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[OrsonTyphanel93]

Backdoor attack via jumps-Diffusion and stochastic processes : BackStockPros

Hello Dear @beat-buesser ! , I recently performed a more specific and complex analysis on stochastic calculations and jumps incorporating a more advanced Bayesian analysis, in order to understand the change in data distribution during a backdoor attack, you will find attached the full code, you can also download in the code the full csv file containing all the details of this Bayesian stochastic analysis.

Description

This research paper presents a comprehensive approach for executing backdoor attacks on audio data. It uses a diffusion model and a Bayesian approach (via stochastic process effects). The effectiveness of the attack method and its discretion are confirmed by evaluation results, which highlight their ability to manipulate the integrity and security of audio systems.

1. Simulation of a continuous change in performance due to backdoor triggering using the Ornstein-Uhlenbeck process
2. Simulate performance fluctuations due to the backdoor trigger using the Ito formula for jump-diffusion
3. Simulate the spread of the backdoor effect using the Black-Scholes to Diffusion method
4. Simulate the spread of the backdoor effect over time using the Kolmogorov-Feller equation

After compilation, please examine the results of the csv file. This file contains information that can help improve understanding in various fields, such as finance, particle physics and chaotic time, where the passage of data to undetectable backdoors, biological simulations etc...

Testing

UPDATE Notebook: a more comprehensible version for those not familiar with Bayesian techniques BackStockPros, notebook complet

code update, this version is correctly optimal it integrates all simulations correctly in Bayesian execution

easy to understand
UPDATE Best (easy to understand )! please consider the following BackStockPros, notebook complet

Reference maybe
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[OrsonTyphanel93]

FAQ

if you have trouble understanding the background of the entire code, focus on this one, I've simplified it to make it more comprehensible to those not familiar with Bayesian mirroring.

Usage/Examples

    def _bayesian_sampling_diffusion_model(
        self,
        T: int,
        alpha: np.ndarray,
        beta: np.ndarray,
        sigma: np.ndarray,
        noise_dist: Callable[[Any], np.ndarray],
        initial_state: np.ndarray,
        time_steps: int,
        theta: float,
        jump_size_dist: Callable[[Any], np.ndarray],
        non_linear_drift: Callable[[float, int], float],
        volatility: np.ndarray,  # Add this line to include the volatility parameter
        jump_volatility: float,
        dt: float = 0.01,
        # Additional parameters to reflect the dynamics of the simulation methods
        performance_fluctuations_prior=None,
        effect_spread_prior=None,
        continuous_performance_change_prior=None,
        backdoor_spread_prior=None
) -> pm.backends.base.MultiTrace:
     """
     Perform Bayesian sampling diffusion for a given time period, alpha, beta, sigma, and noise distribution.
     Incorporates the dynamics of simulate_performance_fluctuations, simulate_effect_spread,
     simulate_continuous_performance_change, and simulate_backdoor_spread.
     """
     assert isinstance(T, int), "Expected T to be an integer"
     assert isinstance(alpha, np.ndarray) and alpha.ndim == 1, "Expected alpha to be a 1D numpy array"
     assert isinstance(beta, np.ndarray) and beta.ndim == 1, "Expected beta to be a 1D numpy array"
     assert isinstance(sigma, np.ndarray) and sigma.ndim == 1, "Expected sigma to be a 1D numpy array"
     assert callable(noise_dist), "Expected noise_dist to be a callable function"
     assert isinstance(initial_state, np.ndarray), "Expected initial_state to be a numpy array"
     assert isinstance(time_steps, int), "Expected time_steps to be an integer"
     assert isinstance(theta, float), "Expected theta to be a float"
     assert callable(jump_size_dist), "Expected jump_size_dist to be a callable function"
     assert callable(non_linear_drift), "Expected non_linear_drift to be a callable function"
     assert isinstance(volatility, np.ndarray) and volatility.ndim == 1, "Expected volatility to be a 1D numpy array"
     assert isinstance(jump_volatility, float), "Expected jump_volatility to be a float"

     try:
        with Model() as model:
            # Define priors based on the simulation outputs
            if performance_fluctuations_prior is not None:
                x_T = Normal('x_T', mu=performance_fluctuations_prior[0], sigma=1)
            elif effect_spread_prior is not None:
                x_T = Normal('x_T', mu=effect_spread_prior[0], sigma=1)
            elif continuous_performance_change_prior is not None:
                x_T = Normal('x_T', mu=continuous_performance_change_prior[0], sigma=1)
            elif backdoor_spread_prior is not None:
                x_T = Normal('x_T', mu=backdoor_spread_prior[0], sigma=1)
            else:
                x_T = Normal('x_T', mu=noise_dist(initial_state), sigma=1)

            # Use the jump-diffusion process to update the state
            for t in range(T - 1, -1, -1):
                z = noise_dist(0) if t > 1 else 0
                x_t_minus_1 = Normal(f'x_{t}', mu=np.sqrt(alpha[t]) * (x_T - np.sqrt(1 - alpha[t]) * noise_dist(beta[t])) + sigma[t] * z, sigma=1)
                x_T = x_t_minus_1

            # Sample from the posterior
            trace = sample(2000, tune=1000, cores=2, chains=2, step=pm.NUTS())

        return trace

     except Exception as e:
            print(f"An error occurred: {e}")
            raise ```