strokeColor": "#D0D0D0",
      "backgroundColor": "#FFFFFF",
      "fillStyle": "solid",
      "strokeWidth": 1.5,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-lower"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 797004526,
      "version": 2,
      "versionNonce": 2005215012,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "index": "ah"
    },
    {
      "id": "state-backward-lower",
      "type": "text",
      "x": 665,
      "y": 926,
      "width": 160,
      "height": 35,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [
        "group-backward-lower"
      ],
      "frameId": null,
      "roundness": null,
      "seed": 2060243242,
      "version": 2,
      "versionNonce": 203377948,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 28,
      "fontFamily": 2,
      "text": "y − h",
      "textAlign": "center",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "y − h",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "ai"
    },
    {
      "id": "backward-footer",
      "type": "text",
      "x": 40,
      "y": 1017,
      "width": 820,
      "height": 27.5,
      "angle": 0,
      "strokeColor": "#212121",
      "backgroundColor": "transparent",
      "fillStyle": "solid",
      "strokeWidth": 2,
      "strokeStyle": "solid",
      "roughness": 0,
      "opacity": 100,
      "groupIds": [],
      "frameId": null,
      "roundness": null,
      "seed": 414723633,
      "version": 2,
      "versionNonce": 1109172900,
      "isDeleted": false,
      "boundElements": [],
      "updated": 1789194852939,
      "link": null,
      "locked": false,
      "fontSize": 22,
      "fontFamily": 2,
      "text": "Hold the target at T fixed; differentiate in y and t.",
      "textAlign": "left",
      "verticalAlign": "middle",
      "containerId": null,
      "originalText": "Hold the target at T fixed; differentiate in y and t.",
      "autoResize": false,
      "lineHeight": 1.25,
      "index": "aj"
    }
  ],
  "appState": {
    "gridSize": null,
    "viewBackgroundColor": "#ffffff",
    "name": "Forward and backward transition-density bookkeeping",
    "exportBackground": true,
    "exportWithDarkMode": false
  },
  "files": {}
}