> For the complete documentation index, see [llms.txt](https://torch-1.gitbook.io/litepaper/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://torch-1.gitbook.io/litepaper/system-mechanics/local-confidence.md).

# Local confidence

Local confidence is a measure from 0 to 1 of how likely a given price range is to be correct at a specific time, based on the distribution of active, stake-weighted bets.

It answers the question: *"Given that a bet resolves at time x, what is the probability that the actual price will fall within a specific price range \[y1, y2]?"*

Local confidence is computed using the current market's kernel density estimate (KDE) of all outstanding bets, weighted by stake and linearly adjusted for decay (i.e. how long ago each bet was placed).

1. A vertical slice of the 2D KDE surface is taken at the desired time x.
2. This slice is normalized so that its total probability sums to 100% (i.e. the price must hit some value at time x).
3. The percentage of this probability mass that falls within the desired price range \[y1, y2] is summed: this is the Local confidence value.

Underlying math:

1. **Raw input data**\
   Each bet is defined by:
   * Resolution time `xᵢ`
   * Price center `yᵢ`
   * Stake size `sᵢ`
   * Time since placed `tᵢ`
2. **Decay-adjusted weight**\
   Each bet’s influence decays over time:\
   \
   $$w\_i = s\_i \cdot e^{-r\_i \cdot t\_i}$$

   \
   Where:

   * `sᵢ` is the stake
   * `rᵢ` is the decay rate (e.g. 1–10% daily)
   * `tᵢ` is time since placement in days
3. **Kernel density estimation**\
   To estimate the smooth probability surface P(x, y):\
   \
   $$P(x, y) = \frac{1}{Z} \sum\_{i=1}^{n} w\_i \cdot K(x - x\_i, y - y\_i)$$<br>

   Where:

   * `K()` is a 2D Gaussian kernel
   * `Z` is a normalization constant

   This gives us a continuous probability field over all time-price combinations.<br>
4. Local confidence formula\
   For a fixed time `x`, the Local confidence for a price range `[y₁, y₂]` is:\
   \
   $$\text{LocalConfidence}*{\[y\_1, y\_2]}(x) = \frac{\int*{y\_1}^{y\_2} P(x, y) , dy}{\int\_{-\infty}^{\infty} P(x, y) , dy}$$

Visually, local confidence looks like a heatmap style probability map as shown above.


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