Thinking Backwards: The "Reversal Blessing" in LLM Multiple-Choice Reasoning

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    "Given a sequence of tokens $x = (x_1, x_2, \\dots, x_T)$, the model expresses the joint probability as a product of conditional probabilities:\n",
    "\n",
    "$$\n",
    "p_{L2R}(x) = \\prod_{t=1}^{T} p_{L2R}(x_{t}\\mid x_{<t}),\n",
    "$$\n",
    "\n",
    "where $x_{<t}$ represents the preceding tokens $(x_{1}, x_{2}, \\dots, x_{t-1})$. This approach can introduce inductive biases and approximation errors that compound at each step, which may not be optimal for all tasks.\n",
    "\n",
    "For training, models typically optimize the log-likelihood of the sequence for numerical stability and easier differentiation:\n",
    "\n",
    "$$\n",
    "\\log p(x) = \\sum_{t=1}^{T} \\log p_{L2R}(x_{t}\\mid x_{<t}).\n",
    "$$"
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